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Branching of Modular Representations of the Alternating Groups
209, 143᎐174 Ž1998. JA987505
JOURNAL OF ALGEBRA ARTICLE NO.
Branching of Modular Representations of the Alternating Groups C. Bessenrodt* Fakultat ¨ fur ¨ Mathematik, Otto-¨ on-Guericke-Uni¨ ersitat ¨ Magdeburg, 39016 Magdeburg, Germany
and J. B. Olsson† Matematisk Institut, Københa¨ ns Uni¨ ersitet, 2100 Copenhagen Ø, Denmark Communicated by G. D. James Received November 5, 1997 DEDICATED TO GERHARD O. MICHLER ON THE OCCASION OF HIS
60TH
BIRTHDAY
Based on Kleshchev’s branching theorems for the p-modular irreducible representations of the symmetric group and on the recent proof of the Mullineux Conjecture, we investigate in this article the corresponding branching problem for the p-modular irreducible representations of the alternating group A n . We obtain information on the socle of the restrictions of such A n-representations to A ny1 as well as on the multiplicities of certain composition factors; furthermore, irreducible A n-representations with irreducible restrictions to A ny1 are studied. 䊚 1998 Academic Press
1. INTRODUCTION The purpose of this paper is to provide information about the restrictions to A ny 1 of modular irreducible representations of the alternating groups A n . The results are based on Kleshchev’s branching theorems Žsee * E-mail: [email protected]. † E-mail: [email protected]. 143 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
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w8x, w9x, w11x. for the symmetric group Conjecture Žsee w2x, w5x, w10x, w13x.. The starting point in this article is for characteristic 0 representations of the irreducible representation of Sn partition of n. Then S < S ny 1 s
and on the proof of the Mullineux the well-known branching theorem the symmetric group S n . Let S be in characteristic 0 labeled by the
[S
_ A
,
A
where the sum is over all removable nodes of Žsee Section 2.. On the other hand, the decomposition of the restriction of S to the alternating group A n is also easy to describe. It depends on the transposition map on partitions, since S m sgn , S ⬘ w6, Sect. 2.5x. Combining these facts, it is quite easy to deduce a branching result for irreducible representations of A n . This is done in Section 2 to give a perspective of our results in positive characteristic. Examples show that the branching phenomena in characteristic p ) 0 are much more complicated. If D is the p-modular irreducible representation of Sn labeled by the p-regular partition of n, then D < S ny 1 may also contain composition factors D , where is not obtained by removing a node from . Moreover, D < S ny 1 is in general not multiplicity-free. The recent progress on this branching problem started with the Jantzen᎐Seitz conjecture w7x describing the class of p-regular partitions for which D < S ny 1 is irreducible. The Jantzen᎐Seitz conjecture was proved by Kleshchev as a consequence of his much stronger modular branching theorems w8x ᎐ w11x Žsee Theorem 3.1.. To exploit Kleshchev’s result for investigations of the representations of the alternating groups, it is necessary to understand the restrictions of the modules D to A n . For p / 2, these depend on the Mullineux map M on p-regular partitions w13x, since D m sgn , D
M
Žsee w2x, w5x, w10x.. In contrast to the situation at characteristic 0, both the branching and restriction to A n are quite complicated in characteristic p, and to deal with the representations of the alternating groups, we have to control both processes simultaneously. This is possible using the residue symbols and signature sequences introduced by the authors in w2x. In the case of characteristic 2 we have to invoke a result of Benson w1x. The paper is organized as follows. In Section 2 we treat branching for representations of A n in characteristic 0. In Section 3 we present the necessary background dealing with positive characteristic, that is, Kleshchev’s results and residue symbols. In the next section, we describe the classification of the modular irreducible A n-representations based on
ALTERNATING GROUPS
145
Benson’s result at p s 2 and on the Mullineux map at p / 2. Section 5 deals with the case of odd characteristic, and Section 6 with the case of characteristic 2. We obtain information on the socle of the restrictions of irreducible A n-representations to A ny1 as well as on the multiplicities of certain composition factors of these restrictions. Furthermore, we describe the labels of those irreducible modules of A n that remain irreducible upon restriction to A ny 1 at characteristic p / 2, and combinatorial conditions for such labels are given at p s 2. For general facts on representations of the symmetric groups, we refer the reader to w6x. Throughout the paper we will always assume that our representations are over fields that are splitting fields for the alternating groups, e.g., one may take the fields to be algebraically closed.
2. BRANCHING OF REPRESENTATIONS OF THE ALTERNATING GROUPS AT CHARACTERISTIC 0 OR LARGE p Let be a partition of n. It is well known that at characteristic 0 resp., at prime characteristic p with p ) n, the character w x resp., the corresponding representation of Sn splits on restriction to A n if and only if is symmetric, i.e., s ⬘. In this case, it splits into two Žvia the transposition Ž12.. conjugate A n-characters 4 q and 4 y. So for a branching formula for A n-characters, one needs to study conjugation properties together with the usual well-known branching of ordinary irreducible S n-representations. First we recall some notations and definitions for partitions. Let s Ž 1 , 2 , . . . , k . be a partition of n, i.e., 1 , . . . , k are integers with 1 G 2 G ⭈⭈⭈ G k ) 0 and Ý kis1 i s n. Then Y Ž . s Ž i , j . g ⺪ = ⺪ <1 F i F k, 1 F j F i 4 ; ⺪ = ⺪ is the Young diagram of , and Ž i, j . g Y Ž . is called a node of . If A s Ž i, j . is a node of and Y Ž . _ Ž i, j .4 is again a Young diagram of a partition, then A is called a remo¨ able node, and _ A denotes the corresponding partition of n y 1. Similarly, if A s Ž i, j . g ⺞ = ⺞ is such that Y Ž . j Ž i, j .4 is the Young diagram of a partition of n q 1, then A is called an indent node of . The Nonsymmetric Case: / ⬘ Assume that has a removable node A such that
_ A s Ž _ A . ⬘ s ⬘ _ B for a suitable node B in ⬘. In this case we say that is almost symmetric.
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Since is not symmetric, the ‘‘conjugate node’’ A⬘ to A does not belong to . Hence for every other removable node C / A in we have
_ C / Ž _ C . ⬘. Even stronger: for every pair of removable nodes C / A, C˜ in , we have
_ C / Ž _ C˜. ⬘. Now, if we assume that there is no node A as above, then we can easily see that also in this case for every pair of removable nodes C, C˜ in , we have
_ C / Ž _ C˜. ⬘. Thus we obtain for the branching of the character 4 : Ži. If is almost symmetric Žwith respect to the node A., then $
4 < A ny 1 s _ A 4 q
Ý _ C4 , C/A
$
where 4 s 4 qq 4 y for a symmetric partition , and the partitions _ C, C / A, are all pairwise nonconjugate, so that the restriction is multiplicity-free. Žii. If is not almost symmetric, then
4 < A ny 1 s Ý _ C 4 , C
where the partitions _ C are all pairwise nonconjugate, so that again the restriction is multiplicity-free. The Symmetric Case: s ⬘ In this case there exists at most one removable node A Žon the main diagonal. with
_ A s Ž _ A . ⬘. Moreover, it is clear that for every other removable node C / A in , we have a removable node C˜ / C in with
_ C s Ž _ C˜. ⬘. Before we can state the restrictions of the two conjugate characters 4 q and 4 y to A ny 1 , we first have to be precise about the choice of our
147
ALTERNATING GROUPS
labeling Žsee w6, Sect. 2.5x.. The only critical conjugacy class of A n where the characters 4 q and 4 y differ are the two classes of cycle type hŽ . s Ž h11 , . . . , h r r ., where h ii denotes the length of the ith principal hook of , and r is the order of the Durfee square of . Let ry1
hŽ .qs Ž 1 ⭈⭈⭈ h11 . Ž h11 q 1 ⭈⭈⭈ h11 q h 22 . ⭈⭈⭈
ž
r
Ý h ii q 1 ⭈⭈⭈ Ý h ii is1
is1
/
be an element in the conjugacy class of cycle type hŽ . and of q-type, and choose the labeling of the characters such that
4 " Ž hŽ .q . s 12 w x Ž hŽ . . "
ž
r
(
w x Ž hŽ . . Ł h ii . is1
/
If s ⬘ has a removable node A on the main diagonal, then h r r s 1, and so hŽ _ A. s Ž h11 , . . . , h ry1 ry1 .. The sign choice for the conjugacy classes in A ny 1 of type Ž h11 , . . . , h ry1 ry1 . is made compatibly with the choice above, and the signs for the characters are chosen as before. Then
_ A4 " Ž hŽ _ A.q . s 12 w _ A x Ž hŽ _ A. . "
ž
(
ry1
w x Ž hŽ . . Ł h ii . is1
/
Hence we have the following branching behavior: Ži. If has a removable node A on the main diagonal, then
4 " < A ny 1 s _ A4 " q
Ý _ C4 , C/A
where C only runs through the removable nodes of above the main diagonal. Žii. If has no removable node on the main diagonal, then
4 " < A ny 1 s Ý _ C 4 , C
where C only runs through the removable nodes of above the main diagonal. In particular, the restrictions are again multiplicity-free. In the discussion above, some of the properties become even more obvious when one uses the Frobenius symbol displaying the principal hooks of rather than working in terms of the parts of .
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3. MODULAR BRANCHING OF Sn-REPRESENTATIONS AND RESIDUE SYMBOLS In this section we collect some definitions and results that are needed in the following sections. Let p be a prime; this is needed in the representation theoretic context, for the combinatorial statements p may be an arbitrary odd integer ) 1. We define a Ž p .-signature sequence X as a sequence X : c1 1 c 2 2 ⭈⭈⭈ c s s , where the c i are residues mod p, and the i are signs, for i s 1, . . . , s. For 0 F i F s and 0 F ␣ F p y 1 we define
␣X Ž i . s
Ý
kFi < c ks ␣ 4
k ;
here we make the conventions that an empty sum is 0 and that q is counted as q1 and y as y1 in the sum. The end ¨ alue ␣X of ␣ in X is then defined to be
␣X s ␣ X Ž s . . Furthermore, we define the peak ¨ alue of ␣ in X to be
␣ Ž X . s max ␣ X Ž i . < 0 F i F s 4 . We call c i g X normal of residue ␣ if ␣ X Ž i . ) ␣ X Ž j . for all j F i y 1 and ␣X Ž i . ) 0. This is only possible when c i i s ␣ q. In this case we also call ␣X Ž i . the height ht c i of c i . Moreover, c i g X is called good of residue ␣ Žfor short also ␣-good. if c i is normal of residue ␣ and i is minimal with
␣X Ž i . s ␣ Ž X . . Note that if c i is good of residue ␣ , then ht c i s ␣ Ž X .. Before we associate these definitions with partitions, we recall some further notation and definitions for partitions. A partition s Ž l 1a1 , l 2a 2 , . . . , l ta t . is called p-regular if 1 F a i F p y 1 for i s 1, . . . , t. The p-regular partitions of n label in a canonical way the p-modular irreducible representations D of S n w6x. The p-residue of a node A s Ž i, j . in the Young diagram of is defined to be the residue modulo p of j y i, denoted res A s j y i Žmod p .. The p-residue diagram of is obtained by writing the p-residue of each node of the Young diagram of in the corresponding place.
ALTERNATING GROUPS
149
EXAMPLE. Take p s 5, s Ž6 2 , 5, 4.. Then the 5-residue diagram of is 0 4 3 2
1 0 4 3
2 1 0 4
3 2 1 0
4 3 2
0 4
For a partition , its node sequence N Ž . is a signature sequence defined as follows. Let A1 , . . . , A l be all of the indent and removable nodes of , read successively from top to bottom and from right to left in the Young diagram of . For each i g 1, . . . , r 4 , let c i be the residue of A i and let the sign i be q if A i is removable and y if A i is an indent node. Then the node sequence is defined to be N Ž . : c1 1 c 2 2 ⭈⭈⭈ c s s . A removable node A of is then called normal Žresp., good . if the corresponding entry in the node sequence is normal Žresp., good.; the height ht A is then the height of this entry. The ␣-peak ¨ alue of is the peak value of ␣ in its node sequence. EXAMPLE. Let p s 5 and s Ž12, 7 2 , 5 3 , 3, 13 .. Below, we have only indicated the removable and indent nodes in the 5-residue diagram of .
Then the node sequence of is N Ž . : 2y 1q 1y 4q 2y 4q 2y 1q 4y 1q 0y, where we have underlined the normal entries in the node sequence. Note that then the corresponding removable nodes of are also normal. These combinatorial concepts play a vital role ˆ in the following theorem due to Kleshchev Žsee w8x ᎐ w11x..
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THEOREM 3.1. Let be a p-regular partition of n, n g ⺞, n G 2. Then the following holds: Ži. Žii. are good. Žiii. Then the
socŽ D < S ny 1 . , [A good D _ A. D < S ny 1 is completely reducible if and only if all normal nodes in Let A be a removable node of such that _ A is p-regular. multiplicity of D _ A in D < S ny 1 is given by D < S ny 1 : D _ A s
½
ht A 0
if A is normal in , else.
As a consequence of this theorem one can determine the p-regular partitions of n for which the restriction D < S ny 1 is irreducible. We recall the following definition: DEFINITION 3.2. Let s Ž l 1a1 , . . . , l ta t . be a p-regular partition, where l 1 ) l 2 ) ⭈⭈⭈ ) l t , 0 - a i - p for i s 1, . . . , t. Then is called a JS-partition if its parts satisfy the congruence l i y l iq1 q a i q a iq1 ' 0 mod p
for 1 F i - t.
The type ␣ of is defined by ␣ ' l 1 y a1 mod p. Note that the type of a JS-partition is just the residue of the unique normal Žand thus good. node at the top corner of . The letters JS are an abbreviation of Jantzen and Seitz; these authors conjectured the equivalence of Ži. m Žiii. in the following corollary Žsee w3x, w8x, w9x.. We denote the partition obtained by removing the ith corner node from by Ž i .. COROLLARY 3.3. With notation as abo¨ e, the following are equi¨ alent: Ži. Žii. in .. Žiii.
D < S ny 1 is irreducible Ž and in this case, D < S ny 1 , D Ž1. .. has exactly one normal node Ž which is then the only good node
is a JS-partition.
For later purposes we also have to introduce a different notation for p-regular partitions, which will allow us to have both the removal of good nodes and the p-analogue of conjugation given by the Mullineux map Ždefined in the next section. under control. Let be a p-regular partition of n. The p-rim of is a part of the rim of w6, p. 56x, which is composed of p-segments. Each p-segment, except possibly the last, contains p points. The first p-segment consists of the first p points of the rim of , starting with the longest row. ŽIf the rim
151
ALTERNATING GROUPS
contains at most p points, it is the entire rim.. The next segment is obtained by starting in the row next below the previous p-segment. This process is continued until the final row is reached. We let a1 be the number of nodes in the p-rim of and let r 1 be the number of rows in ; if a1 is a multiple of p, the p-rim is called p-singular. Removing the p-rim of , we get a p-regular partition of n y a1 , and we let a2 , r 2 be the length of the p-rim and the number of parts of this new partition, respectively. Continuing in this way, we get the sequences of numbers a1 , . . . , a m , r 1 , . . . , rm . The residue symbol R p Ž . of is then defined as R p Ž . s
½
x1 y1
x2 y2
⭈⭈⭈ ⭈⭈⭈
xm , ym
5
where x j is the residue of a mq1yj y rmq1yj modulo p, and y j is the residue of 1 y rmq 1yj modulo p. The partition can be recovered from the residue symbol R p Ž . as the unique p-regular partition with this residue symbol Žsee w2x.. A column xy ii in the residue symbol is called singular if yi s x i q 1 and regular otherwise. Note that a residue symbol can never start with the singular column 01 , as this does not correspond to a p-regular partition. Now we define the Mullineux Žsignature. sequence M Ž . via the residue symbol of . With R p Ž . as above, we set M Ž . s 0y
x1 q x2 q xm q
Ž x1 q 1. y Ž x 2 q 1. y
.. . x q Ž m 1. y
y1 q y2 q .. . ym q
Ž y1 y 1. y Ž y 2 y 1. y
.
Ž ym y 1 . y
Starting with the signature 0 y corresponds to starting with an empty partition at the beginning, which just has the indent node Ž1, 1. of residue 0. In w2x the following result was proved. THEOREM 3.4. p y 1 we ha¨ e
Let be a p-regular partition. Then for all ␣ , 0 F ␣ F
␣ Ž M Ž . . s ␣ Ž N Ž . . . As a consequence, we can recognize the normal and good nodes of also in R p Ž .: COROLLARY 3.5. The following statements are equi¨ alent for a p-regular partition Ž0 F ␣ F p y 1.: Ži. Žii.
has a normal Ž good . node of residue ␣ . M Ž . has a normal Ž good . entry of residue ␣ .
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BESSENRODT AND OLSSON
In w2x the following result on the behavior of residue symbols with respect to removing good nodes was proved: THEOREM 3.6. Suppose that the p-regular partition has a good node A of residue ␣ . Let R p Ž . s
½
x1 y1
x2 y2
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Then for some j, 1 F j F m, one of the following occurs: Ž1.
x j is ␣-good for M Ž . and R p Ž _ A. s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
x2 y2
xj y 1 yj
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Ž2. y j is ␣-good for M Ž ., and in this case if j s 1 and ␣ s 0, then x 1 s 0 and R p Ž _ A. s
½
x2 y2
⭈⭈⭈ ⭈⭈⭈
xm , ym
5
i.e., the first column is remo¨ ed, or else if Ž j, ␣ . / Ž1, 0., then R p Ž _ A. s
½
x1
x2
⭈⭈⭈
xj
⭈⭈⭈
xm
y1
y2
⭈⭈⭈
yj q 1
⭈⭈⭈
ym
5
.
4. MODULAR IRREDUCIBLE A n-REPRESENTATIONS For a classification of the modular irreducible A n-representations, we need to know which modular irreducible Sn-representations split when restricted to A n . For p s 2, the answer was given by Benson w1x: THEOREM 4.1. Let s Ž 1 , . . . , m . & n be a 2-regular partition of n g ⺞, n G 2, D the corresponding modular irreducible Sn-representation. Then the restriction D < A n is reducible if and only if the parts of satisfy the following two conditions Ž where we set mq 1 s 0 if m is odd .: Ži. Žii.
2 jy1 y 2 j F 2 for all j. 2 jy1 q 2 j k 2 Žmod 4. for all j.
If D < A n is reducible, then it splits into two nonisomorphic irreducible sumq y mands, say D < A n , C [ C .
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ALTERNATING GROUPS
We call the 2-regular partitions satisfying the conditions Ži. and Žii. above S-partitions. From Theorem 4.1 we can now deduce the classification of the 2-modular irreducible A n-representations: A complete list of 2-modular irreducible A n-representa-
COROLLARY 4.2. tions is gi¨ en by q
y
C , C C
for & n a 2-regular S-partition, for & n a 2-regular non-S-partition.
For p / 2, the split restrictions are those of modules fixed by tensoring with the sign representation. This case is determined by the Mullineux map, which we describe on the residue symbols: DEFINITION 4.3. Let the residue symbol of the p-regular partition be R p Ž . s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Then the Mullineux conjugate M is the p-regular partition whose residue symbol is R p Ž M . s
½
j s
½
1 y y1 1 y x1
⭈⭈⭈ ⭈⭈⭈
m y ym m y x m ,
5
where 1
if x j q 1 s y j ,
0
otherwise.
As a consequence of Theorem 3.6, we state the following property of good nodes with respect to the Mullineux map: THEOREM 4.4 w2x, w5x. Let be a p-regular partition, and let A be a good node of . Then there exists a good node B of the Mullineux image M such that Ž _ A. M s M _ B. In 1979, Mullineux conjectured that the map M defined above describes the result of tensoring an irreducible representation by the sign representation. After a long struggle this conjecture was finally solved by the work of Kleshchev and Ford and Kleshchev. As an application of his Branching Theorems for modular Sn-representations, Kleshchev had reduced the Mullineux Conjecture to the purely combinatorial statement in Theorem 4.4 on the removal of good nodes from a partition, which was then proved by Ford and Kleshchev.
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THEOREM 4.5.
Let be a p-regular partition. Then D m sgn , D . M
Based on this and Clifford theory, we have a combinatorial description for the splitting of the modular irreducible Sn-representations over A n also for odd primes p Žsee w4x.: THEOREM 4.6. Let p be an odd prime, and let s Ž 1 , . . . , m . & n be a p-regular partition of n, D the corresponding modular irreducible Sn-representation. Then the restriction D < A n is reducible if and only if is a fixed point under the Mullineux map, i.e., M s . If D < A n is reducible, then it q y splits into two nonisomorphic irreducible summands C , C , so D < A n , q y C [C . As before, this implies the classification of the p-modular irreducible A n-representations w4x: COROLLARY 4.7. A complete list of p-modular irreducible A n-representations is gi¨ en by the modules: q
y
C , C
C s C
& n p-regular, s M M
& n p-regular, / M .
5. BRANCHING AND p-CONJUGATION AT ODD p In this section, we always assume that p ) 2; otherwise p is an arbitrary odd integer in the combinatorial statements and has to be a prime in the representation theoretic statements. Recall the phenomena occurring ‘‘at characteristic 0’’ Ždescribed in Section 2. to imagine what to expect now. In the following, we denote by F a field of characteristic p, again assumed to be sufficiently large. First a lemma: LEMMA 5.1.
Let M be an FS n-module. Then soc Ž M < A n . s soc Ž M . < A n .
Proof. Since A n is normal in Sn and of index prime to p, the Jacobson radicals of FS n and FA n are related by J Ž FS n . s J Ž FA n . FS n , due to a result by Villamayor w12, p. 524x. Since the socle of a module is the set of elements annihilated by the Jacobson radical w15, Theorem 1.8.18x, the assertion follows. For convenience, we let C denote the module C s C if is not a o q y Mullineux fixed point, and C g C , C 4 if is a Mullineux fixed point. o
o
155
ALTERNATING GROUPS
PROPOSITION 5.2. D < S ny 1 is completely reducible if and only if C < A ny 1 is completely reducible. o
Proof. This follows immediately from the preceding Lemma and Theorem 4.6. In the proof of the following result we require the content vector of a partition , which is defined to be cŽ . s Ž c 0 , . . . , c py1 ., where c i is the number of nodes in the p-residue diagram of , which are of residue i. The basic fact we need is that with cŽ . as above, the content vector of the Mullineux conjugate is cŽ M . s Ž d 0 , . . . , d py1 ., where d␣ s cy ␣ Žsee w2x, w14x.. THEOREM 5.3 ŽThe almost-p-symmetric case.. Let be p-regular, and assume / M. Furthermore, assume that has a good node A with _ A s Ž _ A. M . Ži. Then all of the columns in the residue symbol of are Mullineuxfixed, except for one column,
¡
x 1yx x y s or x y1 y x
~
¢
with x / 0,
with x / y1,
where the marked entry corresponds to the good node A in the sense of Theorem 3.6. In particular, res A / 0. Žii. There are no normal nodes B and B˜ with B / A / B˜ such that _ B and _ B˜ are p-regular and satisfy M
_ B s Ž _ B˜. . In particular, there are no good nodes B and B˜ different from A for which this holds. Proof. Ži. Theorem 3.6 immediately tells us that all of the columns in the residue symbol of are Mullineux-fixed, except for the one column with the good entry corresponding to the good node A. As itself is not Mullineux-fixed, this exceptional column cannot be a 0-column at the start of the residue symbol, since this would vanish on removing A.
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So let us assume that xy is the exceptional column, and consider first the case where the upper entry x is the good entry corresponding to A. Then the corresponding column x yy 1 in R p Ž _ A. has to be Mullineux-fixed. If this column is regular, then we have y s 1 y x and x / 0, as claimed. If we assume it is singular, then y s x, contradicting the assumption that the upper entry x in R p Ž . was good. Similarly, if the lower entry y in the exceptional column is the good entry corresponding to A, then in the case of the corresponding column y qx 1 for _ A being regular, we obtain y s yx y 1, and we must have x / y1, since otherwise in the Žsingular. column xy s y1 the lower entry is not good. If we assume y qx 1 to be singular, 0 then y s x and the fixed point condition gives x s yx; hence in this case the exceptional column is 00 and hence is Mullineux-fixed, contradicting our assumption. The conditions we have derived above immediately imply that res A / 0. Žii. Let cŽ . s Ž c 0 , c1 , . . . , c py1 . be the content vector of , and let r s res A, so r / 0 by Ži.. Then the condition _ A s Ž _ A. M implies c r y 1 s c pyr and c i s c pyi for all residues i / r, p y r. Now let B, B˜ be nodes of , different from A, with residues i and j, respectively, such that their removal gives p-regular partitions with _ B s Ž _ B˜. M . In the first part of the proof we show that i s r s j; here we do not need the condition that these nodes are normal Žand that A is good.. Since the Mullineux map leaves the content of residue 0 invariant, either none or both of B and B˜ are of residue 0. If i s 0 s j, then c k s c pyk for all residues k / 0, a contradiction, since r / 0. Hence i / 0 / j. We assume first that i / r / j. If j / p y i, then comparing the content at residue i and using the relation above gives c i y 1 s c pyi s c i , a contradiction. If j s p y i, then we obtain c k s c pyk for all residues k, again a contradiction. Hence we only have to consider the case where B / A / B˜ are normal nodes of with residues i s r s j. As these nodes are normal but not good, the r-peak value goes down by 2 on removing these nodes Žthe q1-contribution before the good node is replaced by a y1-contribution.; hence
r Ž _ B˜. s r Ž . y 2 s r Ž Ž _ B .
M
.
s yr Ž _ B . s yr Ž . q Ž B . ,
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ALTERNATING GROUPS
where Ž B . g 0, 14 ; the last equation follows as res B s r / yr Ž r / 0., and hence the effect of removing B can at most be an increase of 1 in the peak values for the residues r " 1. On the other hand, removing a good node only decreases the peak value for the corresponding residue by 1, and the effect on other residues is similar to that above, so
r Ž . y 1 s r Ž _ A . s yr Ž Ž _ A .
M
.
s yr Ž _ A . s yr Ž . q Ž A . , where Ž A. g 0, 14 Žnote that for the last equation we have used again that r / 0 and p is odd, so that r / yr .. Hence we conclude
r Ž . y yr Ž . y 1 s Ž B . q 1 s Ž A . ; thus we deduce Ž B . s 0 and Ž A. s 1. Now by removing a removable node of residue r, we can either create new removable nodes of residue r " 1 or indent nodes of such residues can vanish; the overall effect is in all cases to lift the paths for these residues by 1 at this position. If A and B both are higher Žresp., both are lower. than the good node of such residues, then the corresponding peak value in _ A and _ B is increased in both cases by 1 Žresp., is in both cases unchanged. compared to the peak value in . Since B is normal of the same residue as the good node A, it is higher in than A; hence we also might have an increase in the critical peak value by 1 in _ B and an unchanged peak value when removing A Žbut not the other way around!.. Hence the situation Ž B . s 0 and Ž A. s 1 is impossible, and thus the claim is proved. The theorem above allows us to deduce information on the restrictions $ of the A n-representations; as in characteristic 0, we denote by C the q y M module C [ C if s M Žresp., the module C s C if / M .. THEOREM 5.4. Let be p-regular with / M , and assume that has a good node A with _ A s Ž _ A. M . Then Ži.
$
soc Ž C < A ny 1 . , C _ A [
[C
_ B
,
B/A good
and this socle is multiplicity-free. Žii. Let B be a remo¨ able node of such that _ B is p-regular. Then o the multiplicity of C _ B in C < A ny 1 is gi¨ en by C < A ny 1 : C _ B
o
s
½
if B s A,
ht A ht B q D < S ny 1 : D Ž _ B .
M
else.
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BESSENRODT AND OLSSON
Proof. This follows from the Branching Theorem for modular Sn-representations, the classification of the modular irreducible A n-representations, and part Žii. of Theorem 5.3. Remark. Note that in the case of B / A in part Žii. above, we do not know how to compute the second summand. THEOREM 5.5 ŽThe far-from-p-symmetric case.. Let be p-regular with / M , and assume that has no good node A with _ A s Ž _ A. M . Then there are no good nodes B and B˜ with M
_ B s Ž _ B˜. . Proof. Assume there are good nodes B and B˜ with
Ž ).
M
_ B s Ž _ B˜. .
Then these nodes are different from A, and their residues r s res B, s s res B˜ are different, and both are different from 0. We consider again the p-content vector cŽ . s Ž c 0 , c1 , . . . , c py1 . of , and then compare the content vectors of _ B and Ž _ B˜. M . For s / p y r, this gives the equations c py r s c r y 1, c r s c y p y r and hence a contradiction. Thus we must have s s p y r, i.e., B and B˜ are good nodes with conjugate residues. Next we consider the residue symbols of _ B and Ž _ B˜. M . Let the residue symbol of be R p Ž . s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Since B and B˜ are good Žand not of residue 0., their removal from only has an effect on an entry r Žresp., p y r . in the residue symbol at the ith and jth columns Žsay.. Let us first consider the case where the ith column in is r , y
ALTERNATING GROUPS
159
with the marked entry corresponding to the good node B. If i / j, then Ž). implies that the ith column in R p Ž _ B . satisfies
ry1 y s
¡ yy
if y / 1 q r ,
¢
if y s 1 q r .
~
yr or yr 1yr
In both cases we immediately obtain a contradiction. If i s j, then the ith column in is r yr
,
˜ hence with the second marked entry corresponding to the good node B; this column is Mullineux-fixed. But by Ž)., all other columns in the residue symbol are also Mullineux-fixed, and hence so is , a contradiction. Next we first consider the case where the ith column in is x , r with the marked entry corresponding again to the good node B; note that this column then has to be regular. If i / j, then Ž). implies now that the ith column in R p Ž _ B . satisfies x s yr , yx rq1 immediately giving a contradiction. If i s j, then the ith column in is now yr r
;
hence again this column is Mullineux-fixed. But as before, Ž). implies that all other columns in the residue symbol are also Mullineux-fixed; hence so is , again a contradiction.
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BESSENRODT AND OLSSON
From this we deduce the following. THEOREM 5.6. Let be p-regular with / M , and assume that has no good node A with _ A s Ž _ A. M . Then Ži. soc Ž C < A ny 1 . ,
[C
_ B
,
B good
and this socle is multiplicity-free. Žii. Let B be a remo¨ able node of such that _ B is p-regular. Then the multiplicity of C _ B in C < A ny 1 is gi¨ en by C < A ny 1 : C _ B s ht B q D < S ny 1 : D Ž _ B . THEOREM 5.7 ŽThe p-symmetric case..
M
.
Let be p-regular with s M .
Ži. If N is a remo¨ able node of with _ N s Ž _ N . M , then res N s 0. ˜ Žii. For any good node B there is a good node B˜ with res B s yres B, and M
_ B s Ž _ B˜. . In particular, a good node A in of residue 0 satisfies _ A s Ž _ A. M . Žiii. If has only one good node, then this is of residue 0. Proof. Ži. We consider the content vector cŽ . s Ž c 0 , . . . , c py1 .. As s M , we have c i s c pyi for all i. Let N be a removable node of residue r / 0; if _ N s Ž _ N . M , then this implies c r y 1 s c pyr , and hence a contradiction. Žii. If B is a good node of of residue r, then we have M Ž _ B . s M _ B⬘ s _ B⬘,
where B⬘ is a good node of of conjugate residue p y r. If B is good of residue r s 0, then we must have B⬘ s B. Žiii. follows immediately from Žii.. For the branching of the representations this implies THEOREM 5.8.
Let be p-regular with s M. Then we ha¨ e
Ži. The restrictions of the two conjugate modules C q and C y ha¨ e conjugate socles "
Ž C _ A 0 . [
[
B good res Bg 1, . . . , Ž py1 .r2 4
C _ B ,
161
ALTERNATING GROUPS
where the first summand only occurs if has a good node A 0 of residue 0. In particular, these restrictions are multiplicity-free. Žii. Let B be a normal node of such that _ B is p-regular. Then $
C < A ny 1 : C _ B
o
s
½
2 ht B ht B
if Ž _ B .
M
/ _ B,
if Ž _ B .
M
s _ B.
As has been described in Section 3, we know by Kleshchev’s Branching Theorems which irreducible Sn-modules split on restriction to Sny1. We are now going to use this together with the description of the residue symbols of JS-partitions given in w3x for dealing with the analogue of the Jantzen᎐Seitz question for the alternating groups. We first recall the description of JS-partitions that are fixed under the Mullineux map. THEOREM 5.9. Let be a p-regular partition. Then is a JS-partition and a Mullineux fixed point if and only if its residue symbol, R p Ž . s
x1 y1
½
⭈⭈⭈ ⭈⭈⭈
xk , yk
5
can be constructed iterati¨ ely by the following procedure: x1 0 s . y1 0
½ 5 ½5
If the first k y 1 columns of the residue symbol are already constructed, then we ha¨ e two possibilities for a regular extension, namely xk s yk
½ 5 ½
1 y y ky 1 y ky1 y 1
5
or
½
y ky1 y 1 , 1 y y ky1
5
and if y ky 1 s 1, then we ha¨ e, furthermore, one possibility for a singular extension, namely, xk 0 s . yk 1
½ 5 ½5
The construction rule given in the theorem can be described alternatively by a diagram in the following way. The JS-partitions that are Mullineux fixed points are obtained by adding on columns to the residue symbol according to a walk starting at 0 0
½5
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in the diagram Ž). below Žwe omit the brackets for simplification.; the symbol ; in the diagram means that we have a loop at the corresponding node of the graph.
Since the starting column 0 0
½5 already guarantees a normal node of residue 0, it is clear that the unique good node of such JS partitions is of residue 0, i.e., that they are of type 0. Furthermore, Mullineux fixed JS-partitions have empty or square p-cores, and the p-core only depends on the end of the walk in the diagram above Žsee w3x.. Thus we can give the p-cores of Mullineux fixed JS-partitions by putting the corresponding p-core at the end point of the walk in the diagram:
THEOREM 5.10. Let n G 2, let p be an odd prime, and let be a p-regular partition of n. Then the following are equi¨ alent: Ži. C < A ny 1 is irreducible. Žii. One of the following holds: Ža. D < S ny 1 is irreducible. M Žb. D < S ny 1 , D Ž1. [ D Ž1. and M s but Ž1. M / Ž1.. Žiii. One of the following holds: Ža. is a JS-partition. Žb. The residue symbol R p Ž . is obtained by adding on columns along a walk in the extended diagram Ž)). below, starting at any regular column yxx , with x / 0. o
163
ALTERNATING GROUPS
One of the following holds: Ža. is a JS-partition that is not Mullineux fixed. Žb. R p Ž . is obtained by adding on columns along a walk in the diagram Ž). abo¨ e starting at 00 , or along a walk in the extended diagram Ž)). below, starting at any regular column yxx , with x / 0. Žiii⬘.
Proof. First we assume Ži.. If M / , then D < A ny 1 , C < A ny 1 is irreducible, so this implies immediately that D < S ny 1 is irreducible. q y So we now consider a fixed point M s . Note that C and C are conjugate representations, so by assumption the conjugate modules q y C < A ny 1 and C < A ny 1 are also both irreducible. If D < S ny 1 is irreducible, there is nothing to prove. In the other case we deduce by Proposition 5.2 and Kleshchev’s Branching Theorem: D < S ny 1 , D _ A [ D _ B , where A and B are the two good nodes of . Moreover, since s M , we must have Ž _ A. M s _ B. Since we always have a normal node at the first corner of a partition, giving a composition factor of the restriction in the block corresponding to the removal of the good node of the same residue, the first removable node itself must be good. Hence we obtain D < S ny 1 , D Ž1. [ D Ž1. , M
with Ž1. / Ž1. M . Thus Ži. « Žii. is proved. For Žii. « Žiii. we have to show that a non-JS-partition with s M , Ž1. / Ž1. M , and D < S ny 1 , D Ž1. [ D Ž1.
M
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BESSENRODT AND OLSSON
can be constructed along the diagram Ž)). above. Note that Ž1. M s Ž g . for some g / 1, corresponding to removing a good node at the g th corner of . Consider the residue symbol R p Ž . s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
xk . yk
5
The condition s M amounts to the equations x i s yyi
for regular columns Ž note that x i / Ž p y 1 . r2 . .
x i s 0, yi s 1
for singular columns Ž here we must have i / 1 . .
We know from the decomposition of D < S ny 1 that has only the good nodes _ Ž1. and _ Ž g ., and no further normal nodes w11x. Note also that the two good nodes must have conjugate residues r / yr, say. Now consider the Mullineux sequence for : M Ž . s 0y
x1 q x2 q .. . xkq
Ž x1 q 1. y Ž x 2 q 1. y
Ž yx 1 . q
Žy Ž x1 q 1. . y
y2 q
Ž y 2 q 1. y
Ž x k q 1. y
yk q
Ž yk q 1. y
.
If x 1 s 0, then y 1 q s Žyx 1 . q gives a normal node of residue 0 Žhence the good one of residue 0., contradicting the above properties of our two good nodes. Hence x 1 / 0 is the first good node. Since x 1 / Ž p y 1.r2, we have yx 1 / x 1 q 1, and hence yx 1 s y 1 is the next good node. Since we then have no further normal node, we must have x 2 s x 1 q 1 or yŽ x 1 q 1. or 0. If x 2 s 0, then y 2 s 0 is not possible, but only y 2 s 1, i.e., a singular column, and then the situation for the next column in the residue symbol is as before. Thus we have a certain number of singular columns 01 4 , and then the next regular column xy ii 4 has to satisfy x i 2 s x 1 q 1 or yŽ x 1 q 1., yi 2 s yx i 2 . Then again, we may have a number of singular columns, and then the next regular column xy ii 4 satisfies x i 3 s x i 2 q 1 or yŽ x i 2 q 1., yi 3 s yx i 3 . This repeats, so we obtain indeed a walk in the extended diagram Ž))., starting at a regular column different from 00 . Now to Žiii. « Ži.. 2
2
3
3
Case 1. Assume is a JS-partition with / M . If Ž1. / Ž1. M , then clearly, D < A ny 1 , D Ž1. < A ny 1 , C < A ny 1 is irreducible. So we may assume now that Ž1. s Ž1. M .
165
ALTERNATING GROUPS
Consider the residue symbol R p Ž . s
½
⭈⭈⭈ ⭈⭈⭈
x1 y1
xk . yk
5
By Theorem 3.6 we know that the residue symbol R p Ž Ž1.. of Ž1. can only be one of
½
x1 y 1 y1
⭈⭈⭈ ⭈⭈⭈
xk yk
5 ½
or
½
or ⭈⭈⭈ ⭈⭈⭈
x2 y2
⭈⭈⭈ ⭈⭈⭈
x1 y1 q 1
xk yk
5
xk , yk
5
where
½
x1 y 1 y1
M
5 ½ s
x1 y 1 y1
5
resp.,
x1 y1 q 1
½
M
5 ½ s
x1 y1 q 1
5
resp., x 1 s y 1 s 0, and furthermore, M
x1 y1
x1 , y1
½ 5 ½ 5 /
and all other columns are Mullineux-fixed and hence of the form
½ yxx 5
or
0 . 1
½5
By these conditions, the second and third possibilities for the residue symbol of Ž1. are immediately excluded; in particular, this implies that the type ␣ of satisfies ␣ / 0. So we have the following possibilities now for the first column of R p Ž .: x1 g y1
½ 5 ½½
␣ , ␣ / 0; ␣q1
5
½␣5 0
, ␣ / 0, p y 1;
½ ␣0 5 , ␣ / 0, 1 5 .
Then R p Ž Ž1.. starts with one of
½ ␣␣ 5 ½ ␣ 5 ½ ␣ 5 y1 , q1
y1 , 0
0 , q1
respectively; but the first one is not Mullineux-fixed, the second is only for ␣ s 1, and the third is only for ␣ s p y 1; hence R p Ž . has to start with 10 4 and ␣ s 1, or with p y0 1 4 and ␣ s p y 1.
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BESSENRODT AND OLSSON
The restrictions for the further columns do not allow any regular extensions from these first columns. But also, the construction rules for singular columns do not give Mullineux fixed columns as required. Hence we must have R p Ž . s
1 0
½5
or
½
0 , py1
5
which corresponds to s Ž2. Žresp., s Ž12 .., and then Ži. obviously holds. Case 2. We assume now that is a JS-partition with s M. Then is of type 0, and it is immediately seen that also Ž1. s Ž1. M , and then Ži. holds. Case 3. Assume now that is not a JS-partition, so is constructed along a path in diagram Ž))., in particular, s M . Reversing the arguments for Žii. « Žiii., we have D < S ny 1 , D _ A [ D _ B , where A, B are good nodes of of conjugate residues / 0, and Ž _ A. M o s _ B. Then C < A ny 1 , D _ N < A ny 1 , N g A, B4 , and hence this restriction is irreducible. The equivalence of Žiii. and Žiii⬘. is clear. From the proof of the theorem, one easily deduces the following. COROLLARY 5.11. We assume the notation of the theorem. Then if C < A ny 1 o o is irreducible, we ha¨ e C < A ny 1 , C Ž1. . o
Remarks. Ži. In part Žiii. of the theorem above, it would be nice to have a description in terms of the parts of like the one available for JS-partitions. Žii. We have already observed that JS-partitions have special p-cores w3x; moreover, the ‘‘almost JS-partitions’’ constructed along a path in Ž)). Žresp., also those constructed in a more general fashion in the construction diagram for JS-partitions w3x. have special cores that depend in an easy way on the start and the end of the path in the diagram. 6. BRANCHING AT p s 2 As we have already seen in Section 4, the behavior of the A n-representations at characteristic 2 is very much different from that at odd primes. While in some respects the combinatorics of S-partitions is easier than
ALTERNATING GROUPS
167
that of Mullineux fixed points, the 2-modular representation theory presents more difficulties, since A n is a normal subgroup of index p s 2 in Sn . First we give a description of S-partitions Žsee Theorem 4.1. in terms of residue symbols as an analogue of the description of Mullineux fixed points via residue symbols. PROPOSITION 6.1. The 2-residue symbols of S-partitions are constructed by a walk in the following directed graph, where we start at one of the marked columns and we may loop around any of the three ¨ ertices in the graph:
In other words, the residue symbols start with a regular column, and they ha¨ e no column 10 . Proof. As long as the smallest part of is at least 2, the 2-rim consists of horizontal dominoes and thus is 2-singular. In removing such singular 2-rims from a given S-partition , we again obtain S-partitions. Doing this as long as possible, we reach an S-partition, which we may assume to be of even length by adding a part 0 if necessary. This S-partition then ends on one of the following pairs of parts: 1, 0 2, 1 3, 1. Then the next step in removing the 2-rim leads to the following corresponding columns in the residue symbol: 0 0
1 1
1 , 1
and the partition obtained after the removal is again an S-partition. So S-partitions can be built up along residue symbols. As we can never start a residue symbol with the singular column 01 , and since 10 is not an S-partition, the start has to be at one of the two marked columns. Using the list of end pairs given above, it is then easily seen that at each step all three columns in the diagram above give possible extensions to an S-partition.
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BESSENRODT AND OLSSON
COROLLARY 6.2. Let be an S-partition with residue symbol R 2 Ž .. Let m 0 be the number of columns 00 in R 2 Ž . Ž resp., let m1 be the number of columns 11 .. Then the 2-core of is gi¨ en by
Ž2.
¡Ž2Ž m y m . y 1, 2Ž m y m . y 2, . . . , 2, 1. y m . , 2 Ž m y m . y 1, . . . , 2, 1 . ¢⭋
s~ Ž 2 Ž m
0
1
1
0
0
1
if m 0 y m1 ) 0,
1
if m1 y m 0 ) 0, if m 0 s m1 .
0
Proof. Note that the p-content c s Ž c 0 , c1 , . . . , c py1 . of a partition determines its p-core. The associated ª n-vector is given by ª n s Ž c 0 y c1 , . w x c1 y c 2 , . . . , c py2 y c py1 , c py1 y c0 . By 2 we can easily compute the ª nvector of via its residue symbol. Hence we obtain the 2-core of the partition given via its residue symbol R 2 Ž . as above; we omit the details. Let be a 2-regular partition.
LEMMA 6.3.
Ža. If is an S-partition and A is a remo¨ able node of such that _ A is 2-regular, then we ha¨ e
_ A is an S-partition if and only if res A s 0. Žb. If is not an S-partition, then there is at most one node A in such that _ A is an S-partition, and such a node is of residue 1. Proof. By definition, the parts of S-partitions come in one of the following pairs Žpictured in the 2-residue diagram.: ⭈⭈⭈ ⭈⭈⭈
⭈⭈⭈ ⭈⭈⭈
0 0
⭈⭈⭈ ⭈⭈⭈
1 1
0 1
From this, Ža. and Žb. easily follow. Analogous to the notation at odd characteristic, we let C s C if is o q y not an S-partition and C g C , C 4 if is an S-partition. For the A n-representations at characteristic 2, we can now deduce the following: o
THEOREM 6.4.
Let be a 2-regular partition of n. "
Ža. Assume is an S-partition of n. Then the module socŽ C < A ny 1 . " has a constituent C Ž _ A 0 . if A 0 is a good node of residue 0, and it has a _ A 1 constituent C if A1 is a good node of residue 1. Furthermore, if B is a remo¨ able node with _ B 2-regular, then $
C < A ny 1 : C _ B
o
s
½
ht B 0
if B is normal, else,
and ht B is e¨ en for all normal nodes B of residue 1 such that _ B is 2-regular.
ALTERNATING GROUPS
169
Žb. Assume is not an S-partition. Then the module socŽ C < A ny 1 . has a constituent C _ A 0 if A 0 is a good node of residue 0, and it has a constituent C _ A 1 if A1 is a $ good node of residue 1 with _ A1 not an S-partition Ž resp., a constituent C _A 1 if A1 is a good node of residue 1 with _ A1 an S-partition.. The latter occurs only if is almost an S-partition except for one pair of parts Ž2 k, 2 k y 2., with A1 at the end of the first of these two parts. Furthermore, if B is a remo¨ able node with _ B 2-regular, then C < A ny 1 : C _ B
o
s
½
ht B 0
if B is normal, else.
"
The case C _ B s C _ B arises only if is almost an S-partition except for one pair of parts of the form Ž2 k, 2 k y 2. or Ž2 k, 2 k y 3., with B at the end of the part 2 k normal of residue 1. o
For p s 2, the characterization of the irreducible restrictions is not as complete as for odd primes p, but there are at least very strong combinatorial restrictions. First we compare the A n-situation with the Sn-situation: THEOREM 6.5. equi¨ alent:
Let be a 2-regular partition. Then the following are
Ži. C < A ny 1 is irreducible. Žii. One of the following holds: Ža. D < S ny 1 is irreducible and / Ž2 l, 2 l y 2. if n ' 2 Žmod 4.. o
Ž2.
Žb. D < S ny 1 , D Ž where this denotes a uniserial module with D Ž2. Ž2. . Ž top and socle D , and s 1 , . . . , k . is an S-partition with 1 e¨ en. Proof. First we assume Ži.. Case 1. Suppose is not an S-partition; hence D < A ny 1 , C < A ny 1 is irreducible, and so D < S ny 1 , D Ž1. must be irreducible, i.e., is a JS-partition. Furthermore, as D Ž1. < A ny 1 is irreducible, also Ž1. is not an S-partition. Thus is not of the form Ž2 l, 2 l y 2. for some l ) 1, as was to be proved in this case. Case 2. Suppose is an S-partition; hence q
y
D < An , C [ C , q
y
with C ` C . Now q
y
D < A ny 1 , C < A ny 1 [ C < A ny 1 ,
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BESSENRODT AND OLSSON
a direct sum of two conjugate irreducibles; hence we have one of the following two possibilities: Ža. Žb.
is a JS-partition and Ž1. is an S-partition. D < S ny 1 ; 2 ⭈ D , where is not an S-partition.
Case Ža.. As s Ž 1 , . . . , k . is a JS-partition, we have 1 ' 2 ' ⭈⭈⭈ ' k Žmod 2.. Since is also an S-partition, we have
2 jy1 y 2 j s 2
and
1 ' 2 ' ⭈⭈⭈ ' k ' 1 Ž mod 2 . .
But then Ž1. is always an S-partition, so this condition can be omitted in Ža.. Note also that in this case again, s Ž2 l, 2 l y 2. does not occur. Case Žb.. Since the socle of D < S ny 1 is multiplicity-free, the module D S ny 1 cannot be completely reducible. Hence we must have <
soc Ž D < S ny 1 . , D Ž i. , where s Ž i . s _ A, A being the only good node of . Furthermore, the Branching Theorem tells us that there is no normal node of residue / res A, and there is exactly one other normal node B of residue res A; moreover, for this normal node B, the partition _ B is not p-regular, and we have D < S ny 1 ,
D Ž i. . D Ž i.
Since the first removable node of is always normal, this must be the normal node B. Note also that Ž i . is not an S-partition, since otherwise we have a contradiction to condition Ži. of the theorem. Hence we have 1 y 2 s 1, so resŽ _ Ž1.. s resŽ _ Ž2.., and thus _ Ž2. is also normal, and hence it has to be the good node A. Thus we must have i s 2, i.e., D < S ny 1 ,
D Ž2. , D Ž2.
as claimed. Furthermore, as Ž2. is not an S-partition, the S-partition must start with an even part. Now assume that Žii. is satisfied. Case 1. First suppose that D < S ny 1 is irreducible, so is a JS-partition, but is not of the form Ž2 l, 2 l y 2. for some l ) 1. If is an S-partition, then Ž1. is an S-partition by the same argument as in Case Ža. above.
171
ALTERNATING GROUPS
Hence
Ž D < S . A ny 1
q
ny 1
y
, D < A ny 1 , C < A ny 1 [ C < A ny 1 q
y
, D Ž1. < A ny 1 , C Ž1. [ C Ž1. , q
y
and so both C < A ny 1 and C < A ny 1 are irreducible. If s Ž 1 , . . . , k . is not an S-partition, then Ž1. is also not an S-partition. To see this, assume that Ž1. s Ž 1 y 1, . . . , k . is an S-partition. As is a JS-partition, we have 1 ' 2 ' ⭈⭈⭈ ' k Žmod 2., and hence 1 q 2 ' 2 Žmod 4., 2 jy1 q 2 j k 2 Žmod 4. for j / 1, and 2 jy1 y 2 j s 2 for all j. Hence we must have j ' 0 Žmod 2. for all j, but then 2 jy1 q 2 j ' 2 Žmod 4. for all j, and this implies that has only two parts, i.e., s Ž2 l, 2 l y 2.. But this case was excluded in condition Žii. of the theorem. Having shown all these properties of and Ž1., it is now clear that condition Ži. of the theorem is satisfied. Case 2. We now consider the situation where is an S-partition with even first part 1 and D < S ny 1 ,
D Ž2. . D Ž2.
Here s Ž 1 , . . . , k . must satisfy 1 y 2 s 1, 1 even, and then Ž2. s Ž 1 , 1 y 2, . . . . is not an S-partition. Hence D < A ny 1,
D Ž2. D Ž2.
; 2 ⭈ C Ž2. , A ny 1
and thus q
y
C < A ny 1 , C < A ny 1 , C Ž2. is irreducible, as was to be proved. Remark. Note that the proof of the theorem tells us exactly what the o restriction C < A ny 1 is in the cases when it is irreducible. For a partition s Ž 1 , . . . , m . of n into distinct parts, we define its doubling to be the partition dbl Ž . s
ž
1 q 1 2
,
1 2
,
2 q 1 2
,
2 2
,...,
m q 1 2
,
m 2
/
.
Via the process of regularization Žsee w6x., we then obtain a 2-regular partition dbl 2 Ž . [ dblŽ . R.
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PROPOSITION 6.6. Let p s 2, and let be a 2-regular partition satisfying one Ž and hence both. of the conditions in the theorem abo¨ e. Then is a JS-partition / Ž2 l, 2 l y 2., or it is of the form s dbl 2 Ž ., where is a partition of n into distinct parts satisfying i y iq1 ) 4 for i s 1, . . . , m y 1, and the residues of 1 , 2 , . . . modulo 4 follow a path in the diagram below, starting at 3:
Proof. If is not a JS-partition, we have already seen in the proof of the theorem that has the following properties: Ži. is an S-partition. Žii. 1 even. Žiii. 1 y 2 s 1. Živ. The first two removable nodes are the only normal nodes of . Property Ži. immediately implies that s dbl 2 Ž . for some 2-regular partition s Ž 1 , 2 , . . . . with i y iq1 G 4 and i y iq1 ) 4 if i ' 0 Žmod 4., and i k 2 Žmod 4. for all i. By properties Žii. and Žiii., we have 1 ' 3 Žmod 4.. Consider the sequence r 1 , r 2 , . . . of end residues of the rows of in its 2-residue diagram. By properties Žii. and Živ. we have 0 F j F i < r j s 1 4 y j F i < r j s 0 4 F 2. Since a part i ' 0, 1 or 3 Žmod 4. leads to the residue pairs Ž0, 1., Ž0, 0. resp., Ž1, 1., the condition above easily translates into the admissible walks in the triangle graph above, and it is clear that then i y iq1 ) 4 must hold for all i. Remarks. Ža. There are examples of the phenomenon occurring in part Žii.Žb. of Theorem 6.5 above. For all l we have D Ž2 l , 2 ly1. < S 2 ly 2 , This follows by using, e.g., w1x.
D Ž2 l , 2 ly2. . D Ž2 l , 2 ly2.
173
ALTERNATING GROUPS
There are also such examples with partitions with more than two parts: D Ž6 , 5, 1. < S11 ,
D Ž6, 4, 1. D Ž6, 4, 1.
D Ž6 , 5, 3, 1. < S14 ,
D Ž6, 4, 3, 1. . D Ž6 , 4, 3, 1.
Žb. It is not clear whether the combinatorial condition in the proposition characterizes the partitions giving irreducible A n-restrictions, thus providing an analogue of Žiii. in Theorem 5.10. Žc. By the description of the partitions given in the proposition above, it is clear that we can have the irreducible restrictions of modular A n-representations of the second type only if n ' 0 or 3 Žmod 4., and the corresponding irreducible representations belong to the principal 2-block resp., to the 2-block with 2-core Ž2, 1..
ACKNOWLEDGMENTS We thank the Mathematical Sciences Research Institute for its hospitality during a stay of the authors at the time of the ‘‘Combinatorics Year’’; the research at MSRI was supported in part by National Science Foundation grant DMS 9022140, as well as by the Deutsche Forschungsgemeinschaft Žgrant Be 923r6-1 of the first author.. The authors gratefully acknowledge the support by these institutions and the support by the Danish Natural Science Foundation.
REFERENCES 1. D. Benson, Spin modules for symmetric groups, J. London Math. Soc. (2) 38 Ž1988., 250᎐262. 2. C. Bessenrodt and J. B. Olsson, On residue symbols and the Mullineux Conjecture, J. Algebraic Combin. 7 Ž1998., 227᎐251. 3. C. Bessenrodt and J. B. Olsson, Residue symbols and Jantzen-Seitz partitions, J. Comb. Theory Ser. A 81 Ž1998., 201᎐230. 4. B. Ford, Irreducible representations of the alternating group in odd characteristic, Proc. Amer. Math. Soc. 125 Ž1997., 375᎐380. 5. B. Ford and A. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 Ž1997., 267᎐308. 6. G. James and A. Kerber, ‘‘The Representation Theory of the Symmetric Group,’’ Addison-Wesley, Reading, MA, 1981. 7. J. Jantzen and G. Seitz, On the representation theory of the symmetric group, Proc. London Math. Soc. (3) 65 Ž1992., 475᎐504. 8. A. Kleshchev, Branching rules for modular representations of symmetric groups I, J. Algebra 178 Ž1995., 493᎐511. 9. A. Kleshchev, Branching rules for modular representations of symmetric groups II, J. Reine Angew. Math. 459 Ž1995., 163᎐212.
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10. A. Kleshchev, Branching rules for modular representations of symmetric groups III, J. London Math. Soc. 54 Ž1996., 25᎐38. 11. A. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. Ž 3 . 75 Ž1997., 497᎐558. 12. G. O. Michler, Blocks and centers of group algebras, Lecture Notes in Math. 246 Ž1973., 430᎐465. 13. G. Mullineux, Bijections of p-regular partitions and p-modular irreducibles of symmetric groups, J. London Math. Soc. Ž 2 . 20 Ž1979., 60᎐66. 14. G. Mullineux, On the p-cores of p-regular diagrams, J. London Math. Soc. Ž 2 . Ž1979., 222᎐226. 15. H. Nagao and Y. Tsushima, ‘‘Representations of Finite Groups,’’ Academic Press, San Diego, 1988.
JOURNAL OF ALGEBRA ARTICLE NO.
Branching of Modular Representations of the Alternating Groups C. Bessenrodt* Fakultat ¨ fur ¨ Mathematik, Otto-¨ on-Guericke-Uni¨ ersitat ¨ Magdeburg, 39016 Magdeburg, Germany
and J. B. Olsson† Matematisk Institut, Københa¨ ns Uni¨ ersitet, 2100 Copenhagen Ø, Denmark Communicated by G. D. James Received November 5, 1997 DEDICATED TO GERHARD O. MICHLER ON THE OCCASION OF HIS
60TH
BIRTHDAY
Based on Kleshchev’s branching theorems for the p-modular irreducible representations of the symmetric group and on the recent proof of the Mullineux Conjecture, we investigate in this article the corresponding branching problem for the p-modular irreducible representations of the alternating group A n . We obtain information on the socle of the restrictions of such A n-representations to A ny1 as well as on the multiplicities of certain composition factors; furthermore, irreducible A n-representations with irreducible restrictions to A ny1 are studied. 䊚 1998 Academic Press
1. INTRODUCTION The purpose of this paper is to provide information about the restrictions to A ny 1 of modular irreducible representations of the alternating groups A n . The results are based on Kleshchev’s branching theorems Žsee * E-mail: [email protected]. † E-mail: [email protected]. 143 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
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w8x, w9x, w11x. for the symmetric group Conjecture Žsee w2x, w5x, w10x, w13x.. The starting point in this article is for characteristic 0 representations of the irreducible representation of Sn partition of n. Then S < S ny 1 s
and on the proof of the Mullineux the well-known branching theorem the symmetric group S n . Let S be in characteristic 0 labeled by the
[S
_ A
,
A
where the sum is over all removable nodes of Žsee Section 2.. On the other hand, the decomposition of the restriction of S to the alternating group A n is also easy to describe. It depends on the transposition map on partitions, since S m sgn , S ⬘ w6, Sect. 2.5x. Combining these facts, it is quite easy to deduce a branching result for irreducible representations of A n . This is done in Section 2 to give a perspective of our results in positive characteristic. Examples show that the branching phenomena in characteristic p ) 0 are much more complicated. If D is the p-modular irreducible representation of Sn labeled by the p-regular partition of n, then D < S ny 1 may also contain composition factors D , where is not obtained by removing a node from . Moreover, D < S ny 1 is in general not multiplicity-free. The recent progress on this branching problem started with the Jantzen᎐Seitz conjecture w7x describing the class of p-regular partitions for which D < S ny 1 is irreducible. The Jantzen᎐Seitz conjecture was proved by Kleshchev as a consequence of his much stronger modular branching theorems w8x ᎐ w11x Žsee Theorem 3.1.. To exploit Kleshchev’s result for investigations of the representations of the alternating groups, it is necessary to understand the restrictions of the modules D to A n . For p / 2, these depend on the Mullineux map M on p-regular partitions w13x, since D m sgn , D
M
Žsee w2x, w5x, w10x.. In contrast to the situation at characteristic 0, both the branching and restriction to A n are quite complicated in characteristic p, and to deal with the representations of the alternating groups, we have to control both processes simultaneously. This is possible using the residue symbols and signature sequences introduced by the authors in w2x. In the case of characteristic 2 we have to invoke a result of Benson w1x. The paper is organized as follows. In Section 2 we treat branching for representations of A n in characteristic 0. In Section 3 we present the necessary background dealing with positive characteristic, that is, Kleshchev’s results and residue symbols. In the next section, we describe the classification of the modular irreducible A n-representations based on
ALTERNATING GROUPS
145
Benson’s result at p s 2 and on the Mullineux map at p / 2. Section 5 deals with the case of odd characteristic, and Section 6 with the case of characteristic 2. We obtain information on the socle of the restrictions of irreducible A n-representations to A ny1 as well as on the multiplicities of certain composition factors of these restrictions. Furthermore, we describe the labels of those irreducible modules of A n that remain irreducible upon restriction to A ny 1 at characteristic p / 2, and combinatorial conditions for such labels are given at p s 2. For general facts on representations of the symmetric groups, we refer the reader to w6x. Throughout the paper we will always assume that our representations are over fields that are splitting fields for the alternating groups, e.g., one may take the fields to be algebraically closed.
2. BRANCHING OF REPRESENTATIONS OF THE ALTERNATING GROUPS AT CHARACTERISTIC 0 OR LARGE p Let be a partition of n. It is well known that at characteristic 0 resp., at prime characteristic p with p ) n, the character w x resp., the corresponding representation of Sn splits on restriction to A n if and only if is symmetric, i.e., s ⬘. In this case, it splits into two Žvia the transposition Ž12.. conjugate A n-characters 4 q and 4 y. So for a branching formula for A n-characters, one needs to study conjugation properties together with the usual well-known branching of ordinary irreducible S n-representations. First we recall some notations and definitions for partitions. Let s Ž 1 , 2 , . . . , k . be a partition of n, i.e., 1 , . . . , k are integers with 1 G 2 G ⭈⭈⭈ G k ) 0 and Ý kis1 i s n. Then Y Ž . s Ž i , j . g ⺪ = ⺪ <1 F i F k, 1 F j F i 4 ; ⺪ = ⺪ is the Young diagram of , and Ž i, j . g Y Ž . is called a node of . If A s Ž i, j . is a node of and Y Ž . _ Ž i, j .4 is again a Young diagram of a partition, then A is called a remo¨ able node, and _ A denotes the corresponding partition of n y 1. Similarly, if A s Ž i, j . g ⺞ = ⺞ is such that Y Ž . j Ž i, j .4 is the Young diagram of a partition of n q 1, then A is called an indent node of . The Nonsymmetric Case: / ⬘ Assume that has a removable node A such that
_ A s Ž _ A . ⬘ s ⬘ _ B for a suitable node B in ⬘. In this case we say that is almost symmetric.
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Since is not symmetric, the ‘‘conjugate node’’ A⬘ to A does not belong to . Hence for every other removable node C / A in we have
_ C / Ž _ C . ⬘. Even stronger: for every pair of removable nodes C / A, C˜ in , we have
_ C / Ž _ C˜. ⬘. Now, if we assume that there is no node A as above, then we can easily see that also in this case for every pair of removable nodes C, C˜ in , we have
_ C / Ž _ C˜. ⬘. Thus we obtain for the branching of the character 4 : Ži. If is almost symmetric Žwith respect to the node A., then $
4 < A ny 1 s _ A 4 q
Ý _ C4 , C/A
$
where 4 s 4 qq 4 y for a symmetric partition , and the partitions _ C, C / A, are all pairwise nonconjugate, so that the restriction is multiplicity-free. Žii. If is not almost symmetric, then
4 < A ny 1 s Ý _ C 4 , C
where the partitions _ C are all pairwise nonconjugate, so that again the restriction is multiplicity-free. The Symmetric Case: s ⬘ In this case there exists at most one removable node A Žon the main diagonal. with
_ A s Ž _ A . ⬘. Moreover, it is clear that for every other removable node C / A in , we have a removable node C˜ / C in with
_ C s Ž _ C˜. ⬘. Before we can state the restrictions of the two conjugate characters 4 q and 4 y to A ny 1 , we first have to be precise about the choice of our
147
ALTERNATING GROUPS
labeling Žsee w6, Sect. 2.5x.. The only critical conjugacy class of A n where the characters 4 q and 4 y differ are the two classes of cycle type hŽ . s Ž h11 , . . . , h r r ., where h ii denotes the length of the ith principal hook of , and r is the order of the Durfee square of . Let ry1
hŽ .qs Ž 1 ⭈⭈⭈ h11 . Ž h11 q 1 ⭈⭈⭈ h11 q h 22 . ⭈⭈⭈
ž
r
Ý h ii q 1 ⭈⭈⭈ Ý h ii is1
is1
/
be an element in the conjugacy class of cycle type hŽ . and of q-type, and choose the labeling of the characters such that
4 " Ž hŽ .q . s 12 w x Ž hŽ . . "
ž
r
(
w x Ž hŽ . . Ł h ii . is1
/
If s ⬘ has a removable node A on the main diagonal, then h r r s 1, and so hŽ _ A. s Ž h11 , . . . , h ry1 ry1 .. The sign choice for the conjugacy classes in A ny 1 of type Ž h11 , . . . , h ry1 ry1 . is made compatibly with the choice above, and the signs for the characters are chosen as before. Then
_ A4 " Ž hŽ _ A.q . s 12 w _ A x Ž hŽ _ A. . "
ž
(
ry1
w x Ž hŽ . . Ł h ii . is1
/
Hence we have the following branching behavior: Ži. If has a removable node A on the main diagonal, then
4 " < A ny 1 s _ A4 " q
Ý _ C4 , C/A
where C only runs through the removable nodes of above the main diagonal. Žii. If has no removable node on the main diagonal, then
4 " < A ny 1 s Ý _ C 4 , C
where C only runs through the removable nodes of above the main diagonal. In particular, the restrictions are again multiplicity-free. In the discussion above, some of the properties become even more obvious when one uses the Frobenius symbol displaying the principal hooks of rather than working in terms of the parts of .
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3. MODULAR BRANCHING OF Sn-REPRESENTATIONS AND RESIDUE SYMBOLS In this section we collect some definitions and results that are needed in the following sections. Let p be a prime; this is needed in the representation theoretic context, for the combinatorial statements p may be an arbitrary odd integer ) 1. We define a Ž p .-signature sequence X as a sequence X : c1 1 c 2 2 ⭈⭈⭈ c s s , where the c i are residues mod p, and the i are signs, for i s 1, . . . , s. For 0 F i F s and 0 F ␣ F p y 1 we define
␣X Ž i . s
Ý
kFi < c ks ␣ 4
k ;
here we make the conventions that an empty sum is 0 and that q is counted as q1 and y as y1 in the sum. The end ¨ alue ␣X of ␣ in X is then defined to be
␣X s ␣ X Ž s . . Furthermore, we define the peak ¨ alue of ␣ in X to be
␣ Ž X . s max ␣ X Ž i . < 0 F i F s 4 . We call c i g X normal of residue ␣ if ␣ X Ž i . ) ␣ X Ž j . for all j F i y 1 and ␣X Ž i . ) 0. This is only possible when c i i s ␣ q. In this case we also call ␣X Ž i . the height ht c i of c i . Moreover, c i g X is called good of residue ␣ Žfor short also ␣-good. if c i is normal of residue ␣ and i is minimal with
␣X Ž i . s ␣ Ž X . . Note that if c i is good of residue ␣ , then ht c i s ␣ Ž X .. Before we associate these definitions with partitions, we recall some further notation and definitions for partitions. A partition s Ž l 1a1 , l 2a 2 , . . . , l ta t . is called p-regular if 1 F a i F p y 1 for i s 1, . . . , t. The p-regular partitions of n label in a canonical way the p-modular irreducible representations D of S n w6x. The p-residue of a node A s Ž i, j . in the Young diagram of is defined to be the residue modulo p of j y i, denoted res A s j y i Žmod p .. The p-residue diagram of is obtained by writing the p-residue of each node of the Young diagram of in the corresponding place.
ALTERNATING GROUPS
149
EXAMPLE. Take p s 5, s Ž6 2 , 5, 4.. Then the 5-residue diagram of is 0 4 3 2
1 0 4 3
2 1 0 4
3 2 1 0
4 3 2
0 4
For a partition , its node sequence N Ž . is a signature sequence defined as follows. Let A1 , . . . , A l be all of the indent and removable nodes of , read successively from top to bottom and from right to left in the Young diagram of . For each i g 1, . . . , r 4 , let c i be the residue of A i and let the sign i be q if A i is removable and y if A i is an indent node. Then the node sequence is defined to be N Ž . : c1 1 c 2 2 ⭈⭈⭈ c s s . A removable node A of is then called normal Žresp., good . if the corresponding entry in the node sequence is normal Žresp., good.; the height ht A is then the height of this entry. The ␣-peak ¨ alue of is the peak value of ␣ in its node sequence. EXAMPLE. Let p s 5 and s Ž12, 7 2 , 5 3 , 3, 13 .. Below, we have only indicated the removable and indent nodes in the 5-residue diagram of .
Then the node sequence of is N Ž . : 2y 1q 1y 4q 2y 4q 2y 1q 4y 1q 0y, where we have underlined the normal entries in the node sequence. Note that then the corresponding removable nodes of are also normal. These combinatorial concepts play a vital role ˆ in the following theorem due to Kleshchev Žsee w8x ᎐ w11x..
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BESSENRODT AND OLSSON
THEOREM 3.1. Let be a p-regular partition of n, n g ⺞, n G 2. Then the following holds: Ži. Žii. are good. Žiii. Then the
socŽ D < S ny 1 . , [A good D _ A. D < S ny 1 is completely reducible if and only if all normal nodes in Let A be a removable node of such that _ A is p-regular. multiplicity of D _ A in D < S ny 1 is given by D < S ny 1 : D _ A s
½
ht A 0
if A is normal in , else.
As a consequence of this theorem one can determine the p-regular partitions of n for which the restriction D < S ny 1 is irreducible. We recall the following definition: DEFINITION 3.2. Let s Ž l 1a1 , . . . , l ta t . be a p-regular partition, where l 1 ) l 2 ) ⭈⭈⭈ ) l t , 0 - a i - p for i s 1, . . . , t. Then is called a JS-partition if its parts satisfy the congruence l i y l iq1 q a i q a iq1 ' 0 mod p
for 1 F i - t.
The type ␣ of is defined by ␣ ' l 1 y a1 mod p. Note that the type of a JS-partition is just the residue of the unique normal Žand thus good. node at the top corner of . The letters JS are an abbreviation of Jantzen and Seitz; these authors conjectured the equivalence of Ži. m Žiii. in the following corollary Žsee w3x, w8x, w9x.. We denote the partition obtained by removing the ith corner node from by Ž i .. COROLLARY 3.3. With notation as abo¨ e, the following are equi¨ alent: Ži. Žii. in .. Žiii.
D < S ny 1 is irreducible Ž and in this case, D < S ny 1 , D Ž1. .. has exactly one normal node Ž which is then the only good node
is a JS-partition.
For later purposes we also have to introduce a different notation for p-regular partitions, which will allow us to have both the removal of good nodes and the p-analogue of conjugation given by the Mullineux map Ždefined in the next section. under control. Let be a p-regular partition of n. The p-rim of is a part of the rim of w6, p. 56x, which is composed of p-segments. Each p-segment, except possibly the last, contains p points. The first p-segment consists of the first p points of the rim of , starting with the longest row. ŽIf the rim
151
ALTERNATING GROUPS
contains at most p points, it is the entire rim.. The next segment is obtained by starting in the row next below the previous p-segment. This process is continued until the final row is reached. We let a1 be the number of nodes in the p-rim of and let r 1 be the number of rows in ; if a1 is a multiple of p, the p-rim is called p-singular. Removing the p-rim of , we get a p-regular partition of n y a1 , and we let a2 , r 2 be the length of the p-rim and the number of parts of this new partition, respectively. Continuing in this way, we get the sequences of numbers a1 , . . . , a m , r 1 , . . . , rm . The residue symbol R p Ž . of is then defined as R p Ž . s
½
x1 y1
x2 y2
⭈⭈⭈ ⭈⭈⭈
xm , ym
5
where x j is the residue of a mq1yj y rmq1yj modulo p, and y j is the residue of 1 y rmq 1yj modulo p. The partition can be recovered from the residue symbol R p Ž . as the unique p-regular partition with this residue symbol Žsee w2x.. A column xy ii in the residue symbol is called singular if yi s x i q 1 and regular otherwise. Note that a residue symbol can never start with the singular column 01 , as this does not correspond to a p-regular partition. Now we define the Mullineux Žsignature. sequence M Ž . via the residue symbol of . With R p Ž . as above, we set M Ž . s 0y
x1 q x2 q xm q
Ž x1 q 1. y Ž x 2 q 1. y
.. . x q Ž m 1. y
y1 q y2 q .. . ym q
Ž y1 y 1. y Ž y 2 y 1. y
.
Ž ym y 1 . y
Starting with the signature 0 y corresponds to starting with an empty partition at the beginning, which just has the indent node Ž1, 1. of residue 0. In w2x the following result was proved. THEOREM 3.4. p y 1 we ha¨ e
Let be a p-regular partition. Then for all ␣ , 0 F ␣ F
␣ Ž M Ž . . s ␣ Ž N Ž . . . As a consequence, we can recognize the normal and good nodes of also in R p Ž .: COROLLARY 3.5. The following statements are equi¨ alent for a p-regular partition Ž0 F ␣ F p y 1.: Ži. Žii.
has a normal Ž good . node of residue ␣ . M Ž . has a normal Ž good . entry of residue ␣ .
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BESSENRODT AND OLSSON
In w2x the following result on the behavior of residue symbols with respect to removing good nodes was proved: THEOREM 3.6. Suppose that the p-regular partition has a good node A of residue ␣ . Let R p Ž . s
½
x1 y1
x2 y2
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Then for some j, 1 F j F m, one of the following occurs: Ž1.
x j is ␣-good for M Ž . and R p Ž _ A. s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
x2 y2
xj y 1 yj
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Ž2. y j is ␣-good for M Ž ., and in this case if j s 1 and ␣ s 0, then x 1 s 0 and R p Ž _ A. s
½
x2 y2
⭈⭈⭈ ⭈⭈⭈
xm , ym
5
i.e., the first column is remo¨ ed, or else if Ž j, ␣ . / Ž1, 0., then R p Ž _ A. s
½
x1
x2
⭈⭈⭈
xj
⭈⭈⭈
xm
y1
y2
⭈⭈⭈
yj q 1
⭈⭈⭈
ym
5
.
4. MODULAR IRREDUCIBLE A n-REPRESENTATIONS For a classification of the modular irreducible A n-representations, we need to know which modular irreducible Sn-representations split when restricted to A n . For p s 2, the answer was given by Benson w1x: THEOREM 4.1. Let s Ž 1 , . . . , m . & n be a 2-regular partition of n g ⺞, n G 2, D the corresponding modular irreducible Sn-representation. Then the restriction D < A n is reducible if and only if the parts of satisfy the following two conditions Ž where we set mq 1 s 0 if m is odd .: Ži. Žii.
2 jy1 y 2 j F 2 for all j. 2 jy1 q 2 j k 2 Žmod 4. for all j.
If D < A n is reducible, then it splits into two nonisomorphic irreducible sumq y mands, say D < A n , C [ C .
153
ALTERNATING GROUPS
We call the 2-regular partitions satisfying the conditions Ži. and Žii. above S-partitions. From Theorem 4.1 we can now deduce the classification of the 2-modular irreducible A n-representations: A complete list of 2-modular irreducible A n-representa-
COROLLARY 4.2. tions is gi¨ en by q
y
C , C C
for & n a 2-regular S-partition, for & n a 2-regular non-S-partition.
For p / 2, the split restrictions are those of modules fixed by tensoring with the sign representation. This case is determined by the Mullineux map, which we describe on the residue symbols: DEFINITION 4.3. Let the residue symbol of the p-regular partition be R p Ž . s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Then the Mullineux conjugate M is the p-regular partition whose residue symbol is R p Ž M . s
½
j s
½
1 y y1 1 y x1
⭈⭈⭈ ⭈⭈⭈
m y ym m y x m ,
5
where 1
if x j q 1 s y j ,
0
otherwise.
As a consequence of Theorem 3.6, we state the following property of good nodes with respect to the Mullineux map: THEOREM 4.4 w2x, w5x. Let be a p-regular partition, and let A be a good node of . Then there exists a good node B of the Mullineux image M such that Ž _ A. M s M _ B. In 1979, Mullineux conjectured that the map M defined above describes the result of tensoring an irreducible representation by the sign representation. After a long struggle this conjecture was finally solved by the work of Kleshchev and Ford and Kleshchev. As an application of his Branching Theorems for modular Sn-representations, Kleshchev had reduced the Mullineux Conjecture to the purely combinatorial statement in Theorem 4.4 on the removal of good nodes from a partition, which was then proved by Ford and Kleshchev.
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BESSENRODT AND OLSSON
THEOREM 4.5.
Let be a p-regular partition. Then D m sgn , D . M
Based on this and Clifford theory, we have a combinatorial description for the splitting of the modular irreducible Sn-representations over A n also for odd primes p Žsee w4x.: THEOREM 4.6. Let p be an odd prime, and let s Ž 1 , . . . , m . & n be a p-regular partition of n, D the corresponding modular irreducible Sn-representation. Then the restriction D < A n is reducible if and only if is a fixed point under the Mullineux map, i.e., M s . If D < A n is reducible, then it q y splits into two nonisomorphic irreducible summands C , C , so D < A n , q y C [C . As before, this implies the classification of the p-modular irreducible A n-representations w4x: COROLLARY 4.7. A complete list of p-modular irreducible A n-representations is gi¨ en by the modules: q
y
C , C
C s C
& n p-regular, s M M
& n p-regular, / M .
5. BRANCHING AND p-CONJUGATION AT ODD p In this section, we always assume that p ) 2; otherwise p is an arbitrary odd integer in the combinatorial statements and has to be a prime in the representation theoretic statements. Recall the phenomena occurring ‘‘at characteristic 0’’ Ždescribed in Section 2. to imagine what to expect now. In the following, we denote by F a field of characteristic p, again assumed to be sufficiently large. First a lemma: LEMMA 5.1.
Let M be an FS n-module. Then soc Ž M < A n . s soc Ž M . < A n .
Proof. Since A n is normal in Sn and of index prime to p, the Jacobson radicals of FS n and FA n are related by J Ž FS n . s J Ž FA n . FS n , due to a result by Villamayor w12, p. 524x. Since the socle of a module is the set of elements annihilated by the Jacobson radical w15, Theorem 1.8.18x, the assertion follows. For convenience, we let C denote the module C s C if is not a o q y Mullineux fixed point, and C g C , C 4 if is a Mullineux fixed point. o
o
155
ALTERNATING GROUPS
PROPOSITION 5.2. D < S ny 1 is completely reducible if and only if C < A ny 1 is completely reducible. o
Proof. This follows immediately from the preceding Lemma and Theorem 4.6. In the proof of the following result we require the content vector of a partition , which is defined to be cŽ . s Ž c 0 , . . . , c py1 ., where c i is the number of nodes in the p-residue diagram of , which are of residue i. The basic fact we need is that with cŽ . as above, the content vector of the Mullineux conjugate is cŽ M . s Ž d 0 , . . . , d py1 ., where d␣ s cy ␣ Žsee w2x, w14x.. THEOREM 5.3 ŽThe almost-p-symmetric case.. Let be p-regular, and assume / M. Furthermore, assume that has a good node A with _ A s Ž _ A. M . Ži. Then all of the columns in the residue symbol of are Mullineuxfixed, except for one column,
¡
x 1yx x y s or x y1 y x
~
¢
with x / 0,
with x / y1,
where the marked entry corresponds to the good node A in the sense of Theorem 3.6. In particular, res A / 0. Žii. There are no normal nodes B and B˜ with B / A / B˜ such that _ B and _ B˜ are p-regular and satisfy M
_ B s Ž _ B˜. . In particular, there are no good nodes B and B˜ different from A for which this holds. Proof. Ži. Theorem 3.6 immediately tells us that all of the columns in the residue symbol of are Mullineux-fixed, except for the one column with the good entry corresponding to the good node A. As itself is not Mullineux-fixed, this exceptional column cannot be a 0-column at the start of the residue symbol, since this would vanish on removing A.
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BESSENRODT AND OLSSON
So let us assume that xy is the exceptional column, and consider first the case where the upper entry x is the good entry corresponding to A. Then the corresponding column x yy 1 in R p Ž _ A. has to be Mullineux-fixed. If this column is regular, then we have y s 1 y x and x / 0, as claimed. If we assume it is singular, then y s x, contradicting the assumption that the upper entry x in R p Ž . was good. Similarly, if the lower entry y in the exceptional column is the good entry corresponding to A, then in the case of the corresponding column y qx 1 for _ A being regular, we obtain y s yx y 1, and we must have x / y1, since otherwise in the Žsingular. column xy s y1 the lower entry is not good. If we assume y qx 1 to be singular, 0 then y s x and the fixed point condition gives x s yx; hence in this case the exceptional column is 00 and hence is Mullineux-fixed, contradicting our assumption. The conditions we have derived above immediately imply that res A / 0. Žii. Let cŽ . s Ž c 0 , c1 , . . . , c py1 . be the content vector of , and let r s res A, so r / 0 by Ži.. Then the condition _ A s Ž _ A. M implies c r y 1 s c pyr and c i s c pyi for all residues i / r, p y r. Now let B, B˜ be nodes of , different from A, with residues i and j, respectively, such that their removal gives p-regular partitions with _ B s Ž _ B˜. M . In the first part of the proof we show that i s r s j; here we do not need the condition that these nodes are normal Žand that A is good.. Since the Mullineux map leaves the content of residue 0 invariant, either none or both of B and B˜ are of residue 0. If i s 0 s j, then c k s c pyk for all residues k / 0, a contradiction, since r / 0. Hence i / 0 / j. We assume first that i / r / j. If j / p y i, then comparing the content at residue i and using the relation above gives c i y 1 s c pyi s c i , a contradiction. If j s p y i, then we obtain c k s c pyk for all residues k, again a contradiction. Hence we only have to consider the case where B / A / B˜ are normal nodes of with residues i s r s j. As these nodes are normal but not good, the r-peak value goes down by 2 on removing these nodes Žthe q1-contribution before the good node is replaced by a y1-contribution.; hence
r Ž _ B˜. s r Ž . y 2 s r Ž Ž _ B .
M
.
s yr Ž _ B . s yr Ž . q Ž B . ,
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ALTERNATING GROUPS
where Ž B . g 0, 14 ; the last equation follows as res B s r / yr Ž r / 0., and hence the effect of removing B can at most be an increase of 1 in the peak values for the residues r " 1. On the other hand, removing a good node only decreases the peak value for the corresponding residue by 1, and the effect on other residues is similar to that above, so
r Ž . y 1 s r Ž _ A . s yr Ž Ž _ A .
M
.
s yr Ž _ A . s yr Ž . q Ž A . , where Ž A. g 0, 14 Žnote that for the last equation we have used again that r / 0 and p is odd, so that r / yr .. Hence we conclude
r Ž . y yr Ž . y 1 s Ž B . q 1 s Ž A . ; thus we deduce Ž B . s 0 and Ž A. s 1. Now by removing a removable node of residue r, we can either create new removable nodes of residue r " 1 or indent nodes of such residues can vanish; the overall effect is in all cases to lift the paths for these residues by 1 at this position. If A and B both are higher Žresp., both are lower. than the good node of such residues, then the corresponding peak value in _ A and _ B is increased in both cases by 1 Žresp., is in both cases unchanged. compared to the peak value in . Since B is normal of the same residue as the good node A, it is higher in than A; hence we also might have an increase in the critical peak value by 1 in _ B and an unchanged peak value when removing A Žbut not the other way around!.. Hence the situation Ž B . s 0 and Ž A. s 1 is impossible, and thus the claim is proved. The theorem above allows us to deduce information on the restrictions $ of the A n-representations; as in characteristic 0, we denote by C the q y M module C [ C if s M Žresp., the module C s C if / M .. THEOREM 5.4. Let be p-regular with / M , and assume that has a good node A with _ A s Ž _ A. M . Then Ži.
$
soc Ž C < A ny 1 . , C _ A [
[C
_ B
,
B/A good
and this socle is multiplicity-free. Žii. Let B be a remo¨ able node of such that _ B is p-regular. Then o the multiplicity of C _ B in C < A ny 1 is gi¨ en by C < A ny 1 : C _ B
o
s
½
if B s A,
ht A ht B q D < S ny 1 : D Ž _ B .
M
else.
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BESSENRODT AND OLSSON
Proof. This follows from the Branching Theorem for modular Sn-representations, the classification of the modular irreducible A n-representations, and part Žii. of Theorem 5.3. Remark. Note that in the case of B / A in part Žii. above, we do not know how to compute the second summand. THEOREM 5.5 ŽThe far-from-p-symmetric case.. Let be p-regular with / M , and assume that has no good node A with _ A s Ž _ A. M . Then there are no good nodes B and B˜ with M
_ B s Ž _ B˜. . Proof. Assume there are good nodes B and B˜ with
Ž ).
M
_ B s Ž _ B˜. .
Then these nodes are different from A, and their residues r s res B, s s res B˜ are different, and both are different from 0. We consider again the p-content vector cŽ . s Ž c 0 , c1 , . . . , c py1 . of , and then compare the content vectors of _ B and Ž _ B˜. M . For s / p y r, this gives the equations c py r s c r y 1, c r s c y p y r and hence a contradiction. Thus we must have s s p y r, i.e., B and B˜ are good nodes with conjugate residues. Next we consider the residue symbols of _ B and Ž _ B˜. M . Let the residue symbol of be R p Ž . s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
xm . ym
5
Since B and B˜ are good Žand not of residue 0., their removal from only has an effect on an entry r Žresp., p y r . in the residue symbol at the ith and jth columns Žsay.. Let us first consider the case where the ith column in is r , y
ALTERNATING GROUPS
159
with the marked entry corresponding to the good node B. If i / j, then Ž). implies that the ith column in R p Ž _ B . satisfies
ry1 y s
¡ yy
if y / 1 q r ,
¢
if y s 1 q r .
~
yr or yr 1yr
In both cases we immediately obtain a contradiction. If i s j, then the ith column in is r yr
,
˜ hence with the second marked entry corresponding to the good node B; this column is Mullineux-fixed. But by Ž)., all other columns in the residue symbol are also Mullineux-fixed, and hence so is , a contradiction. Next we first consider the case where the ith column in is x , r with the marked entry corresponding again to the good node B; note that this column then has to be regular. If i / j, then Ž). implies now that the ith column in R p Ž _ B . satisfies x s yr , yx rq1 immediately giving a contradiction. If i s j, then the ith column in is now yr r
;
hence again this column is Mullineux-fixed. But as before, Ž). implies that all other columns in the residue symbol are also Mullineux-fixed; hence so is , again a contradiction.
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BESSENRODT AND OLSSON
From this we deduce the following. THEOREM 5.6. Let be p-regular with / M , and assume that has no good node A with _ A s Ž _ A. M . Then Ži. soc Ž C < A ny 1 . ,
[C
_ B
,
B good
and this socle is multiplicity-free. Žii. Let B be a remo¨ able node of such that _ B is p-regular. Then the multiplicity of C _ B in C < A ny 1 is gi¨ en by C < A ny 1 : C _ B s ht B q D < S ny 1 : D Ž _ B . THEOREM 5.7 ŽThe p-symmetric case..
M
.
Let be p-regular with s M .
Ži. If N is a remo¨ able node of with _ N s Ž _ N . M , then res N s 0. ˜ Žii. For any good node B there is a good node B˜ with res B s yres B, and M
_ B s Ž _ B˜. . In particular, a good node A in of residue 0 satisfies _ A s Ž _ A. M . Žiii. If has only one good node, then this is of residue 0. Proof. Ži. We consider the content vector cŽ . s Ž c 0 , . . . , c py1 .. As s M , we have c i s c pyi for all i. Let N be a removable node of residue r / 0; if _ N s Ž _ N . M , then this implies c r y 1 s c pyr , and hence a contradiction. Žii. If B is a good node of of residue r, then we have M Ž _ B . s M _ B⬘ s _ B⬘,
where B⬘ is a good node of of conjugate residue p y r. If B is good of residue r s 0, then we must have B⬘ s B. Žiii. follows immediately from Žii.. For the branching of the representations this implies THEOREM 5.8.
Let be p-regular with s M. Then we ha¨ e
Ži. The restrictions of the two conjugate modules C q and C y ha¨ e conjugate socles "
Ž C _ A 0 . [
[
B good res Bg 1, . . . , Ž py1 .r2 4
C _ B ,
161
ALTERNATING GROUPS
where the first summand only occurs if has a good node A 0 of residue 0. In particular, these restrictions are multiplicity-free. Žii. Let B be a normal node of such that _ B is p-regular. Then $
C < A ny 1 : C _ B
o
s
½
2 ht B ht B
if Ž _ B .
M
/ _ B,
if Ž _ B .
M
s _ B.
As has been described in Section 3, we know by Kleshchev’s Branching Theorems which irreducible Sn-modules split on restriction to Sny1. We are now going to use this together with the description of the residue symbols of JS-partitions given in w3x for dealing with the analogue of the Jantzen᎐Seitz question for the alternating groups. We first recall the description of JS-partitions that are fixed under the Mullineux map. THEOREM 5.9. Let be a p-regular partition. Then is a JS-partition and a Mullineux fixed point if and only if its residue symbol, R p Ž . s
x1 y1
½
⭈⭈⭈ ⭈⭈⭈
xk , yk
5
can be constructed iterati¨ ely by the following procedure: x1 0 s . y1 0
½ 5 ½5
If the first k y 1 columns of the residue symbol are already constructed, then we ha¨ e two possibilities for a regular extension, namely xk s yk
½ 5 ½
1 y y ky 1 y ky1 y 1
5
or
½
y ky1 y 1 , 1 y y ky1
5
and if y ky 1 s 1, then we ha¨ e, furthermore, one possibility for a singular extension, namely, xk 0 s . yk 1
½ 5 ½5
The construction rule given in the theorem can be described alternatively by a diagram in the following way. The JS-partitions that are Mullineux fixed points are obtained by adding on columns to the residue symbol according to a walk starting at 0 0
½5
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BESSENRODT AND OLSSON
in the diagram Ž). below Žwe omit the brackets for simplification.; the symbol ; in the diagram means that we have a loop at the corresponding node of the graph.
Since the starting column 0 0
½5 already guarantees a normal node of residue 0, it is clear that the unique good node of such JS partitions is of residue 0, i.e., that they are of type 0. Furthermore, Mullineux fixed JS-partitions have empty or square p-cores, and the p-core only depends on the end of the walk in the diagram above Žsee w3x.. Thus we can give the p-cores of Mullineux fixed JS-partitions by putting the corresponding p-core at the end point of the walk in the diagram:
THEOREM 5.10. Let n G 2, let p be an odd prime, and let be a p-regular partition of n. Then the following are equi¨ alent: Ži. C < A ny 1 is irreducible. Žii. One of the following holds: Ža. D < S ny 1 is irreducible. M Žb. D < S ny 1 , D Ž1. [ D Ž1. and M s but Ž1. M / Ž1.. Žiii. One of the following holds: Ža. is a JS-partition. Žb. The residue symbol R p Ž . is obtained by adding on columns along a walk in the extended diagram Ž)). below, starting at any regular column yxx , with x / 0. o
163
ALTERNATING GROUPS
One of the following holds: Ža. is a JS-partition that is not Mullineux fixed. Žb. R p Ž . is obtained by adding on columns along a walk in the diagram Ž). abo¨ e starting at 00 , or along a walk in the extended diagram Ž)). below, starting at any regular column yxx , with x / 0. Žiii⬘.
Proof. First we assume Ži.. If M / , then D < A ny 1 , C < A ny 1 is irreducible, so this implies immediately that D < S ny 1 is irreducible. q y So we now consider a fixed point M s . Note that C and C are conjugate representations, so by assumption the conjugate modules q y C < A ny 1 and C < A ny 1 are also both irreducible. If D < S ny 1 is irreducible, there is nothing to prove. In the other case we deduce by Proposition 5.2 and Kleshchev’s Branching Theorem: D < S ny 1 , D _ A [ D _ B , where A and B are the two good nodes of . Moreover, since s M , we must have Ž _ A. M s _ B. Since we always have a normal node at the first corner of a partition, giving a composition factor of the restriction in the block corresponding to the removal of the good node of the same residue, the first removable node itself must be good. Hence we obtain D < S ny 1 , D Ž1. [ D Ž1. , M
with Ž1. / Ž1. M . Thus Ži. « Žii. is proved. For Žii. « Žiii. we have to show that a non-JS-partition with s M , Ž1. / Ž1. M , and D < S ny 1 , D Ž1. [ D Ž1.
M
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BESSENRODT AND OLSSON
can be constructed along the diagram Ž)). above. Note that Ž1. M s Ž g . for some g / 1, corresponding to removing a good node at the g th corner of . Consider the residue symbol R p Ž . s
½
x1 y1
⭈⭈⭈ ⭈⭈⭈
xk . yk
5
The condition s M amounts to the equations x i s yyi
for regular columns Ž note that x i / Ž p y 1 . r2 . .
x i s 0, yi s 1
for singular columns Ž here we must have i / 1 . .
We know from the decomposition of D < S ny 1 that has only the good nodes _ Ž1. and _ Ž g ., and no further normal nodes w11x. Note also that the two good nodes must have conjugate residues r / yr, say. Now consider the Mullineux sequence for : M Ž . s 0y
x1 q x2 q .. . xkq
Ž x1 q 1. y Ž x 2 q 1. y
Ž yx 1 . q
Žy Ž x1 q 1. . y
y2 q
Ž y 2 q 1. y
Ž x k q 1. y
yk q
Ž yk q 1. y
.
If x 1 s 0, then y 1 q s Žyx 1 . q gives a normal node of residue 0 Žhence the good one of residue 0., contradicting the above properties of our two good nodes. Hence x 1 / 0 is the first good node. Since x 1 / Ž p y 1.r2, we have yx 1 / x 1 q 1, and hence yx 1 s y 1 is the next good node. Since we then have no further normal node, we must have x 2 s x 1 q 1 or yŽ x 1 q 1. or 0. If x 2 s 0, then y 2 s 0 is not possible, but only y 2 s 1, i.e., a singular column, and then the situation for the next column in the residue symbol is as before. Thus we have a certain number of singular columns 01 4 , and then the next regular column xy ii 4 has to satisfy x i 2 s x 1 q 1 or yŽ x 1 q 1., yi 2 s yx i 2 . Then again, we may have a number of singular columns, and then the next regular column xy ii 4 satisfies x i 3 s x i 2 q 1 or yŽ x i 2 q 1., yi 3 s yx i 3 . This repeats, so we obtain indeed a walk in the extended diagram Ž))., starting at a regular column different from 00 . Now to Žiii. « Ži.. 2
2
3
3
Case 1. Assume is a JS-partition with / M . If Ž1. / Ž1. M , then clearly, D < A ny 1 , D Ž1. < A ny 1 , C < A ny 1 is irreducible. So we may assume now that Ž1. s Ž1. M .
165
ALTERNATING GROUPS
Consider the residue symbol R p Ž . s
½
⭈⭈⭈ ⭈⭈⭈
x1 y1
xk . yk
5
By Theorem 3.6 we know that the residue symbol R p Ž Ž1.. of Ž1. can only be one of
½
x1 y 1 y1
⭈⭈⭈ ⭈⭈⭈
xk yk
5 ½
or
½
or ⭈⭈⭈ ⭈⭈⭈
x2 y2
⭈⭈⭈ ⭈⭈⭈
x1 y1 q 1
xk yk
5
xk , yk
5
where
½
x1 y 1 y1
M
5 ½ s
x1 y 1 y1
5
resp.,
x1 y1 q 1
½
M
5 ½ s
x1 y1 q 1
5
resp., x 1 s y 1 s 0, and furthermore, M
x1 y1
x1 , y1
½ 5 ½ 5 /
and all other columns are Mullineux-fixed and hence of the form
½ yxx 5
or
0 . 1
½5
By these conditions, the second and third possibilities for the residue symbol of Ž1. are immediately excluded; in particular, this implies that the type ␣ of satisfies ␣ / 0. So we have the following possibilities now for the first column of R p Ž .: x1 g y1
½ 5 ½½
␣ , ␣ / 0; ␣q1
5
½␣5 0
, ␣ / 0, p y 1;
½ ␣0 5 , ␣ / 0, 1 5 .
Then R p Ž Ž1.. starts with one of
½ ␣␣ 5 ½ ␣ 5 ½ ␣ 5 y1 , q1
y1 , 0
0 , q1
respectively; but the first one is not Mullineux-fixed, the second is only for ␣ s 1, and the third is only for ␣ s p y 1; hence R p Ž . has to start with 10 4 and ␣ s 1, or with p y0 1 4 and ␣ s p y 1.
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BESSENRODT AND OLSSON
The restrictions for the further columns do not allow any regular extensions from these first columns. But also, the construction rules for singular columns do not give Mullineux fixed columns as required. Hence we must have R p Ž . s
1 0
½5
or
½
0 , py1
5
which corresponds to s Ž2. Žresp., s Ž12 .., and then Ži. obviously holds. Case 2. We assume now that is a JS-partition with s M. Then is of type 0, and it is immediately seen that also Ž1. s Ž1. M , and then Ži. holds. Case 3. Assume now that is not a JS-partition, so is constructed along a path in diagram Ž))., in particular, s M . Reversing the arguments for Žii. « Žiii., we have D < S ny 1 , D _ A [ D _ B , where A, B are good nodes of of conjugate residues / 0, and Ž _ A. M o s _ B. Then C < A ny 1 , D _ N < A ny 1 , N g A, B4 , and hence this restriction is irreducible. The equivalence of Žiii. and Žiii⬘. is clear. From the proof of the theorem, one easily deduces the following. COROLLARY 5.11. We assume the notation of the theorem. Then if C < A ny 1 o o is irreducible, we ha¨ e C < A ny 1 , C Ž1. . o
Remarks. Ži. In part Žiii. of the theorem above, it would be nice to have a description in terms of the parts of like the one available for JS-partitions. Žii. We have already observed that JS-partitions have special p-cores w3x; moreover, the ‘‘almost JS-partitions’’ constructed along a path in Ž)). Žresp., also those constructed in a more general fashion in the construction diagram for JS-partitions w3x. have special cores that depend in an easy way on the start and the end of the path in the diagram. 6. BRANCHING AT p s 2 As we have already seen in Section 4, the behavior of the A n-representations at characteristic 2 is very much different from that at odd primes. While in some respects the combinatorics of S-partitions is easier than
ALTERNATING GROUPS
167
that of Mullineux fixed points, the 2-modular representation theory presents more difficulties, since A n is a normal subgroup of index p s 2 in Sn . First we give a description of S-partitions Žsee Theorem 4.1. in terms of residue symbols as an analogue of the description of Mullineux fixed points via residue symbols. PROPOSITION 6.1. The 2-residue symbols of S-partitions are constructed by a walk in the following directed graph, where we start at one of the marked columns and we may loop around any of the three ¨ ertices in the graph:
In other words, the residue symbols start with a regular column, and they ha¨ e no column 10 . Proof. As long as the smallest part of is at least 2, the 2-rim consists of horizontal dominoes and thus is 2-singular. In removing such singular 2-rims from a given S-partition , we again obtain S-partitions. Doing this as long as possible, we reach an S-partition, which we may assume to be of even length by adding a part 0 if necessary. This S-partition then ends on one of the following pairs of parts: 1, 0 2, 1 3, 1. Then the next step in removing the 2-rim leads to the following corresponding columns in the residue symbol: 0 0
1 1
1 , 1
and the partition obtained after the removal is again an S-partition. So S-partitions can be built up along residue symbols. As we can never start a residue symbol with the singular column 01 , and since 10 is not an S-partition, the start has to be at one of the two marked columns. Using the list of end pairs given above, it is then easily seen that at each step all three columns in the diagram above give possible extensions to an S-partition.
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BESSENRODT AND OLSSON
COROLLARY 6.2. Let be an S-partition with residue symbol R 2 Ž .. Let m 0 be the number of columns 00 in R 2 Ž . Ž resp., let m1 be the number of columns 11 .. Then the 2-core of is gi¨ en by
Ž2.
¡Ž2Ž m y m . y 1, 2Ž m y m . y 2, . . . , 2, 1. y m . , 2 Ž m y m . y 1, . . . , 2, 1 . ¢⭋
s~ Ž 2 Ž m
0
1
1
0
0
1
if m 0 y m1 ) 0,
1
if m1 y m 0 ) 0, if m 0 s m1 .
0
Proof. Note that the p-content c s Ž c 0 , c1 , . . . , c py1 . of a partition determines its p-core. The associated ª n-vector is given by ª n s Ž c 0 y c1 , . w x c1 y c 2 , . . . , c py2 y c py1 , c py1 y c0 . By 2 we can easily compute the ª nvector of via its residue symbol. Hence we obtain the 2-core of the partition given via its residue symbol R 2 Ž . as above; we omit the details. Let be a 2-regular partition.
LEMMA 6.3.
Ža. If is an S-partition and A is a remo¨ able node of such that _ A is 2-regular, then we ha¨ e
_ A is an S-partition if and only if res A s 0. Žb. If is not an S-partition, then there is at most one node A in such that _ A is an S-partition, and such a node is of residue 1. Proof. By definition, the parts of S-partitions come in one of the following pairs Žpictured in the 2-residue diagram.: ⭈⭈⭈ ⭈⭈⭈
⭈⭈⭈ ⭈⭈⭈
0 0
⭈⭈⭈ ⭈⭈⭈
1 1
0 1
From this, Ža. and Žb. easily follow. Analogous to the notation at odd characteristic, we let C s C if is o q y not an S-partition and C g C , C 4 if is an S-partition. For the A n-representations at characteristic 2, we can now deduce the following: o
THEOREM 6.4.
Let be a 2-regular partition of n. "
Ža. Assume is an S-partition of n. Then the module socŽ C < A ny 1 . " has a constituent C Ž _ A 0 . if A 0 is a good node of residue 0, and it has a _ A 1 constituent C if A1 is a good node of residue 1. Furthermore, if B is a remo¨ able node with _ B 2-regular, then $
C < A ny 1 : C _ B
o
s
½
ht B 0
if B is normal, else,
and ht B is e¨ en for all normal nodes B of residue 1 such that _ B is 2-regular.
ALTERNATING GROUPS
169
Žb. Assume is not an S-partition. Then the module socŽ C < A ny 1 . has a constituent C _ A 0 if A 0 is a good node of residue 0, and it has a constituent C _ A 1 if A1 is a $ good node of residue 1 with _ A1 not an S-partition Ž resp., a constituent C _A 1 if A1 is a good node of residue 1 with _ A1 an S-partition.. The latter occurs only if is almost an S-partition except for one pair of parts Ž2 k, 2 k y 2., with A1 at the end of the first of these two parts. Furthermore, if B is a remo¨ able node with _ B 2-regular, then C < A ny 1 : C _ B
o
s
½
ht B 0
if B is normal, else.
"
The case C _ B s C _ B arises only if is almost an S-partition except for one pair of parts of the form Ž2 k, 2 k y 2. or Ž2 k, 2 k y 3., with B at the end of the part 2 k normal of residue 1. o
For p s 2, the characterization of the irreducible restrictions is not as complete as for odd primes p, but there are at least very strong combinatorial restrictions. First we compare the A n-situation with the Sn-situation: THEOREM 6.5. equi¨ alent:
Let be a 2-regular partition. Then the following are
Ži. C < A ny 1 is irreducible. Žii. One of the following holds: Ža. D < S ny 1 is irreducible and / Ž2 l, 2 l y 2. if n ' 2 Žmod 4.. o
Ž2.
Žb. D < S ny 1 , D Ž where this denotes a uniserial module with D Ž2. Ž2. . Ž top and socle D , and s 1 , . . . , k . is an S-partition with 1 e¨ en. Proof. First we assume Ži.. Case 1. Suppose is not an S-partition; hence D < A ny 1 , C < A ny 1 is irreducible, and so D < S ny 1 , D Ž1. must be irreducible, i.e., is a JS-partition. Furthermore, as D Ž1. < A ny 1 is irreducible, also Ž1. is not an S-partition. Thus is not of the form Ž2 l, 2 l y 2. for some l ) 1, as was to be proved in this case. Case 2. Suppose is an S-partition; hence q
y
D < An , C [ C , q
y
with C ` C . Now q
y
D < A ny 1 , C < A ny 1 [ C < A ny 1 ,
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a direct sum of two conjugate irreducibles; hence we have one of the following two possibilities: Ža. Žb.
is a JS-partition and Ž1. is an S-partition. D < S ny 1 ; 2 ⭈ D , where is not an S-partition.
Case Ža.. As s Ž 1 , . . . , k . is a JS-partition, we have 1 ' 2 ' ⭈⭈⭈ ' k Žmod 2.. Since is also an S-partition, we have
2 jy1 y 2 j s 2
and
1 ' 2 ' ⭈⭈⭈ ' k ' 1 Ž mod 2 . .
But then Ž1. is always an S-partition, so this condition can be omitted in Ža.. Note also that in this case again, s Ž2 l, 2 l y 2. does not occur. Case Žb.. Since the socle of D < S ny 1 is multiplicity-free, the module D S ny 1 cannot be completely reducible. Hence we must have <
soc Ž D < S ny 1 . , D Ž i. , where s Ž i . s _ A, A being the only good node of . Furthermore, the Branching Theorem tells us that there is no normal node of residue / res A, and there is exactly one other normal node B of residue res A; moreover, for this normal node B, the partition _ B is not p-regular, and we have D < S ny 1 ,
D Ž i. . D Ž i.
Since the first removable node of is always normal, this must be the normal node B. Note also that Ž i . is not an S-partition, since otherwise we have a contradiction to condition Ži. of the theorem. Hence we have 1 y 2 s 1, so resŽ _ Ž1.. s resŽ _ Ž2.., and thus _ Ž2. is also normal, and hence it has to be the good node A. Thus we must have i s 2, i.e., D < S ny 1 ,
D Ž2. , D Ž2.
as claimed. Furthermore, as Ž2. is not an S-partition, the S-partition must start with an even part. Now assume that Žii. is satisfied. Case 1. First suppose that D < S ny 1 is irreducible, so is a JS-partition, but is not of the form Ž2 l, 2 l y 2. for some l ) 1. If is an S-partition, then Ž1. is an S-partition by the same argument as in Case Ža. above.
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ALTERNATING GROUPS
Hence
Ž D < S . A ny 1
q
ny 1
y
, D < A ny 1 , C < A ny 1 [ C < A ny 1 q
y
, D Ž1. < A ny 1 , C Ž1. [ C Ž1. , q
y
and so both C < A ny 1 and C < A ny 1 are irreducible. If s Ž 1 , . . . , k . is not an S-partition, then Ž1. is also not an S-partition. To see this, assume that Ž1. s Ž 1 y 1, . . . , k . is an S-partition. As is a JS-partition, we have 1 ' 2 ' ⭈⭈⭈ ' k Žmod 2., and hence 1 q 2 ' 2 Žmod 4., 2 jy1 q 2 j k 2 Žmod 4. for j / 1, and 2 jy1 y 2 j s 2 for all j. Hence we must have j ' 0 Žmod 2. for all j, but then 2 jy1 q 2 j ' 2 Žmod 4. for all j, and this implies that has only two parts, i.e., s Ž2 l, 2 l y 2.. But this case was excluded in condition Žii. of the theorem. Having shown all these properties of and Ž1., it is now clear that condition Ži. of the theorem is satisfied. Case 2. We now consider the situation where is an S-partition with even first part 1 and D < S ny 1 ,
D Ž2. . D Ž2.
Here s Ž 1 , . . . , k . must satisfy 1 y 2 s 1, 1 even, and then Ž2. s Ž 1 , 1 y 2, . . . . is not an S-partition. Hence D < A ny 1,
D Ž2. D Ž2.
; 2 ⭈ C Ž2. , A ny 1
and thus q
y
C < A ny 1 , C < A ny 1 , C Ž2. is irreducible, as was to be proved. Remark. Note that the proof of the theorem tells us exactly what the o restriction C < A ny 1 is in the cases when it is irreducible. For a partition s Ž 1 , . . . , m . of n into distinct parts, we define its doubling to be the partition dbl Ž . s
ž
1 q 1 2
,
1 2
,
2 q 1 2
,
2 2
,...,
m q 1 2
,
m 2
/
.
Via the process of regularization Žsee w6x., we then obtain a 2-regular partition dbl 2 Ž . [ dblŽ . R.
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PROPOSITION 6.6. Let p s 2, and let be a 2-regular partition satisfying one Ž and hence both. of the conditions in the theorem abo¨ e. Then is a JS-partition / Ž2 l, 2 l y 2., or it is of the form s dbl 2 Ž ., where is a partition of n into distinct parts satisfying i y iq1 ) 4 for i s 1, . . . , m y 1, and the residues of 1 , 2 , . . . modulo 4 follow a path in the diagram below, starting at 3:
Proof. If is not a JS-partition, we have already seen in the proof of the theorem that has the following properties: Ži. is an S-partition. Žii. 1 even. Žiii. 1 y 2 s 1. Živ. The first two removable nodes are the only normal nodes of . Property Ži. immediately implies that s dbl 2 Ž . for some 2-regular partition s Ž 1 , 2 , . . . . with i y iq1 G 4 and i y iq1 ) 4 if i ' 0 Žmod 4., and i k 2 Žmod 4. for all i. By properties Žii. and Žiii., we have 1 ' 3 Žmod 4.. Consider the sequence r 1 , r 2 , . . . of end residues of the rows of in its 2-residue diagram. By properties Žii. and Živ. we have 0 F j F i < r j s 1 4 y j F i < r j s 0 4 F 2. Since a part i ' 0, 1 or 3 Žmod 4. leads to the residue pairs Ž0, 1., Ž0, 0. resp., Ž1, 1., the condition above easily translates into the admissible walks in the triangle graph above, and it is clear that then i y iq1 ) 4 must hold for all i. Remarks. Ža. There are examples of the phenomenon occurring in part Žii.Žb. of Theorem 6.5 above. For all l we have D Ž2 l , 2 ly1. < S 2 ly 2 , This follows by using, e.g., w1x.
D Ž2 l , 2 ly2. . D Ž2 l , 2 ly2.
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ALTERNATING GROUPS
There are also such examples with partitions with more than two parts: D Ž6 , 5, 1. < S11 ,
D Ž6, 4, 1. D Ž6, 4, 1.
D Ž6 , 5, 3, 1. < S14 ,
D Ž6, 4, 3, 1. . D Ž6 , 4, 3, 1.
Žb. It is not clear whether the combinatorial condition in the proposition characterizes the partitions giving irreducible A n-restrictions, thus providing an analogue of Žiii. in Theorem 5.10. Žc. By the description of the partitions given in the proposition above, it is clear that we can have the irreducible restrictions of modular A n-representations of the second type only if n ' 0 or 3 Žmod 4., and the corresponding irreducible representations belong to the principal 2-block resp., to the 2-block with 2-core Ž2, 1..
ACKNOWLEDGMENTS We thank the Mathematical Sciences Research Institute for its hospitality during a stay of the authors at the time of the ‘‘Combinatorics Year’’; the research at MSRI was supported in part by National Science Foundation grant DMS 9022140, as well as by the Deutsche Forschungsgemeinschaft Žgrant Be 923r6-1 of the first author.. The authors gratefully acknowledge the support by these institutions and the support by the Danish Natural Science Foundation.
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10. A. Kleshchev, Branching rules for modular representations of symmetric groups III, J. London Math. Soc. 54 Ž1996., 25᎐38. 11. A. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. Ž 3 . 75 Ž1997., 497᎐558. 12. G. O. Michler, Blocks and centers of group algebras, Lecture Notes in Math. 246 Ž1973., 430᎐465. 13. G. Mullineux, Bijections of p-regular partitions and p-modular irreducibles of symmetric groups, J. London Math. Soc. Ž 2 . 20 Ž1979., 60᎐66. 14. G. Mullineux, On the p-cores of p-regular diagrams, J. London Math. Soc. Ž 2 . Ž1979., 222᎐226. 15. H. Nagao and Y. Tsushima, ‘‘Representations of Finite Groups,’’ Academic Press, San Diego, 1988.