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Decomposition Numbers ofSp(4, q)
199, 544]555 Ž1998. JA977189
JOURNAL OF ALGEBRA ARTICLE NO.
Decomposition Numbers of SpŽ 4, q . Tetsuro Okuyama Laboratory of Mathematics, Hokkaido Uni¨ ersity of Education, Asahikawa Campus, Hokumoncho 9, Asahikawa, 070, Japan
and Katsushi Waki Department of Information Science, Hirosaki Uni¨ ersity, 3, Bunkyo-cho, Hirosaki, 036 Japan Communicated by Walter Feit Received December 10, 1996
1. INTRODUCTION AND NOTATION Let p be a prime and q s p n. Decomposition numbers of the symplectic group SpŽ4, q . in characteristics other than p are almost determined by White w6, 7x. But in case the characteristic divides q q 1, one column in the decomposition matrix of the principal block is only determined up to one parameter. We shall determine it in this paper ŽTheorem 2.3, below.. The ordinary characters and conjugacy classes of SpŽ4, q . are given in Srinivasan w4x and Enomoto w3x. We shall denote by r an odd prime which divides q q 1, and r d will denote the r-part of q q 1. We denote by G the symplectic group SpŽ4, q ., which is determined by the matrix 0 0 Js y1 0
0 0 0 y1 544
0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
1 0 0 0
0 1 . 0 0
0
DECOMPOSITION NUMBERS OF
SpŽ4, q .
545
So G s X g GLŽ4, q .< t XJX s J 4 . We put 1 Ž l , a , m , b . s ya 0 0
0 1 0 0
l b 1 0
la q b m , where l , a , m , b g GF Ž q . . a 1
0
Then the set U s Ž l, a , m , b .< l, a , m , b g GF Ž q .4 forms a Sylow psubgroup of G and e s Ž0, 0, 0, 0. is a unit element of G. A diagonal subgroup H s diagŽ a, b, ay1 , by1 .< a, b g GF Ž q .=4 is a torus of G. The Borel subgroup B is the semi-direct product of U by H. Generators of the Weyl group of G are 0 y1 ss 0 0
1 0 0 0
0 0 0 y1
0 0 1 0
0
and
0 0 ts 1 0
0 1 0 0
y1 0 0 0
0 0 . 0 1
0
The group G has two non-conjugate maximal parabolic subgroups Gs s ² B, s : and Gt s ² B, t :. To calculate the unknown parameter in the decomposition matrix for the principal r-block, we need to investigate restrictions of irreducible characters of G to Gs . We shall discuss characters of Gs in Section 4. We also need to give a module theoretical argument which is our main tool to obtain the result. This is done in Section 6 and a proof of our theorem is given in Section 7. Let Ž R, K, k s RrŽp .. be the splitting r-modular system.
2. MAIN THEOREM THEOREM 2.1 ŽWhite w6, 7x.. The principal r-block B0 Ž G . has fi¨ e irreducible Brauer characters k G s w 0 , w , ws , w t , wst 4 and a part of the decomposition matrix for B0 Ž G . is as follows. The unknown entry a is an integer satisfying 1 F a F Ž q y 1.r2 Žor qr2 if p s 2.. Degrees
Chars.
w0
1 q Ž q y 1. 2r2 q Ž q 2 q 1.r2 q Ž q 2 q 1.r2 q4
1G h xs xt x st
1
w
ws
wt
wst
1 1
1
1 1 1 1
1
a
1
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OKUYAMA AND WAKI
Remark 2.2. In the above we changed the notation in w6, 7x, because we intended to give a unified argument both for q odd and even. In the notations there h s u 10 Ž x 5 ., x s s u 11Ž x 3 ., x t s u 12 Ž x4 ., and x st s u 13 Ž x6 .. The remaining part of the decomposition matrix can be read from the above. The inequality for a is improved as a - r d by Waki w8x. The next is our main theorem in this paper. THEOREM 2.3.
If r d s 3 then a s 1 else a s 2.
From White w6x and Enomoto w3x, we can get the following lemma. LEMMA 2.4. Ž1. 1 BG s 1 G q x s q x t q x st q 2 j . Ž2. 1 G G s 1 G q x s q j . s Ž3. 1 G G s 1 G q x t q j . t The character j is an irreducible character of degree q Ž q q 1. 2r2, and is of r-defect 0 and x st is the Steinberg character of G.
3. PARABOLIC SUBGROUPS OF G Let us denote t y1 w Ax s A
ž
0
0 , A
/
where A g GL Ž 2, q . ,
Us s Ž l , 0, m , b . < l , m , b g GF Ž q . 4 ; U, U0 s Ž 0, 0, m , 0 . < m g GF Ž q . 4 ; Us , L s s w A x < A g GL Ž 2, q . 4 . A maximal parabolic subgroup Gs is a semi-direct product of a unipotent radical Us by a Levi complement L s . We denote by Bs an intersection of B and L s . All elements of Bs are represented by wŽ 0a ab .x where a, b in GF Ž q .= and a in GF Ž q .. From Proposition 2.7.3Ži. in w2x, we can determine distinguished double coset representatives of parabolic subgroups. LEMMA 3.1. Ž1. The double coset representati¨ es of B _ GrGs , Gt _ GrGs , and Gs _ GrGs are e, t, st, tst 4 , e, st 4 , and e, t, tst 4 , respecti¨ ely.
DECOMPOSITION NUMBERS OF
SpŽ4, q .
547
Ž2. The following hold: B t l Gs s Gst l Gs s Bs h Ž 0, 0, m , b . < m , b g GF Ž q . 4 B st l Gs s Gtst l Gs s Bs h Ž 0, 0, m , 0 . < m g GF Ž q . 4 B t st l Gs s Bs ,
Gst st l Gs s L s .
4. CHARACTERS OF Gs Let an element x be Ž ac db . in GLŽ2, q .. We identify x with w x x in L s . We also use Ž l, m , b . to denote the element Ž l, 0, m , b . in Us . The action of x on Us is given as x
Ž l , m , b . s Ž a2l q 2 ac b q c 2m , b 2l q 2 bd b q d 2m , abl q Ž ad q bc . b q cd m . .
Ž ).
Let s be a non-trivial irreducible character of the additive group GF Ž q . defined by
s Ž a . s e 2 ip trŽ a .r p ,
a g GF Ž q . ,
ny 1
where trŽ a . s a q a p q ??? qa p . We pick up an element n in GF Ž q 2 .=_ GF Ž q . such that its order is q q 1. Let a polynomial X 2 y uX q 1 in GF Ž q .w X x be the minimal polynomial of n over GF Ž q .. We define a linear character r of Us to be r Ž l, m , b . s s Ž l q m q u b . and IL sŽ r . s x g L s < r x s r 4 . LEMMA 4.1. 2 a b .< 2 0 1 Ž1. IL Ž r . s Ž yb 4 . This is a dihea q ub a q uab q b s 1 i 1 0 s dral group of order 2Ž q q 1.. Ž2. Each Gs-conjugate character of r is non-tri¨ ial on U0 . Ž3. If a linear character t of Us is not Gs-conjugate to r , there is a Gs-conjugate character of t which is tri¨ ial on U0 . Moreo¨ er if t is not tri¨ ial, IG sŽt . is an r 9-group.
¦ž /;
Proof. From Ž)., r x s r if and only if a2 q uab q b 2 s 1, d 2 s 1, and 2Ž ac q bd . q uŽ bc q ad . s u. In particular, for r x 0 s r if and only if Ž c 0 , d 0 . s Ž0, 1. or Ž u, y1.. In general, 2 2 a b . Ž . y s Ž yb a q ub , where a q uab q b s 1 is in IL s r . Thus
c 2 q ucd q x 0 s Ž c10 d00 ., an element Ž1. follows
548
OKUYAMA AND WAKI
since xyy1 is in IL sŽ r . and is of the form Ž c10 d00 .. Similarly since Ž0, m , 0. x s Ž c 2m , d 2m , cd m ., r x Ž0, m , 0. s s ŽŽ c 2 q ucd q d 2 . m .. If r x is trivial on U0 then c 2 q ucd q d 2 s 0. This contradicts the fact that the polynomial X 2 y uX q 1 is irreducible over GF Ž q .. Thus Ž2. is proved. We define linear characters t 1 and t 2 of Us to be t 1Ž l, m , b . [ s Ž b . and t 2 Ž l, m , b . [ s Ž l.. Moreover if q is odd, let t 2X be a linear character of Us such that t 2X Ž l, m , b . [ s Žgl. where g is a generator of the multiplicative group GF Ž q .=. Then it is easy to check that a set of representatives of Gs conjugacy classes of linear characters of Us is either 1, r , t 1 , t 2 4 Ž q:even. or 1, r , t 1 , t 2 , t 2X 4 Ž q:odd.. From this, Ž3. is proved. u. Let us denote K s s IL sŽ r .. Two elements x s s Ž 01 10 . and ys s Ž y1 0 1 are generators of K s . Since Us is an abelian group, r can be extended to the 0 . character rq of K s Us so that rqs 1 K s on K s . We define z to be Ž y1 0 y1 . In case q is odd, we define linear characters 1q, 1y, 19, 10 4 of K s as
1q
1y
19
xs
1
y1 y1
ys
1
y1
1
10 1 y1
Let characters rq, ry, r 9, and r 0 of K s Us be the product with rq and the four characters above. In case q is even, characters r ", with r " Ž x s . s "1, are defined in the same way. Note that these linear characters are all of the linear composition factors in r K sUs . If q is odd, the elements x s and ys aren’t K s-conjugate but they are conjugate in L s . Let v be an element of L s , such that x sv s ys . e, v 4 is a set of double coset representatives of K s _ L srBs . Moreover K s l Bs s ² ys , z : and K sv l Bs s ² x s , z :v s ² ys , z :. If q is even, L s s K s Bs and K s l Bs s ² ys :. LEMMA 4.2.
Ž 1. Ž 2. Ž 3.
Gs rq Ž ys . s
Gs ry Ž ys . s
½
½
2
Ž q y 1 . r2 q Ž q y 1 . r2 2
y Ž q y 1 . r2 yq Ž q y 1 . r2
Ž q : odd . Ž q : e¨ en . Ž q : odd . Ž q : e¨ en .
r 9G s Ž ys . s r 0 G s Ž ys . s 0.
Proof. For any g in Bs , ys is in ² ys , z : g if and only if ysg s ys . The order of CB sŽ ys . is Ž q y 1. 2 Ž q : odd. and q Ž q y 1. Ž q : even.. Let u be a
DECOMPOSITION NUMBERS OF
SpŽ4, q .
549
character of K s Us . Since u G sB s s u K sL sB s,
¡ Ž q y 1.
u G s Ž ys . s
~
¢
2
4 q Ž q y 1. 2
Ž u Ž ys . q u Ž x s . .
Ž q : odd.
u Ž ys .
Ž q : even . .
The lemma is immediate from this.
5. BLOCKS OF Gs AND K s Since Us is a normal r 9-subgroup of Gs and IG sŽ r . s K s Us , we can get one to one correspondence between blocks of RGs covering r and those blocks of Rw K s Us x by Clifford’s theorem ŽFong’s reduction.. Indeed the corresponding blocks are Morita equivalent and the equivalence is given by inductions of modules. Moreover since Us is abelian and K s Us has a complement, a character r is extended to a character rq of K s Us Žsee Section 4.. Thus the set of blocks of Rw K s Us x, which cover r , is in one to one correspondence with the set of blocks of RK s . The corresponding blocks are also Morita equivalent and the equivalence is given by tensoring Rw K s x-modules with the R-form of rq. From now on, let b 0 be the principal block of RK s . We define a block b1 of Rw K s Us x and a block b of RGs to be corresponding to the block b 0 in the above correspondence. Since K s is a dihedral group, whose order is 2Ž q q 1., block b 0 is isomorphic to a group ring of a dihedral group of order 2 r d as algebras. The block b 0 contains two simple modules. We denote by Rq the & trivial module of b 0 and& by Ry the other one. We define b1-modules R " and b-modules Y "s R " G s to be corresponding to b 0-modules R ". If h " are ordinary characters of& Y ", then they are irreducible. Since r " are ordinary characters of R " , h "s r " G s. The characters h " are irreducible as Brauer characters and we denote their characters by w ". Now we can prove the following lemma for the character h of G. LEMMA 5.1. hG s s hy. Proof. Since h Ž e . s q Ž q y 1. 2r2 and h Ž0, m , 0. s yq Ž q y 1.r2 for all m in GF Ž q .=, hU0 s Ž q Ž q y 1.r2. = Ž Q y 1U0 . where Q is the regular character of U0 w3, 4x. From Lemma 4.1Ž3., ŽhUs, t . s 0 if t is not Gs-conjugate to r . Since r has q Ž q y 1. 2r2 Gs-conjugate characters, ŽhU , r . s 1, the character hG is one of the induced characters s s
550
OKUYAMA AND WAKI
Gs Gs Gs rq , ry , r 9G s , r 0 G s 4 by the argument in Section 4. So hG s s ry s hy using Lemma 4.2 and the value h Ž ys . w3, 4x.
6. STRUCTURE OF b-MODULES We denote by R G an RG-trivial module. We will use the following three exact sequences of RG-modules for the next two lemmas.
Ž s . 0 ª X sX ª R G sG ª R G ª 0 Ž t . 0 ª X tX ª R G tG ª R G ª 0 Ž st . 0 ª X st ª R BG ª R G sG [ R G tG ª R G ª 0. Ž s . Žresp. Ž t .. is the sequence obtained by the canonical epimorphism from R G sG Žresp. R G tG . to R G . Ž st . is a well known sequence as a realization of the Steinberg character and the character of X st is x st . The ordinary character of X sX is x s q j from Lemma 2.4. Since j is defect 0, X sX is a direct sum of X s and Z where the character of X s is x s and Z is a simple projective module corresponding to j . Similarly X tX is a direct sum of X t with character x t and Z. We will use the notations in Section 5: the block b1 of K s Us , the block b of Gs , b-modules Y ", and the ordinary characters h " of Y ". For an RG-module X, we denote the b-part of XG s by X b and B0 Ž Gs .-part of XG s by X B 0 ŽG s . where B0 Ž Gs . is the principal block of Gs . Let us denote a projective cover of an RGs-module X by QŽ X . and a kernel of the essential epimorphism from QŽ X . to X by V Ž X .. LEMMA 6.1. ti¨ e.
Ž1. X sb s Yq, X s B ŽG . s R G , and the other summand of X sG is projec0 s s s
Ž2. X t b s 0, X t B ŽG . s V Ž R G ., and the other summand of X t G is 0 s s s projecti¨ e. Proof. The exact sequence Ž s . is split when it is restricted to Gs . From X Lemma 3.1, X sG s R B t l G sG s [ R L sG s. We denote B t l Gs by B9. Notice s that B9 is an r 9-subgroup. Then for any g in Gs ,
ž1
B9
Gs
Us ,
r g s Ž 1 B 9G s , r G s .
/
s Ž 1 B 9 , r G sB 9 . s 1B9 ,
ž
s
[
xgUs_GsrB 9
[
xgUs_GsrB 9
ž1
B9 ,
r Ux sx l B 9B 9
/
r Ux sx l B 9B 9 .
/
DECOMPOSITION NUMBERS OF
SpŽ4, q .
551
Since U0 is a subgroup of B9, Lemma 4.1Ž2. implies the right hand side of the equation is 0. This means that the b-part of R B 9G s is 0. Since & Gs K s Us Gs R L s K sUs s R K s , the b1-part of R L s K sUs is Rq . Thus the b-part of R L sG s is Yq Žsee the argument in Section 5.. Because Z is projective, this Yq is a summand of X sG s. Moreover by Lemma 3.1 the B0 Ž Gs .-part of R B 9G s and R L sG s are R B 9UsG s s R BG s and R L sUsG s s R G s since Ker B0 Ž Gs . > Us . R B 9UsG s s R BG s must be a summand of Z. The module R L sUsG s s R G s is the B0 Ž Gs .-part of X sG s as Ž1 BG s , x s . s 1 w3, 4x. From Lemma 3.1, R G tG G s s R BG s [ R B s t l G sG s. Thus the exact sequence Ž t . is a projective resolution of R G when it is restricted to Gs . The b-part s of R BG s s QŽ R G s . is 0. U0 ; B st l Gs by Lemma 3.1. Thus we can show the b-part of R B s t l G sG s is 0 in the same way as in case R B 9G s. Thus the b-part of X t is also 0. Both B0 Ž Gs .-parts of R BG s and R B s t l G sG s are QŽ R G s . as Ker B0 Ž Gs . > Us . One of the two is a summand of Z and the B0 Ž Gs .-part of X t is V Ž R G s .. The rest of proof is immediate from Lemma 4.1Ž3.. LEMMA 6.2.
X st b s
½
V Ž Yq . ,
q : e¨ en
V Ž Yq . [ Q Ž Yq . ,
q : odd.
Proof. When we restrict the exact sequence Ž st . to Gs , a first map Žan epimorphism to R G . is split and a second map can be seen as a projective resolution. From Lemma 3.1 and Lemma 6.1, the b-part of the above sequence is the following. 0 ª X st b ª R B sG sb ª Yqª 0. By using the notation of Section 4,
R Bs
Gs K s Us
¡ ¢R
R s~
Bs l K s Bs l K s
K s Us
[ R B sv y 1 l K sK sUs ,
q : odd
K s Us
,
q : even,
y1
where Bs l K s s ² ys , z : and Bsv l K s s ² x s , z :. From the argument in Section 5, the b-part of R B sG s is QŽ Yq . [ QŽ Yq . Ž q : odd. or QŽ Yq . Ž q : even.. Thus the lemma is proved. Since h is irreducible as a Brauer character, an R-free RG-module Y with character h is uniquely determined. The next lemma is immediate from Lemma 5.1. LEMMA 6.3. Yb s Yy.
552
OKUYAMA AND WAKI
7. PROOF OF THE THEOREM The notation w 0 , ws , w t , wst , w 4 and wq, wy 4 for the characters considered in Sections 2 and 5 will be used to denote the modules which afford the characters. We also use the notation for the blocks over R in Section 5 to denote the corresponding blocks over k. Note that all simple modules in the principal block B0 Ž G . are self dual. LEMMA 7.1. Let a kG-module M be a non-projecti¨ e part of a direct sum decomposition of ws m w t . Then M is self dual and its composition factors are aw q wst . Proof. From Lemma 2.4 and Theorem 2.1ŽWhite., it is easy to prove that non-projective parts of k G sG and k G tG are uniserial modules, whose Loewy series are
w0 ws w0
w0 wt . w0
and
From Lemma 3.1, k G sG m k G tG is projective. Thus
w0 w0 ws m w t s P Ž w 0 . [ P , w0 w0 where P Ž w 0 . is a projective cover of the trivial kG-module and P is a projective kG-module which does not have P Ž w 0 . as a direct summand. By tensoring the following two exact sequences
w0 w 0 ª w 0 ª ws ª 0 ª 0 ws w0
w0 w 0 ª w 0 ª w t ª 0 ª 0, wt w0
and
we obtain a new exact sequence
w0 w0 w0 w0 f w w 0 ª w 0 ª ws [ w t ª ws m w t ª 0 m 0 ª 0. ws wt w0 w0 w0 w0 Let us denote a kernel of f by V. Since ws and w t are not isomorphic, the Loewy structure of V is
w0
ws
w0
wt
w0
DECOMPOSITION NUMBERS OF
SpŽ4, q .
553
Thus a module
w0 w0 ws m w t is isomorphic to Vy1 Ž V .Žs P Ž w 0 .rV . [ P from the exact sequence f
0 ª V ª P Ž w0 . [ P ª
w0 w0 ws m w t ª 0.
Moreover by using two exact sequences 0 ª ws ª
w0 ws ª w 0 ª 0
and
0 ª wt ª
w0 w t ª w 0 ª 0,
we have a new exact sequence 0 ª ws m w t ª
w0 w0 w0 w0 g ws m w t ª ws [ w t ª w 0 ª 0.
Let us denote a kernel of g by W. A Loewy structure of W is
ws
w0
wt .
In particular a top of W is w 0 . Thus from the exact sequence 0 ª ws m w t h ª P [ Vy1 Ž V . ª W ª 0, the module ws m w t is isomorphic to P [ y1 ŽKer h l V Ž V ... The module Ker h l Vy1 Ž V . is a subfactor of P Ž w 0 . and its composition factors are aw q wst from Theorem 2.1. Let us denote by X a kG-module XrX Žp . for an RG-module X and by H Ž S . a kG-module RadŽ QŽ S ..rSocŽ QŽ S .. for a simple kG-module S. LEMMA 7.2. Ž1. wsG s wq[ proj. s Ž2. w t G s H Ž k G . [ proj. s s Ž3. Ž ws m w t . G s H Ž wq . [ proj. s Proof. Since X ss ww0s , Ž1. follows from Lemma 6.1Ž1.. Since X ts ww 0t , Ž2. follows from Lemma 6.1Ž2.. By tensoring an AR-sequence 0 ª V Ž k G s . ª H Ž k G s . [ QŽ k G s . ª Vy1 Ž k G s . ª 0 by wq, we can construct an ARsequence of Vy1 Ž wq . since dim k wq, is an r 9-number Žsee w1x.. Thus H Ž k G s . m wq is isomorphic to the direct sum of H Ž wq . and some projective modules. So Ž3. follows from Ž1. and Ž2..
554 LEMMA 7.3. Then
OKUYAMA AND WAKI
Let us denote by M a non-projecti¨ e module in Lemma 7.1.
Mb s
½
H Ž wq . ,
q : e¨ en
H Ž wq . [ Q Ž wq . ,
q : odd.
Proof. Let us use an exact sequence Ž st .9: 0 ª X stª k BG ª k G sG [ k G tG ª k G ª 0 which is obtained from Ž st .. From the proof of Lemma 7.1, X st has a submodule X whose Loewy structure is
ws
w0
wt
and the module M is isomorphic to X strX. From Lemmas 6.2 and 6.1, Mb s H Ž wq . if q is even and Mb s H Ž wq . [ QŽ wq . or V Ž wq . [ Vy1 Ž wq . if q is odd. The case Mb s V Ž wq . [ Vy1 Ž wq . does not occur by Lemma 7.2Ž3.. Proof of the Main Theorem. We shall calculate the variable a by an investigation of the structure of M. Since
wq wy wq Q Ž wq . s . , .. wy wq
both a top and the socle of Mb have only one copy of wy. Let us denote by M0 the minimal submodule of M, which contains wst as a composition factor. Since the multiplicity of wst is one, the module M0 can be determined uniquely. Note that a top of M0 is wst . First of all, we show that MrM0 / 0. If M is equal to M0 , the top of M is isomorphic to wst . From the self-duality of M, the socle of M is also wst . This means that M0 s wst s M. But this contradicts the fact a is not zero. Next we prove that MrM0 s w . All composition factors of MrM0 are isomorphic to w . We know that Ext 1k G sŽ wy, wy . s 0, and the multiplicity of wy in a top of Ž MrM0 . b s MbrM0 b is one. Thus from Lemma 6.3 Ž w b s wy ., we can get MrM0 s w . Finally, we show that Rad M0 s w if Rad M0 / 0. This follows by the same argument as in the above because composition factors of a module Rad M0 are all w and the multiplicity of wy in the socle of Mb is one.
DECOMPOSITION NUMBERS OF
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555
If r d s 3, H Ž wq ., is wy. So Rad M0 s 0. The self-duality of M shows M s w [ wst . If r d is bigger than 3, H Ž wq . is uniserial, whose composition length is longer than 3. If Rad M0 s 0, then M s w [ wst and Mb has wys w b as a direct summand. This contradicts Lemma 7.3. Thus
w M s wst . w
REFERENCES 1. D. J. Benson and J. F. Carlson, Nilpotent elements in the Green ring, J. Algebra 104 Ž1986., 329]350. 2. R. W. Carter, Finite groups of Lie type, Pure Appl. Math. Ž1985.. 3. H. Enomoto, The characters of the finite symplectic group SpŽ4, q ., q s 2 f , Osaka J. Math. 9 Ž1972., 75]94. 4. B. Srinivasan, The characters of the finite symplectic group SpŽ4, q ., Trans. Amer. Math. Soc. 131 Ž1968., 488]525. 5. D. L. White, ‘‘The 2-Blocks and Decomposition Numbers of SpŽ4, q ., q Odd,’’ Ph.D. Thesis, Yale University, 1987. 6. D. L. White, Decomposition numbers of SpŽ4, q . for primes dividing q " 1, J. Algebra 132 Ž1990., 488]500. 7. D. L. White, Decomposition numbers of Sp4 Ž2 a . in odd characteristics, J. Algebra 177 Ž1995., 264]276. 8. K. Waki, A note on the decomposition numbers of SpŽ4, q ., J. Algebra 186 Ž1996., 105]112.
JOURNAL OF ALGEBRA ARTICLE NO.
Decomposition Numbers of SpŽ 4, q . Tetsuro Okuyama Laboratory of Mathematics, Hokkaido Uni¨ ersity of Education, Asahikawa Campus, Hokumoncho 9, Asahikawa, 070, Japan
and Katsushi Waki Department of Information Science, Hirosaki Uni¨ ersity, 3, Bunkyo-cho, Hirosaki, 036 Japan Communicated by Walter Feit Received December 10, 1996
1. INTRODUCTION AND NOTATION Let p be a prime and q s p n. Decomposition numbers of the symplectic group SpŽ4, q . in characteristics other than p are almost determined by White w6, 7x. But in case the characteristic divides q q 1, one column in the decomposition matrix of the principal block is only determined up to one parameter. We shall determine it in this paper ŽTheorem 2.3, below.. The ordinary characters and conjugacy classes of SpŽ4, q . are given in Srinivasan w4x and Enomoto w3x. We shall denote by r an odd prime which divides q q 1, and r d will denote the r-part of q q 1. We denote by G the symplectic group SpŽ4, q ., which is determined by the matrix 0 0 Js y1 0
0 0 0 y1 544
0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
1 0 0 0
0 1 . 0 0
0
DECOMPOSITION NUMBERS OF
SpŽ4, q .
545
So G s X g GLŽ4, q .< t XJX s J 4 . We put 1 Ž l , a , m , b . s ya 0 0
0 1 0 0
l b 1 0
la q b m , where l , a , m , b g GF Ž q . . a 1
0
Then the set U s Ž l, a , m , b .< l, a , m , b g GF Ž q .4 forms a Sylow psubgroup of G and e s Ž0, 0, 0, 0. is a unit element of G. A diagonal subgroup H s diagŽ a, b, ay1 , by1 .< a, b g GF Ž q .=4 is a torus of G. The Borel subgroup B is the semi-direct product of U by H. Generators of the Weyl group of G are 0 y1 ss 0 0
1 0 0 0
0 0 0 y1
0 0 1 0
0
and
0 0 ts 1 0
0 1 0 0
y1 0 0 0
0 0 . 0 1
0
The group G has two non-conjugate maximal parabolic subgroups Gs s ² B, s : and Gt s ² B, t :. To calculate the unknown parameter in the decomposition matrix for the principal r-block, we need to investigate restrictions of irreducible characters of G to Gs . We shall discuss characters of Gs in Section 4. We also need to give a module theoretical argument which is our main tool to obtain the result. This is done in Section 6 and a proof of our theorem is given in Section 7. Let Ž R, K, k s RrŽp .. be the splitting r-modular system.
2. MAIN THEOREM THEOREM 2.1 ŽWhite w6, 7x.. The principal r-block B0 Ž G . has fi¨ e irreducible Brauer characters k G s w 0 , w , ws , w t , wst 4 and a part of the decomposition matrix for B0 Ž G . is as follows. The unknown entry a is an integer satisfying 1 F a F Ž q y 1.r2 Žor qr2 if p s 2.. Degrees
Chars.
w0
1 q Ž q y 1. 2r2 q Ž q 2 q 1.r2 q Ž q 2 q 1.r2 q4
1G h xs xt x st
1
w
ws
wt
wst
1 1
1
1 1 1 1
1
a
1
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Remark 2.2. In the above we changed the notation in w6, 7x, because we intended to give a unified argument both for q odd and even. In the notations there h s u 10 Ž x 5 ., x s s u 11Ž x 3 ., x t s u 12 Ž x4 ., and x st s u 13 Ž x6 .. The remaining part of the decomposition matrix can be read from the above. The inequality for a is improved as a - r d by Waki w8x. The next is our main theorem in this paper. THEOREM 2.3.
If r d s 3 then a s 1 else a s 2.
From White w6x and Enomoto w3x, we can get the following lemma. LEMMA 2.4. Ž1. 1 BG s 1 G q x s q x t q x st q 2 j . Ž2. 1 G G s 1 G q x s q j . s Ž3. 1 G G s 1 G q x t q j . t The character j is an irreducible character of degree q Ž q q 1. 2r2, and is of r-defect 0 and x st is the Steinberg character of G.
3. PARABOLIC SUBGROUPS OF G Let us denote t y1 w Ax s A
ž
0
0 , A
/
where A g GL Ž 2, q . ,
Us s Ž l , 0, m , b . < l , m , b g GF Ž q . 4 ; U, U0 s Ž 0, 0, m , 0 . < m g GF Ž q . 4 ; Us , L s s w A x < A g GL Ž 2, q . 4 . A maximal parabolic subgroup Gs is a semi-direct product of a unipotent radical Us by a Levi complement L s . We denote by Bs an intersection of B and L s . All elements of Bs are represented by wŽ 0a ab .x where a, b in GF Ž q .= and a in GF Ž q .. From Proposition 2.7.3Ži. in w2x, we can determine distinguished double coset representatives of parabolic subgroups. LEMMA 3.1. Ž1. The double coset representati¨ es of B _ GrGs , Gt _ GrGs , and Gs _ GrGs are e, t, st, tst 4 , e, st 4 , and e, t, tst 4 , respecti¨ ely.
DECOMPOSITION NUMBERS OF
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547
Ž2. The following hold: B t l Gs s Gst l Gs s Bs h Ž 0, 0, m , b . < m , b g GF Ž q . 4 B st l Gs s Gtst l Gs s Bs h Ž 0, 0, m , 0 . < m g GF Ž q . 4 B t st l Gs s Bs ,
Gst st l Gs s L s .
4. CHARACTERS OF Gs Let an element x be Ž ac db . in GLŽ2, q .. We identify x with w x x in L s . We also use Ž l, m , b . to denote the element Ž l, 0, m , b . in Us . The action of x on Us is given as x
Ž l , m , b . s Ž a2l q 2 ac b q c 2m , b 2l q 2 bd b q d 2m , abl q Ž ad q bc . b q cd m . .
Ž ).
Let s be a non-trivial irreducible character of the additive group GF Ž q . defined by
s Ž a . s e 2 ip trŽ a .r p ,
a g GF Ž q . ,
ny 1
where trŽ a . s a q a p q ??? qa p . We pick up an element n in GF Ž q 2 .=_ GF Ž q . such that its order is q q 1. Let a polynomial X 2 y uX q 1 in GF Ž q .w X x be the minimal polynomial of n over GF Ž q .. We define a linear character r of Us to be r Ž l, m , b . s s Ž l q m q u b . and IL sŽ r . s x g L s < r x s r 4 . LEMMA 4.1. 2 a b .< 2 0 1 Ž1. IL Ž r . s Ž yb 4 . This is a dihea q ub a q uab q b s 1 i 1 0 s dral group of order 2Ž q q 1.. Ž2. Each Gs-conjugate character of r is non-tri¨ ial on U0 . Ž3. If a linear character t of Us is not Gs-conjugate to r , there is a Gs-conjugate character of t which is tri¨ ial on U0 . Moreo¨ er if t is not tri¨ ial, IG sŽt . is an r 9-group.
¦ž /;
Proof. From Ž)., r x s r if and only if a2 q uab q b 2 s 1, d 2 s 1, and 2Ž ac q bd . q uŽ bc q ad . s u. In particular, for r x 0 s r if and only if Ž c 0 , d 0 . s Ž0, 1. or Ž u, y1.. In general, 2 2 a b . Ž . y s Ž yb a q ub , where a q uab q b s 1 is in IL s r . Thus
c 2 q ucd q x 0 s Ž c10 d00 ., an element Ž1. follows
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OKUYAMA AND WAKI
since xyy1 is in IL sŽ r . and is of the form Ž c10 d00 .. Similarly since Ž0, m , 0. x s Ž c 2m , d 2m , cd m ., r x Ž0, m , 0. s s ŽŽ c 2 q ucd q d 2 . m .. If r x is trivial on U0 then c 2 q ucd q d 2 s 0. This contradicts the fact that the polynomial X 2 y uX q 1 is irreducible over GF Ž q .. Thus Ž2. is proved. We define linear characters t 1 and t 2 of Us to be t 1Ž l, m , b . [ s Ž b . and t 2 Ž l, m , b . [ s Ž l.. Moreover if q is odd, let t 2X be a linear character of Us such that t 2X Ž l, m , b . [ s Žgl. where g is a generator of the multiplicative group GF Ž q .=. Then it is easy to check that a set of representatives of Gs conjugacy classes of linear characters of Us is either 1, r , t 1 , t 2 4 Ž q:even. or 1, r , t 1 , t 2 , t 2X 4 Ž q:odd.. From this, Ž3. is proved. u. Let us denote K s s IL sŽ r .. Two elements x s s Ž 01 10 . and ys s Ž y1 0 1 are generators of K s . Since Us is an abelian group, r can be extended to the 0 . character rq of K s Us so that rqs 1 K s on K s . We define z to be Ž y1 0 y1 . In case q is odd, we define linear characters 1q, 1y, 19, 10 4 of K s as
1q
1y
19
xs
1
y1 y1
ys
1
y1
1
10 1 y1
Let characters rq, ry, r 9, and r 0 of K s Us be the product with rq and the four characters above. In case q is even, characters r ", with r " Ž x s . s "1, are defined in the same way. Note that these linear characters are all of the linear composition factors in r K sUs . If q is odd, the elements x s and ys aren’t K s-conjugate but they are conjugate in L s . Let v be an element of L s , such that x sv s ys . e, v 4 is a set of double coset representatives of K s _ L srBs . Moreover K s l Bs s ² ys , z : and K sv l Bs s ² x s , z :v s ² ys , z :. If q is even, L s s K s Bs and K s l Bs s ² ys :. LEMMA 4.2.
Ž 1. Ž 2. Ž 3.
Gs rq Ž ys . s
Gs ry Ž ys . s
½
½
2
Ž q y 1 . r2 q Ž q y 1 . r2 2
y Ž q y 1 . r2 yq Ž q y 1 . r2
Ž q : odd . Ž q : e¨ en . Ž q : odd . Ž q : e¨ en .
r 9G s Ž ys . s r 0 G s Ž ys . s 0.
Proof. For any g in Bs , ys is in ² ys , z : g if and only if ysg s ys . The order of CB sŽ ys . is Ž q y 1. 2 Ž q : odd. and q Ž q y 1. Ž q : even.. Let u be a
DECOMPOSITION NUMBERS OF
SpŽ4, q .
549
character of K s Us . Since u G sB s s u K sL sB s,
¡ Ž q y 1.
u G s Ž ys . s
~
¢
2
4 q Ž q y 1. 2
Ž u Ž ys . q u Ž x s . .
Ž q : odd.
u Ž ys .
Ž q : even . .
The lemma is immediate from this.
5. BLOCKS OF Gs AND K s Since Us is a normal r 9-subgroup of Gs and IG sŽ r . s K s Us , we can get one to one correspondence between blocks of RGs covering r and those blocks of Rw K s Us x by Clifford’s theorem ŽFong’s reduction.. Indeed the corresponding blocks are Morita equivalent and the equivalence is given by inductions of modules. Moreover since Us is abelian and K s Us has a complement, a character r is extended to a character rq of K s Us Žsee Section 4.. Thus the set of blocks of Rw K s Us x, which cover r , is in one to one correspondence with the set of blocks of RK s . The corresponding blocks are also Morita equivalent and the equivalence is given by tensoring Rw K s x-modules with the R-form of rq. From now on, let b 0 be the principal block of RK s . We define a block b1 of Rw K s Us x and a block b of RGs to be corresponding to the block b 0 in the above correspondence. Since K s is a dihedral group, whose order is 2Ž q q 1., block b 0 is isomorphic to a group ring of a dihedral group of order 2 r d as algebras. The block b 0 contains two simple modules. We denote by Rq the & trivial module of b 0 and& by Ry the other one. We define b1-modules R " and b-modules Y "s R " G s to be corresponding to b 0-modules R ". If h " are ordinary characters of& Y ", then they are irreducible. Since r " are ordinary characters of R " , h "s r " G s. The characters h " are irreducible as Brauer characters and we denote their characters by w ". Now we can prove the following lemma for the character h of G. LEMMA 5.1. hG s s hy. Proof. Since h Ž e . s q Ž q y 1. 2r2 and h Ž0, m , 0. s yq Ž q y 1.r2 for all m in GF Ž q .=, hU0 s Ž q Ž q y 1.r2. = Ž Q y 1U0 . where Q is the regular character of U0 w3, 4x. From Lemma 4.1Ž3., ŽhUs, t . s 0 if t is not Gs-conjugate to r . Since r has q Ž q y 1. 2r2 Gs-conjugate characters, ŽhU , r . s 1, the character hG is one of the induced characters s s
550
OKUYAMA AND WAKI
Gs Gs Gs rq , ry , r 9G s , r 0 G s 4 by the argument in Section 4. So hG s s ry s hy using Lemma 4.2 and the value h Ž ys . w3, 4x.
6. STRUCTURE OF b-MODULES We denote by R G an RG-trivial module. We will use the following three exact sequences of RG-modules for the next two lemmas.
Ž s . 0 ª X sX ª R G sG ª R G ª 0 Ž t . 0 ª X tX ª R G tG ª R G ª 0 Ž st . 0 ª X st ª R BG ª R G sG [ R G tG ª R G ª 0. Ž s . Žresp. Ž t .. is the sequence obtained by the canonical epimorphism from R G sG Žresp. R G tG . to R G . Ž st . is a well known sequence as a realization of the Steinberg character and the character of X st is x st . The ordinary character of X sX is x s q j from Lemma 2.4. Since j is defect 0, X sX is a direct sum of X s and Z where the character of X s is x s and Z is a simple projective module corresponding to j . Similarly X tX is a direct sum of X t with character x t and Z. We will use the notations in Section 5: the block b1 of K s Us , the block b of Gs , b-modules Y ", and the ordinary characters h " of Y ". For an RG-module X, we denote the b-part of XG s by X b and B0 Ž Gs .-part of XG s by X B 0 ŽG s . where B0 Ž Gs . is the principal block of Gs . Let us denote a projective cover of an RGs-module X by QŽ X . and a kernel of the essential epimorphism from QŽ X . to X by V Ž X .. LEMMA 6.1. ti¨ e.
Ž1. X sb s Yq, X s B ŽG . s R G , and the other summand of X sG is projec0 s s s
Ž2. X t b s 0, X t B ŽG . s V Ž R G ., and the other summand of X t G is 0 s s s projecti¨ e. Proof. The exact sequence Ž s . is split when it is restricted to Gs . From X Lemma 3.1, X sG s R B t l G sG s [ R L sG s. We denote B t l Gs by B9. Notice s that B9 is an r 9-subgroup. Then for any g in Gs ,
ž1
B9
Gs
Us ,
r g s Ž 1 B 9G s , r G s .
/
s Ž 1 B 9 , r G sB 9 . s 1B9 ,
ž
s
[
xgUs_GsrB 9
[
xgUs_GsrB 9
ž1
B9 ,
r Ux sx l B 9B 9
/
r Ux sx l B 9B 9 .
/
DECOMPOSITION NUMBERS OF
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551
Since U0 is a subgroup of B9, Lemma 4.1Ž2. implies the right hand side of the equation is 0. This means that the b-part of R B 9G s is 0. Since & Gs K s Us Gs R L s K sUs s R K s , the b1-part of R L s K sUs is Rq . Thus the b-part of R L sG s is Yq Žsee the argument in Section 5.. Because Z is projective, this Yq is a summand of X sG s. Moreover by Lemma 3.1 the B0 Ž Gs .-part of R B 9G s and R L sG s are R B 9UsG s s R BG s and R L sUsG s s R G s since Ker B0 Ž Gs . > Us . R B 9UsG s s R BG s must be a summand of Z. The module R L sUsG s s R G s is the B0 Ž Gs .-part of X sG s as Ž1 BG s , x s . s 1 w3, 4x. From Lemma 3.1, R G tG G s s R BG s [ R B s t l G sG s. Thus the exact sequence Ž t . is a projective resolution of R G when it is restricted to Gs . The b-part s of R BG s s QŽ R G s . is 0. U0 ; B st l Gs by Lemma 3.1. Thus we can show the b-part of R B s t l G sG s is 0 in the same way as in case R B 9G s. Thus the b-part of X t is also 0. Both B0 Ž Gs .-parts of R BG s and R B s t l G sG s are QŽ R G s . as Ker B0 Ž Gs . > Us . One of the two is a summand of Z and the B0 Ž Gs .-part of X t is V Ž R G s .. The rest of proof is immediate from Lemma 4.1Ž3.. LEMMA 6.2.
X st b s
½
V Ž Yq . ,
q : e¨ en
V Ž Yq . [ Q Ž Yq . ,
q : odd.
Proof. When we restrict the exact sequence Ž st . to Gs , a first map Žan epimorphism to R G . is split and a second map can be seen as a projective resolution. From Lemma 3.1 and Lemma 6.1, the b-part of the above sequence is the following. 0 ª X st b ª R B sG sb ª Yqª 0. By using the notation of Section 4,
R Bs
Gs K s Us
¡ ¢R
R s~
Bs l K s Bs l K s
K s Us
[ R B sv y 1 l K sK sUs ,
q : odd
K s Us
,
q : even,
y1
where Bs l K s s ² ys , z : and Bsv l K s s ² x s , z :. From the argument in Section 5, the b-part of R B sG s is QŽ Yq . [ QŽ Yq . Ž q : odd. or QŽ Yq . Ž q : even.. Thus the lemma is proved. Since h is irreducible as a Brauer character, an R-free RG-module Y with character h is uniquely determined. The next lemma is immediate from Lemma 5.1. LEMMA 6.3. Yb s Yy.
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OKUYAMA AND WAKI
7. PROOF OF THE THEOREM The notation w 0 , ws , w t , wst , w 4 and wq, wy 4 for the characters considered in Sections 2 and 5 will be used to denote the modules which afford the characters. We also use the notation for the blocks over R in Section 5 to denote the corresponding blocks over k. Note that all simple modules in the principal block B0 Ž G . are self dual. LEMMA 7.1. Let a kG-module M be a non-projecti¨ e part of a direct sum decomposition of ws m w t . Then M is self dual and its composition factors are aw q wst . Proof. From Lemma 2.4 and Theorem 2.1ŽWhite., it is easy to prove that non-projective parts of k G sG and k G tG are uniserial modules, whose Loewy series are
w0 ws w0
w0 wt . w0
and
From Lemma 3.1, k G sG m k G tG is projective. Thus
w0 w0 ws m w t s P Ž w 0 . [ P , w0 w0 where P Ž w 0 . is a projective cover of the trivial kG-module and P is a projective kG-module which does not have P Ž w 0 . as a direct summand. By tensoring the following two exact sequences
w0 w 0 ª w 0 ª ws ª 0 ª 0 ws w0
w0 w 0 ª w 0 ª w t ª 0 ª 0, wt w0
and
we obtain a new exact sequence
w0 w0 w0 w0 f w w 0 ª w 0 ª ws [ w t ª ws m w t ª 0 m 0 ª 0. ws wt w0 w0 w0 w0 Let us denote a kernel of f by V. Since ws and w t are not isomorphic, the Loewy structure of V is
w0
ws
w0
wt
w0
DECOMPOSITION NUMBERS OF
SpŽ4, q .
553
Thus a module
w0 w0 ws m w t is isomorphic to Vy1 Ž V .Žs P Ž w 0 .rV . [ P from the exact sequence f
0 ª V ª P Ž w0 . [ P ª
w0 w0 ws m w t ª 0.
Moreover by using two exact sequences 0 ª ws ª
w0 ws ª w 0 ª 0
and
0 ª wt ª
w0 w t ª w 0 ª 0,
we have a new exact sequence 0 ª ws m w t ª
w0 w0 w0 w0 g ws m w t ª ws [ w t ª w 0 ª 0.
Let us denote a kernel of g by W. A Loewy structure of W is
ws
w0
wt .
In particular a top of W is w 0 . Thus from the exact sequence 0 ª ws m w t h ª P [ Vy1 Ž V . ª W ª 0, the module ws m w t is isomorphic to P [ y1 ŽKer h l V Ž V ... The module Ker h l Vy1 Ž V . is a subfactor of P Ž w 0 . and its composition factors are aw q wst from Theorem 2.1. Let us denote by X a kG-module XrX Žp . for an RG-module X and by H Ž S . a kG-module RadŽ QŽ S ..rSocŽ QŽ S .. for a simple kG-module S. LEMMA 7.2. Ž1. wsG s wq[ proj. s Ž2. w t G s H Ž k G . [ proj. s s Ž3. Ž ws m w t . G s H Ž wq . [ proj. s Proof. Since X ss ww0s , Ž1. follows from Lemma 6.1Ž1.. Since X ts ww 0t , Ž2. follows from Lemma 6.1Ž2.. By tensoring an AR-sequence 0 ª V Ž k G s . ª H Ž k G s . [ QŽ k G s . ª Vy1 Ž k G s . ª 0 by wq, we can construct an ARsequence of Vy1 Ž wq . since dim k wq, is an r 9-number Žsee w1x.. Thus H Ž k G s . m wq is isomorphic to the direct sum of H Ž wq . and some projective modules. So Ž3. follows from Ž1. and Ž2..
554 LEMMA 7.3. Then
OKUYAMA AND WAKI
Let us denote by M a non-projecti¨ e module in Lemma 7.1.
Mb s
½
H Ž wq . ,
q : e¨ en
H Ž wq . [ Q Ž wq . ,
q : odd.
Proof. Let us use an exact sequence Ž st .9: 0 ª X stª k BG ª k G sG [ k G tG ª k G ª 0 which is obtained from Ž st .. From the proof of Lemma 7.1, X st has a submodule X whose Loewy structure is
ws
w0
wt
and the module M is isomorphic to X strX. From Lemmas 6.2 and 6.1, Mb s H Ž wq . if q is even and Mb s H Ž wq . [ QŽ wq . or V Ž wq . [ Vy1 Ž wq . if q is odd. The case Mb s V Ž wq . [ Vy1 Ž wq . does not occur by Lemma 7.2Ž3.. Proof of the Main Theorem. We shall calculate the variable a by an investigation of the structure of M. Since
wq wy wq Q Ž wq . s . , .. wy wq
both a top and the socle of Mb have only one copy of wy. Let us denote by M0 the minimal submodule of M, which contains wst as a composition factor. Since the multiplicity of wst is one, the module M0 can be determined uniquely. Note that a top of M0 is wst . First of all, we show that MrM0 / 0. If M is equal to M0 , the top of M is isomorphic to wst . From the self-duality of M, the socle of M is also wst . This means that M0 s wst s M. But this contradicts the fact a is not zero. Next we prove that MrM0 s w . All composition factors of MrM0 are isomorphic to w . We know that Ext 1k G sŽ wy, wy . s 0, and the multiplicity of wy in a top of Ž MrM0 . b s MbrM0 b is one. Thus from Lemma 6.3 Ž w b s wy ., we can get MrM0 s w . Finally, we show that Rad M0 s w if Rad M0 / 0. This follows by the same argument as in the above because composition factors of a module Rad M0 are all w and the multiplicity of wy in the socle of Mb is one.
DECOMPOSITION NUMBERS OF
SpŽ4, q .
555
If r d s 3, H Ž wq ., is wy. So Rad M0 s 0. The self-duality of M shows M s w [ wst . If r d is bigger than 3, H Ž wq . is uniserial, whose composition length is longer than 3. If Rad M0 s 0, then M s w [ wst and Mb has wys w b as a direct summand. This contradicts Lemma 7.3. Thus
w M s wst . w
REFERENCES 1. D. J. Benson and J. F. Carlson, Nilpotent elements in the Green ring, J. Algebra 104 Ž1986., 329]350. 2. R. W. Carter, Finite groups of Lie type, Pure Appl. Math. Ž1985.. 3. H. Enomoto, The characters of the finite symplectic group SpŽ4, q ., q s 2 f , Osaka J. Math. 9 Ž1972., 75]94. 4. B. Srinivasan, The characters of the finite symplectic group SpŽ4, q ., Trans. Amer. Math. Soc. 131 Ž1968., 488]525. 5. D. L. White, ‘‘The 2-Blocks and Decomposition Numbers of SpŽ4, q ., q Odd,’’ Ph.D. Thesis, Yale University, 1987. 6. D. L. White, Decomposition numbers of SpŽ4, q . for primes dividing q " 1, J. Algebra 132 Ž1990., 488]500. 7. D. L. White, Decomposition numbers of Sp4 Ž2 a . in odd characteristics, J. Algebra 177 Ž1995., 264]276. 8. K. Waki, A note on the decomposition numbers of SpŽ4, q ., J. Algebra 186 Ž1996., 105]112.