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Characters of the covering group of the Higman-Sims group
JOURNAL
OF ALGEBRA
33, 135-143
Characters
(1975)
of the Covering Group Higman-Sims Group ARUNAS
Department
of the
RUDVALIS*
of Mathematics, University of Massachusetts, Amherst, Massachusetts 01002 Communicated Received
by Walter October
Feit
17, 1973
The conjugacy classes of elements and the faithful irreducible characters of the covering group I? of the Higman-Sims group are determined. The faithful irreducible characters of the 2-fold proper covering of the Mathieu group n/r,, are also determined. Notable among the faithful irreducible characters of H is a character of degree 56 afforded by a rational representation. This is the faithful character of least degree and also the only faithful irreducible character of l? afforded by a real representation.
1. INTRODUCTION J. McKay and D. Wales [4] have shown that the Higman-Sims sporadic simple group H of order 2s . 32 . 53 . 7 . 11 has Schur multiplier of order 2. Here the conjugacy classes of elements of the covering group fi of H and also the faithful irreducible characters of l? are determined. J. S. Frame [3] has determined the conjugacy classes and characters of H and free use is made of his work. S. Magliveras [5] has determined the maximal subgroups of H and among these are ones isomorphic to the Mathieu group Maa , the symmetric group Zs of degree eight, and PZUa(S), the extension of the simple group PSU,(5) by its field automorphism. Examination of the permutation characters afforded by these subgroups reveals that every element of H is conjugate to an element of at least one of these three subgroups. The conjugacy classes of elements of H and of these three subgroups are given in Table I. The inverse images in l? of these subgroups are, respectively: a2, (the 2-fold proper covering of M22), and two groups which are here denoted s and H. If G (respectively, G) denotes either one of S or P (respectively Zs or PZUa(S)), then G/G’ N 2, and G/Z(G) ‘v G so that G is a nonsplit central * Research
supported
in part
by National
Science
135 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
Foundation
grant
NSF
GP-37818.
2932537 . 11 11 11 7 52 23325 2s3 2=3 * 5 23 26 24 24 24 3 *5 223 * 5a 225 2$39 223z 223 233 . 5 225 225 225 2a53
I C&)1
100 1 1 2 5 10 2 20 8 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0
rr
-
-
-
-
1 11, 112 71 + 7, 5 3 6 2 42 41 8
class
Conjugacy
176 0 0 1 1 5 1 16 0 4 0 2 2 0 6 2 12 3 1 16 1 1 1 1
77
P=J3(5)
Classes
1
5b
8, 8,
4, -
6, 2,
7 5, 3
10, 5,
2% 2%
4,
2, 6, 12,
1%
-
-
class
--
I
71
-
17
-
-
1s
classes
of Its Subgroup+
+
+ + + + + + +
split fuse
+ -
-10, -10, +5,
-2,
-5t -1 -3 -6
4a
4,
4,
-2, +2a
+3 $1
2nd
in A
-4,
&
-t 50
+2,
3rd
Powers
2b
$2,
+4, +4,
+3
5th
found in Frame’s paper (3). The permutation characters embeddings of the conjugacy classes of elements of these splits (fuses) in l?. Finally, in the last three columns the one can read off from this table that the inverse images
-
-
35 135 125 162 + 1223 1323 + 232 -+- 126 134 144
153 + 1232 1223 + 26 1422 + 24 42 + 224 1 224 8
4
of H and Certain
1100 0 0 1 0 11 3 60 16 4 2 0 0 1 IO 2 32 5 1 40 0 0 0 0
TABLE of Elements
Q In Table I the conjugacy classes of elements of H are given in the order they are afforded by subgroups isomorphic to M2, , PZU,(S), and & are given as well as the subgroups in the classes of H. In the next column a + (-) indicates that the class classes to which certain powers of elements of Z? belong are given. For instance,
5,
1%
2% 2%
12 4,
6
2b
lob
15 56
80
80
8,
4, 46
6, 2,
3
1 11, lib -I 5,
x
M22
The
s $ t:
t; o\
CHARACTERS
OF THE
COVERING
GROUP
137
extension of G by 2, . The groups G are easy to work with if use is made of the canonical homomorphisms w G * G/G’ ‘V Z, and $J: G ---f G/Z(G) P G to describe their elements. To each element g of G there correspond two elements of G, namely the two elements of +-l(g), which are here denoted g and Zg (Z being a generator of Z(G)), and these two elements necessarily have distinct images under 7~. Consequently, there is an isomorphism mapping any element g of G to the ordered pair (n(g), 4(g)), the multiplication of ordered pairs being componentwise. Assuming that n: G + (i), where i is a fixed square root of - 1, the first component n(g) is I or - 1 if g is in #+(G’) and is i or -i if g is in I,-‘(G - G’). Thus, the elements of G are described very conveniently as &g for g in G’ and +g for g in G - G’. The irreducible characters of G are also easily described. The irreducible characters of G with Z(G) in th en k ernels are just the characters of G, to obtain the characters of G which are faithful on Z(G) one must multiply each of these nonfaithful characters by the one-dimensional character 4 of G defined by 4(g) = r(g). These characters play an important role here because all the irreducible components of characters of A induced from characters of G faithful on Z(G) are faithful on Z(H) and thus on all of A.
2.
CONJUGACY
CLASSES
OF ELEMENTS
OF I?
The inverse image in fi of an element h of H consists of two elements of l? which (by abuse of notation) are denoted h and xh (z, which is frequently denoted by - 1, being a generator of Z(B)). If the elements h and zh are (respectively are not) conjugate in I? then the class of H containing h is said to fuse (respectively split) in l?. Determining the conjugacy classes of elements of I? is thus simply a matter of determining for each class of elements of Ii whether it fuses or splits in A. The classes corresponding to elements of odd order of H must split in fi for if x is any such element of H then only one of its two inverse images in I? has odd order so these two elements cannot possibly be conjugate in l?. Therefore, attention may be devoted to elements of even order. It is worth mentioning at this point that if x is a faithful irreducible character of E? then x(zh) = -x(h) so that if h and zh are conjugate in H (i.e., the class of H containing h fuses in A) then X(h) =
x(zh) = 0. First, consider those classes of elements of H which are conjugate to elements of the subgroup lVIz,. Recall that the inverse image of Mz2 in E? is the 2-fold proper covering h?lz2 . The table of faithful irreducible characters worked out by this author from information in the paper [2] of of rjr,z > Burgoyne and Fong, is given in Table II. The only class of M,, which fuses in fizz is the class of elements of order 4 which have centralizer of order 24 in
138
ARUNAS
RUDVALIS
Ma, . The corresponding class in H has centralizer in H of order 26 and, of course, fuses in fi. Inducing a faithful character of degree 10 of iI&, up to I? and computing its inner product with itself one obtains an integer if, and only if, the classes of elements of order 2 and 6 of H which are represented in iI&, split in fi and this integer is 2 or 1 according as the remaining class of elements of order 4 of H which is represented in MZ2 splits or fuses in A. TABLE Faithful
2.2r3=5.1.11 2.2’3 2.2*3= 2.5 24 2.25 2.7 2.7 2.23 2.223 2.11 2.11 a = (-1
31
xt2 *3 *5 42 &4i +7, 17,
f8 +6 fll, +11, +
d-)/2,
Irreducible
10 -2 1 0 0 2 u 0” 1 -1 -1 B =
10 -2 1 0 0 2 or N 0 1 -1 -1 (-1
Characters
56 8 2 1 0 0 0 0 0 2 1 l-l
120 8 3 0 0 0 1 1 0 -1 -1
$- d--11)/2,
II of &rrz
126 -6 0 1 0 -2 0 0 0 0 ,kl ,iJ
126 -6 0 l-l 0 -2 0 0 0 0 fi fi x(-x)
E Z&Z,,
154 -2 1
154 210 330 -2 -10 -2 1 3 -3 -1 0 0 0 0 0 0 -2 -2 2 2 0 0 0 1 0 0 0 l-l 2i -2i 0 0 1 1 -1 1 0 0 1 0 0 0 10
=
-x(x)
440 8 -1 0 0 0 -1 0 -1 0 0
forallxEfir,.
In Table II the faithful irreducible characters of ii?,, are given, while in Table III those of A are given. In both cases the notation fx refers to a pair of classes of elements, the members of one class being the “negatives” of the members of the other. The given character values are always for +x as the values for --x are given by the equation x(-x) = -x(x) for all x E G for all faithful irreducible characters x of G, where G is Msa and H in the respective tables.
Next, consider the characters of l? induced from the characters 4 and 4 of P. Since the character of H induced from the trivial character of PZUs(5) is doubly transitive, the inner product with itself of the induced character C$t A is at most 2 and it equals 2 only if every class of H represented in PZUs(5) splits in fi. However, it has been shown above that the class of elements of order 4 of H with centralizer of order 26 fuses in I? and since this class is represented in PZUs(5) th e ab ove inner product must be 1. Therecharacters of fore, the induced characters 4 1 l? and $ T fi are irreducible degree 176 of fi. In order to show that these characters are not equal one merely computes the inner product with itself of their sum (4 + 6) f fi. Since (4 + c$) vanishes on those classes of P which are not in the inverse image of the simple subgroup PSUs(5) of PZU,(S), the induced character
CHARACTERS
OF
THE
COVERING
GROUP
139
and its inner product with itself are relatively easy to compute. This inner product must be an integer, in fact an even integer, and this occurs precisely when the class of elements of order 10 of H represented in PSUa(5) splits in fi while the class of elements of order 8 of H represented in PSU,(5) fuses in I?. At this point the information concerning the splitting or fusing in A of those elements of H represented in PSUa(5) is complete. Next, observe that an irreducible character of degree 20 of PZUa(5) vanishes on all classes of elements of PZUa(5) not in PSU,(S) except the pair of mutually inverse classes of elements of order 20. Inducing up to I? either of the faithful irreducible characters of degree 20 of P and computing its inner product with itself one finds that the classes of i? represented in the inverse image of PSU,(S) contribute 3/2 while the contribution from the remaining elements of p is either l/2 or 0 according as the classes of elements of order 20 of H split or fuse in l?. Since this inner product must be an integer one concludes that these classes must split in fi and that the induced character of degree 3520 has exactly two irreducible components. If y is an element of order 5 of H such that C,(y) = (y) x A, , then the involutions centralizing y are represented in PZU,(5) but not in PSU,(S). These involutions lift to elements of order 4 in E? because they lift to elements of order 4 in p. Consequently, the inverse image of C,(y) in fi must be (y) x S&(5) and since all elements of order 4 are conjugate in S&(5) this class of involutions of H must fuse in H. At this point further progress can be made by computing the inner product with itself of either of the above irreducible characters of degree 176. This inner product must be 1 and this occurs if, and only if, the class of elements of order 10 of H represented in PiX7,(5) but not in PSU,(S) fuses in a, while the classes of elements of order 12 and 4 which are represented in PZU,(S) but not in PSU,(S) split in A, and exactly one of the classes of elements of orders 6 and 8 which are represented in PB7,(5) but not in PSU,(S) splits in A. This ambiguity is settled by considering an irreducible character of degree 28 of PZU,(5) since it vanishes on all elements of order 8 but not on elements of order 6. Inducing up to fi any faithful irreducible character of degree 28 of p one obtains a faithful character of l? whose inner product with itself is an integer if, and only if, the above class of elements of order 6 of H splits in F?. Therefore, the above class of elements of order 8 of H fuses in A. This completes the information concerning the splitting or fusing in I? of those classes of elements of H which are represented in PEU,(S). At this point doubt remains only concerning those elements of order 8 of H which are represented in M,, and the elements of order 4 which are their squares and which have centralizer in H of order 2s. These problems are settled by computing the faithful character of degree 1100 of fi induced from the character 4 of S and determining under what conditions its inner product
140
ARUNAS
RUDVALIS
with itself in I? is an integer. The computation of this induced character is a nontrivial matter due to the considerable fusion of classes of elements of Zs in H as indicated in Table I. This fusion creates the following complications: Assume x and y are two elements of 2s which are not conjugate in L’s but are conjugate in H and assume that this class splits in fi. Assume for the moment that both x and y are even permutations of Zs . Then (in A) x is conjugate to exactly one of y and -y and it is not a priori obvious which one. If y is an odd permutation then x is conjugate to exactly one of iy and -iy. If both x and y are odd permutations then ix is conjugate to exactly one of iy and -iy. Settling these complications is accomplished by restricting to S one of the above irreducible characters of degree 176. In the nontrivial cases it is found that 1223 is conjugate to -26, l422 is conjugate to -24, and 1323i is conjugate to -23% and to 126i. In the remaining nontrivial case 42 is conjugate either to 224i or to -224i and it turns out not to matter which as the corresponding class in H fuses in I? so that 42 is conjugate to both. The inner product with itself of the above induced character of degree 1100 of I? turns out to be an integer if, and only if, the class of elements of order 8 of H represented in M,, and L’s splits in fi while the class of elements of order 4 which are their squares fuses in I?. This completes the determination of the classes of elements of A. The information about the splitting or fusing of classes of elements of H in I? is given in the first column of Table I.
3. FAITHFUL
IRREDUCIBLE
CHARACTERS
OF i?
It is now clear that the character of degree 1000 of fi induced from either faithful irreducible character of degree 10 of Mz2 is irreducible. Also there are the two complex conjugate irreducible characters of degree 176 induced from the characters 4 and C$of P. Each of the characters of degree 3520 of I? induced from faithful irreducible characters of degree 20 of P has the above irreducible faithful character of degree 1000 as a component and subtracting this off one obtains an irreducible character of degree 2520 which is not equal to its algebraic conjugate in Q(d5). Also, the induced character of degree 1100 of l? mentioned immediately above has one of the above irreducible characters of degree 176 as a component and subtracting this off one obtains an irreducible character of degree 924 which is not equal to its complex conjugate. With one exception the remaining faithful irreducible characters of fi are easily found by inducing up to fi various faithful irreducible characters of a22 and H and splitting off previously known components to obtain new irreducibles. The exception is a character of degree 1848 which is found in the 3-block of defect one whose other two characters are the above complex
CHARACTERS
OF
THE
COVERING
GROUP
141
conjugate characters of degree 924. To see this observe that if w is an element of order 3 of I? then C~(w) = (w) x G, where G is the group obtained from Zs by the construction described in the Introduction for .Z* and PZUa(5). In fact, C,(w) is clearly the inverse image in S of the centralizer of a 3-cycle character of degree a in S, . Th e group G has only one faithful irreducible multiple of 3 (i.e., of full 3-defect) because ,Ys has only one such character. Thus, by Brauer’s First Main Theorem (1, Theorem IOB), there is a unique 3-block of defect one consisting of faithful characters of H. Furthermore, since w is conjugate to all other generators of (w) (i.e., to w-l) this block must consist of three 3-rational characters. Now two of these are the above complex conjugate characters of degree 924 so the third must be an irreducible character of degree 1848 which coincides with the sum of the two characters of degree 924 on 3-regular elements (i.e., ones of order prime to 3) and with the negative of either of the characters of degree 924 on 3-singular elements (i.e., ones of order a multiple of 3). This determines the character of degree 1848 on all classes of elements of I?. To find the remaining faithful irreducible characters of Z? one may use the following sequence of inductions and decompositions: 2gp”
= 176 + 924 + 1848 + l9tJ,
21@
= 176 + 924 + 1980 + 616 2
56fig
= 176 + 176 + 1980 + 1980 + 616 + 616 + 56 -’
56P” = 56 + 616 + 616 + 1980 + 1980 + (2304 + 2304) 84#
= 1000 + 2520 + 2520’ + (2304 + 2304) + 924 + 1980 + 1232,
105P”
= 2520 + 2520’ + (2304 + 2304) + 1980 + 1980 + 1848 + 1232 + 1792,
126fifi
= 1000 + 2520 + 2520’ + 1792 + 1232 + 1232 + 2304.
In the above, the expression (2304 + 2304) refers to a compound character with exactly two irreducible components which are eventually found to be a complex conjugate pair of characters of degree 2304 but which in the form (2304 + 2304) is used simply as a compound character that can be extracted from certain induced characters. For the characters ‘of degrees 1980, 616, 1232, and 2304 above, complex conjugation gives another character of the same degree. This completes the determination of the faithful irreducible characters of Z?. These characters are displayed in Table III below.
Index
rt4, *20, &20a 110, +5,
33% A12
It1 &II, fllb *7 255, f3 +6, &2, f8, *15 1%
X
0
1
0
T” 1
B = (-1 i- Gii)/2,
I2 1
176 176 0 0 0 0 1 1 1 1 5 5 1 1 16 16 0 0 0 0 6 6 3i -3i -i i 16i -162’ i -i
56 1 1 0 1 2 2 8 0 2 -4 0 0 0 0 0 -2 6 0
-lE -9
616 0 0 0 1 4 0 24 0 -1 -4 0 2i 16i i
0 =
x(-x)
.
-
III
x(x)
0
924 924 0 0 0 0 0 0 -1 -1 6 6 -2 -2 4 4 2i -2i 1 1 4 4 0 0 0 0 24i -24i -i i . -1” -I2 -1 -1
Characters
TABLE
0
-T” -9
616 0 0 0 1 4 0 24 0 -1 -4 0 -2i -16i -i
Irreducible
= Z,Hi
-
-1
1792 -1 -1 0 2 -8 0 0 0 2 2 0 0 0 0 0 0 -8
S
-1
1848 0 0 0 -2 -6 2 8 0 -1 8 0 0 0 0 0 -2 -2 0
T” 5
0
2304 /3 p 1 -1 0 0 0 0 0 -6 0 0 0 0 0 0 4
2304 2520 2520 1 1 p 8 I 1 1 0 0 -1 0 0 0 0 0 o o 0 0 -24 -24 0 0 0 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 d5 -45 0 45 d5 0 1 1 4 -5 -5 -.-- _ 0 -1 -1
x(-x) = -x(x), all faithful last row of Table III gives
0
lZ 5
1980 1980 0 0 0 0 -1 -1 0 0 0 0 0 0 36 36 -2i 2i 0 0 0 0 0 0 0 0 24i -24i -i i
since by the equation = 2 / C,(x)l. The
for all x E A and all x abcve.
1000 1232 1232 -1 0 0 -1 0 0 -1 0 0 0 2 2 10 -1 -1 2 -1 -1 -40 -16 -16 0 0 0 -1 0 -1 0 2 2 0 -3i 3i -i i 0 0 16i -16i 0 i -i . . 0 -I2 3 0 7 0 7 --1 0 0
of fi
In Table III the classes of A arising from classes of H which fuse in X? have been deleted irreducible characters of I? vanish on all such classes. For the remaining classes / C&(&x)1 the Schur-Frobenius index for each faithful irreducible character of A.
S-F
2 . 2932537 . 11 11 2. 2. 11 2. 7 2. 52 2 . 23325 2 . 233 2 . 283 . 5 2 . 24 2. 3.5 2 . 2=3 ’ 52 2 . 2a3a 2 . 2%3 2 * 2s3 . 5 2.2e 5 2.2= 5 2.2= 5 2 . 22 53
I WI
Faithful
CHARACTERS
OF THE
COVERING
4. CONCLUDING
GROUP
143
REMARKS
McKay and Wales (4) mention that fi can have no faithful ordinary character of degree less than 56. It follows from the above that E? has a faithful irreducible character of degree 56 which is afforded by a rational representation. No other faithful irreducible character of A is afforded by any real representation. The rationality of the 56-dimensional representation shows that it is associated with a 56-dimensional rational lattice. This lattice could conceivably have A as a proper subgroup of its full group of automorphisms but determining this automorphism group appears to be a very difficult problem. It is known that the 2-fold proper covering 6 of the simple group PSU,(2) of order 215365 . 7 11 has a 56-dimensional faithful rational representation which remains irreducible when restricted to a subgroup of 0 isomorphic to ii?Iz, . This restriction coincides with the restriction to A&z of the above 56-dimensional representation of A. This suggests the possibility that there might be a very large (and presumably new) perfect finite group with a 56-dimensional rational representation generated by the above representations of i? and I? and such that the intersection of 0 and fi in this group is isomorphic to tiz, . Th e author is presently exploring this and some other related problems.
REFERENCES I. R. BRAUER, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. 2. 63 (1956), 406-444. 2. N. BURCOYNE AND P. FONG, A correction to ‘The Schur multipliers of the Mathieu groups,’ Nagoya Math. J. 31 (1968), 297-304. 3. J. S. FRAME, Computation of characters of the H&man-Sims group and its Automorphism group, J. Algebra 20 (1972), 32&349. 4. JOHN MCKAY AND DAVID WALES, The multiplier of the H&man-Sims simple group, Bull. London Math. Sot. 3 (1971), 283-285. 5. SPYROS S. MAGLIVERAS, The subgroup structure of the H&man-Sims simple group, Bull. AIMS 77 (1971), 535-539.
481/33/r-10
OF ALGEBRA
33, 135-143
Characters
(1975)
of the Covering Group Higman-Sims Group ARUNAS
Department
of the
RUDVALIS*
of Mathematics, University of Massachusetts, Amherst, Massachusetts 01002 Communicated Received
by Walter October
Feit
17, 1973
The conjugacy classes of elements and the faithful irreducible characters of the covering group I? of the Higman-Sims group are determined. The faithful irreducible characters of the 2-fold proper covering of the Mathieu group n/r,, are also determined. Notable among the faithful irreducible characters of H is a character of degree 56 afforded by a rational representation. This is the faithful character of least degree and also the only faithful irreducible character of l? afforded by a real representation.
1. INTRODUCTION J. McKay and D. Wales [4] have shown that the Higman-Sims sporadic simple group H of order 2s . 32 . 53 . 7 . 11 has Schur multiplier of order 2. Here the conjugacy classes of elements of the covering group fi of H and also the faithful irreducible characters of l? are determined. J. S. Frame [3] has determined the conjugacy classes and characters of H and free use is made of his work. S. Magliveras [5] has determined the maximal subgroups of H and among these are ones isomorphic to the Mathieu group Maa , the symmetric group Zs of degree eight, and PZUa(S), the extension of the simple group PSU,(5) by its field automorphism. Examination of the permutation characters afforded by these subgroups reveals that every element of H is conjugate to an element of at least one of these three subgroups. The conjugacy classes of elements of H and of these three subgroups are given in Table I. The inverse images in l? of these subgroups are, respectively: a2, (the 2-fold proper covering of M22), and two groups which are here denoted s and H. If G (respectively, G) denotes either one of S or P (respectively Zs or PZUa(S)), then G/G’ N 2, and G/Z(G) ‘v G so that G is a nonsplit central * Research
supported
in part
by National
Science
135 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
Foundation
grant
NSF
GP-37818.
2932537 . 11 11 11 7 52 23325 2s3 2=3 * 5 23 26 24 24 24 3 *5 223 * 5a 225 2$39 223z 223 233 . 5 225 225 225 2a53
I C&)1
100 1 1 2 5 10 2 20 8 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0
rr
-
-
-
-
1 11, 112 71 + 7, 5 3 6 2 42 41 8
class
Conjugacy
176 0 0 1 1 5 1 16 0 4 0 2 2 0 6 2 12 3 1 16 1 1 1 1
77
P=J3(5)
Classes
1
5b
8, 8,
4, -
6, 2,
7 5, 3
10, 5,
2% 2%
4,
2, 6, 12,
1%
-
-
class
--
I
71
-
17
-
-
1s
classes
of Its Subgroup+
+
+ + + + + + +
split fuse
+ -
-10, -10, +5,
-2,
-5t -1 -3 -6
4a
4,
4,
-2, +2a
+3 $1
2nd
in A
-4,
&
-t 50
+2,
3rd
Powers
2b
$2,
+4, +4,
+3
5th
found in Frame’s paper (3). The permutation characters embeddings of the conjugacy classes of elements of these splits (fuses) in l?. Finally, in the last three columns the one can read off from this table that the inverse images
-
-
35 135 125 162 + 1223 1323 + 232 -+- 126 134 144
153 + 1232 1223 + 26 1422 + 24 42 + 224 1 224 8
4
of H and Certain
1100 0 0 1 0 11 3 60 16 4 2 0 0 1 IO 2 32 5 1 40 0 0 0 0
TABLE of Elements
Q In Table I the conjugacy classes of elements of H are given in the order they are afforded by subgroups isomorphic to M2, , PZU,(S), and & are given as well as the subgroups in the classes of H. In the next column a + (-) indicates that the class classes to which certain powers of elements of Z? belong are given. For instance,
5,
1%
2% 2%
12 4,
6
2b
lob
15 56
80
80
8,
4, 46
6, 2,
3
1 11, lib -I 5,
x
M22
The
s $ t:
t; o\
CHARACTERS
OF THE
COVERING
GROUP
137
extension of G by 2, . The groups G are easy to work with if use is made of the canonical homomorphisms w G * G/G’ ‘V Z, and $J: G ---f G/Z(G) P G to describe their elements. To each element g of G there correspond two elements of G, namely the two elements of +-l(g), which are here denoted g and Zg (Z being a generator of Z(G)), and these two elements necessarily have distinct images under 7~. Consequently, there is an isomorphism mapping any element g of G to the ordered pair (n(g), 4(g)), the multiplication of ordered pairs being componentwise. Assuming that n: G + (i), where i is a fixed square root of - 1, the first component n(g) is I or - 1 if g is in #+(G’) and is i or -i if g is in I,-‘(G - G’). Thus, the elements of G are described very conveniently as &g for g in G’ and +g for g in G - G’. The irreducible characters of G are also easily described. The irreducible characters of G with Z(G) in th en k ernels are just the characters of G, to obtain the characters of G which are faithful on Z(G) one must multiply each of these nonfaithful characters by the one-dimensional character 4 of G defined by 4(g) = r(g). These characters play an important role here because all the irreducible components of characters of A induced from characters of G faithful on Z(G) are faithful on Z(H) and thus on all of A.
2.
CONJUGACY
CLASSES
OF ELEMENTS
OF I?
The inverse image in fi of an element h of H consists of two elements of l? which (by abuse of notation) are denoted h and xh (z, which is frequently denoted by - 1, being a generator of Z(B)). If the elements h and zh are (respectively are not) conjugate in I? then the class of H containing h is said to fuse (respectively split) in l?. Determining the conjugacy classes of elements of I? is thus simply a matter of determining for each class of elements of Ii whether it fuses or splits in A. The classes corresponding to elements of odd order of H must split in fi for if x is any such element of H then only one of its two inverse images in I? has odd order so these two elements cannot possibly be conjugate in l?. Therefore, attention may be devoted to elements of even order. It is worth mentioning at this point that if x is a faithful irreducible character of E? then x(zh) = -x(h) so that if h and zh are conjugate in H (i.e., the class of H containing h fuses in A) then X(h) =
x(zh) = 0. First, consider those classes of elements of H which are conjugate to elements of the subgroup lVIz,. Recall that the inverse image of Mz2 in E? is the 2-fold proper covering h?lz2 . The table of faithful irreducible characters worked out by this author from information in the paper [2] of of rjr,z > Burgoyne and Fong, is given in Table II. The only class of M,, which fuses in fizz is the class of elements of order 4 which have centralizer of order 24 in
138
ARUNAS
RUDVALIS
Ma, . The corresponding class in H has centralizer in H of order 26 and, of course, fuses in fi. Inducing a faithful character of degree 10 of iI&, up to I? and computing its inner product with itself one obtains an integer if, and only if, the classes of elements of order 2 and 6 of H which are represented in iI&, split in fi and this integer is 2 or 1 according as the remaining class of elements of order 4 of H which is represented in MZ2 splits or fuses in A. TABLE Faithful
2.2r3=5.1.11 2.2’3 2.2*3= 2.5 24 2.25 2.7 2.7 2.23 2.223 2.11 2.11 a = (-1
31
xt2 *3 *5 42 &4i +7, 17,
f8 +6 fll, +11, +
d-)/2,
Irreducible
10 -2 1 0 0 2 u 0” 1 -1 -1 B =
10 -2 1 0 0 2 or N 0 1 -1 -1 (-1
Characters
56 8 2 1 0 0 0 0 0 2 1 l-l
120 8 3 0 0 0 1 1 0 -1 -1
$- d--11)/2,
II of &rrz
126 -6 0 1 0 -2 0 0 0 0 ,kl ,iJ
126 -6 0 l-l 0 -2 0 0 0 0 fi fi x(-x)
E Z&Z,,
154 -2 1
154 210 330 -2 -10 -2 1 3 -3 -1 0 0 0 0 0 0 -2 -2 2 2 0 0 0 1 0 0 0 l-l 2i -2i 0 0 1 1 -1 1 0 0 1 0 0 0 10
=
-x(x)
440 8 -1 0 0 0 -1 0 -1 0 0
forallxEfir,.
In Table II the faithful irreducible characters of ii?,, are given, while in Table III those of A are given. In both cases the notation fx refers to a pair of classes of elements, the members of one class being the “negatives” of the members of the other. The given character values are always for +x as the values for --x are given by the equation x(-x) = -x(x) for all x E G for all faithful irreducible characters x of G, where G is Msa and H in the respective tables.
Next, consider the characters of l? induced from the characters 4 and 4 of P. Since the character of H induced from the trivial character of PZUs(5) is doubly transitive, the inner product with itself of the induced character C$t A is at most 2 and it equals 2 only if every class of H represented in PZUs(5) splits in fi. However, it has been shown above that the class of elements of order 4 of H with centralizer of order 26 fuses in I? and since this class is represented in PZUs(5) th e ab ove inner product must be 1. Therecharacters of fore, the induced characters 4 1 l? and $ T fi are irreducible degree 176 of fi. In order to show that these characters are not equal one merely computes the inner product with itself of their sum (4 + 6) f fi. Since (4 + c$) vanishes on those classes of P which are not in the inverse image of the simple subgroup PSUs(5) of PZU,(S), the induced character
CHARACTERS
OF
THE
COVERING
GROUP
139
and its inner product with itself are relatively easy to compute. This inner product must be an integer, in fact an even integer, and this occurs precisely when the class of elements of order 10 of H represented in PSUa(5) splits in fi while the class of elements of order 8 of H represented in PSU,(5) fuses in I?. At this point the information concerning the splitting or fusing in A of those elements of H represented in PSUa(5) is complete. Next, observe that an irreducible character of degree 20 of PZUa(5) vanishes on all classes of elements of PZUa(5) not in PSU,(S) except the pair of mutually inverse classes of elements of order 20. Inducing up to I? either of the faithful irreducible characters of degree 20 of P and computing its inner product with itself one finds that the classes of i? represented in the inverse image of PSU,(S) contribute 3/2 while the contribution from the remaining elements of p is either l/2 or 0 according as the classes of elements of order 20 of H split or fuse in l?. Since this inner product must be an integer one concludes that these classes must split in fi and that the induced character of degree 3520 has exactly two irreducible components. If y is an element of order 5 of H such that C,(y) = (y) x A, , then the involutions centralizing y are represented in PZU,(5) but not in PSU,(S). These involutions lift to elements of order 4 in E? because they lift to elements of order 4 in p. Consequently, the inverse image of C,(y) in fi must be (y) x S&(5) and since all elements of order 4 are conjugate in S&(5) this class of involutions of H must fuse in H. At this point further progress can be made by computing the inner product with itself of either of the above irreducible characters of degree 176. This inner product must be 1 and this occurs if, and only if, the class of elements of order 10 of H represented in PiX7,(5) but not in PSU,(S) fuses in a, while the classes of elements of order 12 and 4 which are represented in PZU,(S) but not in PSU,(S) split in A, and exactly one of the classes of elements of orders 6 and 8 which are represented in PB7,(5) but not in PSU,(S) splits in A. This ambiguity is settled by considering an irreducible character of degree 28 of PZU,(5) since it vanishes on all elements of order 8 but not on elements of order 6. Inducing up to fi any faithful irreducible character of degree 28 of p one obtains a faithful character of l? whose inner product with itself is an integer if, and only if, the above class of elements of order 6 of H splits in F?. Therefore, the above class of elements of order 8 of H fuses in A. This completes the information concerning the splitting or fusing in I? of those classes of elements of H which are represented in PEU,(S). At this point doubt remains only concerning those elements of order 8 of H which are represented in M,, and the elements of order 4 which are their squares and which have centralizer in H of order 2s. These problems are settled by computing the faithful character of degree 1100 of fi induced from the character 4 of S and determining under what conditions its inner product
140
ARUNAS
RUDVALIS
with itself in I? is an integer. The computation of this induced character is a nontrivial matter due to the considerable fusion of classes of elements of Zs in H as indicated in Table I. This fusion creates the following complications: Assume x and y are two elements of 2s which are not conjugate in L’s but are conjugate in H and assume that this class splits in fi. Assume for the moment that both x and y are even permutations of Zs . Then (in A) x is conjugate to exactly one of y and -y and it is not a priori obvious which one. If y is an odd permutation then x is conjugate to exactly one of iy and -iy. If both x and y are odd permutations then ix is conjugate to exactly one of iy and -iy. Settling these complications is accomplished by restricting to S one of the above irreducible characters of degree 176. In the nontrivial cases it is found that 1223 is conjugate to -26, l422 is conjugate to -24, and 1323i is conjugate to -23% and to 126i. In the remaining nontrivial case 42 is conjugate either to 224i or to -224i and it turns out not to matter which as the corresponding class in H fuses in I? so that 42 is conjugate to both. The inner product with itself of the above induced character of degree 1100 of I? turns out to be an integer if, and only if, the class of elements of order 8 of H represented in M,, and L’s splits in fi while the class of elements of order 4 which are their squares fuses in I?. This completes the determination of the classes of elements of A. The information about the splitting or fusing of classes of elements of H in I? is given in the first column of Table I.
3. FAITHFUL
IRREDUCIBLE
CHARACTERS
OF i?
It is now clear that the character of degree 1000 of fi induced from either faithful irreducible character of degree 10 of Mz2 is irreducible. Also there are the two complex conjugate irreducible characters of degree 176 induced from the characters 4 and C$of P. Each of the characters of degree 3520 of I? induced from faithful irreducible characters of degree 20 of P has the above irreducible faithful character of degree 1000 as a component and subtracting this off one obtains an irreducible character of degree 2520 which is not equal to its algebraic conjugate in Q(d5). Also, the induced character of degree 1100 of l? mentioned immediately above has one of the above irreducible characters of degree 176 as a component and subtracting this off one obtains an irreducible character of degree 924 which is not equal to its complex conjugate. With one exception the remaining faithful irreducible characters of fi are easily found by inducing up to fi various faithful irreducible characters of a22 and H and splitting off previously known components to obtain new irreducibles. The exception is a character of degree 1848 which is found in the 3-block of defect one whose other two characters are the above complex
CHARACTERS
OF
THE
COVERING
GROUP
141
conjugate characters of degree 924. To see this observe that if w is an element of order 3 of I? then C~(w) = (w) x G, where G is the group obtained from Zs by the construction described in the Introduction for .Z* and PZUa(5). In fact, C,(w) is clearly the inverse image in S of the centralizer of a 3-cycle character of degree a in S, . Th e group G has only one faithful irreducible multiple of 3 (i.e., of full 3-defect) because ,Ys has only one such character. Thus, by Brauer’s First Main Theorem (1, Theorem IOB), there is a unique 3-block of defect one consisting of faithful characters of H. Furthermore, since w is conjugate to all other generators of (w) (i.e., to w-l) this block must consist of three 3-rational characters. Now two of these are the above complex conjugate characters of degree 924 so the third must be an irreducible character of degree 1848 which coincides with the sum of the two characters of degree 924 on 3-regular elements (i.e., ones of order prime to 3) and with the negative of either of the characters of degree 924 on 3-singular elements (i.e., ones of order a multiple of 3). This determines the character of degree 1848 on all classes of elements of I?. To find the remaining faithful irreducible characters of Z? one may use the following sequence of inductions and decompositions: 2gp”
= 176 + 924 + 1848 + l9tJ,
21@
= 176 + 924 + 1980 + 616 2
56fig
= 176 + 176 + 1980 + 1980 + 616 + 616 + 56 -’
56P” = 56 + 616 + 616 + 1980 + 1980 + (2304 + 2304) 84#
= 1000 + 2520 + 2520’ + (2304 + 2304) + 924 + 1980 + 1232,
105P”
= 2520 + 2520’ + (2304 + 2304) + 1980 + 1980 + 1848 + 1232 + 1792,
126fifi
= 1000 + 2520 + 2520’ + 1792 + 1232 + 1232 + 2304.
In the above, the expression (2304 + 2304) refers to a compound character with exactly two irreducible components which are eventually found to be a complex conjugate pair of characters of degree 2304 but which in the form (2304 + 2304) is used simply as a compound character that can be extracted from certain induced characters. For the characters ‘of degrees 1980, 616, 1232, and 2304 above, complex conjugation gives another character of the same degree. This completes the determination of the faithful irreducible characters of Z?. These characters are displayed in Table III below.
Index
rt4, *20, &20a 110, +5,
33% A12
It1 &II, fllb *7 255, f3 +6, &2, f8, *15 1%
X
0
1
0
T” 1
B = (-1 i- Gii)/2,
I2 1
176 176 0 0 0 0 1 1 1 1 5 5 1 1 16 16 0 0 0 0 6 6 3i -3i -i i 16i -162’ i -i
56 1 1 0 1 2 2 8 0 2 -4 0 0 0 0 0 -2 6 0
-lE -9
616 0 0 0 1 4 0 24 0 -1 -4 0 2i 16i i
0 =
x(-x)
.
-
III
x(x)
0
924 924 0 0 0 0 0 0 -1 -1 6 6 -2 -2 4 4 2i -2i 1 1 4 4 0 0 0 0 24i -24i -i i . -1” -I2 -1 -1
Characters
TABLE
0
-T” -9
616 0 0 0 1 4 0 24 0 -1 -4 0 -2i -16i -i
Irreducible
= Z,Hi
-
-1
1792 -1 -1 0 2 -8 0 0 0 2 2 0 0 0 0 0 0 -8
S
-1
1848 0 0 0 -2 -6 2 8 0 -1 8 0 0 0 0 0 -2 -2 0
T” 5
0
2304 /3 p 1 -1 0 0 0 0 0 -6 0 0 0 0 0 0 4
2304 2520 2520 1 1 p 8 I 1 1 0 0 -1 0 0 0 0 0 o o 0 0 -24 -24 0 0 0 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 d5 -45 0 45 d5 0 1 1 4 -5 -5 -.-- _ 0 -1 -1
x(-x) = -x(x), all faithful last row of Table III gives
0
lZ 5
1980 1980 0 0 0 0 -1 -1 0 0 0 0 0 0 36 36 -2i 2i 0 0 0 0 0 0 0 0 24i -24i -i i
since by the equation = 2 / C,(x)l. The
for all x E A and all x abcve.
1000 1232 1232 -1 0 0 -1 0 0 -1 0 0 0 2 2 10 -1 -1 2 -1 -1 -40 -16 -16 0 0 0 -1 0 -1 0 2 2 0 -3i 3i -i i 0 0 16i -16i 0 i -i . . 0 -I2 3 0 7 0 7 --1 0 0
of fi
In Table III the classes of A arising from classes of H which fuse in X? have been deleted irreducible characters of I? vanish on all such classes. For the remaining classes / C&(&x)1 the Schur-Frobenius index for each faithful irreducible character of A.
S-F
2 . 2932537 . 11 11 2. 2. 11 2. 7 2. 52 2 . 23325 2 . 233 2 . 283 . 5 2 . 24 2. 3.5 2 . 2=3 ’ 52 2 . 2a3a 2 . 2%3 2 * 2s3 . 5 2.2e 5 2.2= 5 2.2= 5 2 . 22 53
I WI
Faithful
CHARACTERS
OF THE
COVERING
4. CONCLUDING
GROUP
143
REMARKS
McKay and Wales (4) mention that fi can have no faithful ordinary character of degree less than 56. It follows from the above that E? has a faithful irreducible character of degree 56 which is afforded by a rational representation. No other faithful irreducible character of A is afforded by any real representation. The rationality of the 56-dimensional representation shows that it is associated with a 56-dimensional rational lattice. This lattice could conceivably have A as a proper subgroup of its full group of automorphisms but determining this automorphism group appears to be a very difficult problem. It is known that the 2-fold proper covering 6 of the simple group PSU,(2) of order 215365 . 7 11 has a 56-dimensional faithful rational representation which remains irreducible when restricted to a subgroup of 0 isomorphic to ii?Iz, . This restriction coincides with the restriction to A&z of the above 56-dimensional representation of A. This suggests the possibility that there might be a very large (and presumably new) perfect finite group with a 56-dimensional rational representation generated by the above representations of i? and I? and such that the intersection of 0 and fi in this group is isomorphic to tiz, . Th e author is presently exploring this and some other related problems.
REFERENCES I. R. BRAUER, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. 2. 63 (1956), 406-444. 2. N. BURCOYNE AND P. FONG, A correction to ‘The Schur multipliers of the Mathieu groups,’ Nagoya Math. J. 31 (1968), 297-304. 3. J. S. FRAME, Computation of characters of the H&man-Sims group and its Automorphism group, J. Algebra 20 (1972), 32&349. 4. JOHN MCKAY AND DAVID WALES, The multiplier of the H&man-Sims simple group, Bull. London Math. Sot. 3 (1971), 283-285. 5. SPYROS S. MAGLIVERAS, The subgroup structure of the H&man-Sims simple group, Bull. AIMS 77 (1971), 535-539.
481/33/r-10