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A Note on the Decomposition Numbers ofSp(4, q)
JOURNAL OF ALGEBRA ARTICLE NO.
186, 105]112 Ž1996.
0364
A Note on the Decomposition Numbers of SpŽ 4, q . Katsushi Waki Department of Information Science, Hirosaki Uni¨ ersity, 3, Bunkyo-cho, Hirosaki, 036 Japan Communicated by Walter Feit Received December 18, 1995
1. INTRODUCTION Let p be a prime and G the symplectic group SpŽ4, q . where q s p n. The decomposition numbers of G in characteristics other than p are almost determined by White w5, 6x. But in case the characteristic divides q q 1, one position in the decomposition matrix of the principal block remains as a variable. We will determine this variable a under some conditions. All calculations of the scalar products in Section 6 are done by ‘‘Mathematica’’ w3x.
2. NOTATION An odd prime r which divides q q 1 is fixed in this paper. Thus there are numbers d and s such that q q 1 s r d s and s is not divisible by r. Let P be the parabolic subgroup of G. The order of P is q 4 Ž q y 1. 2 Ž q q 1.. The subgroup H of G which is denoted by K in Srinivasan w2x is isomorphic to SLŽ2, q . = SLŽ2, q .. The order of H is q 2 Ž q 2 y 1. 2 . We can see the fusion map between G and H in Table III. We use the same notation of White w5, 6x for the ordinary characters of G. The irreducible ordinary characters of P can be found in Enomoto w1x. The ordinary characters and conjugacy classes of G are given in Srinivasan w2x and Enomoto w1x. The group Cn denotes the cyclic group of order n. 105 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
106
KATSUSHI WAKI
3. A RESULT OF WHITE THEOREM 3.1 ŽWhite w5, 6x.. The decomposition matrix for the principal r-block is as follows. The unknown entry a is an integer satisfying 1 F a F Ž q y 1.r2 Ž or qr2 if p s 2.. Degrees
Chars.
No. of chars.
1 q Ž q y 1. 2r2 q Ž q 2 q 1.r2 q Ž q 2 q 1.r2 q4 Ž q 2 q 1.Ž q y 1. 2 Ž q 2 q 1.Ž q y 1. q Ž q 2 q 1.Ž q y 1. Ž q 2 q 1.Ž q y 1. q Ž q 2 q 1.Ž q y 1.
1G Ž x 1 . u 10 Ž x 5 . u 11 Ž x 3 . u 12 Ž x4 . u 13 Ž x6 . x4 Ž x 19 . x6 Ž x 9 . x 7 Ž x 10 . j 1 Ž x 13 . j 1X Ž x 14 .
1 1 1 1 1
1
a ay2 1 ay1 1 ay1
1
1 1 1 1
1 1
1 1 1 1
1 1 1 1 1 Ž r d y 1.Ž r d y 3.r8 Ž r d y 1.r2 Ž r d y 1.r2 Ž r d y 1.r2 Ž r d y 1.r2
The parentheses in the table mean the name of characters in case p s 2.
4. A MAIN RESULT THEOREM 4.1. Ži. Žii.
In Theorem 3.1, we ha¨ e the following.
If p is e¨ en than a F Ž r d y 1.r2. If p is odd then a F Ž r d y 1. 2rŽ r d q 1..
COROLLARY 4.1. From Theorem 4.1, we can determine the ¨ alue of a in the two special cases. Ži. If a 3-Sylow subgroup of SpŽ4, q . is isomorphic to the elementary abelian group of order 3 2 , then a s 1 for r s 3. Žii. If a 5-Sylow subgroup of SpŽ4, q . is isomorphic to the elementary abelian group of order 5 2 , then a s 2 for r s 5. Proof. It is easy to check that an r-Sylow to Cr d = Cr d . Thus the assumption means r d a F 1 if r d s 3. Since a G 1, a must be 1. If a F 2 and Theorem 4.1Žii. shows a F 8r3. a G 2. Thus corollary is proved.
subgroup of G is isomorphic is 3 or 5. From Theorem 4.1, r d s 5, Theorem 4.1Ži. shows In case r d is bigger than 3,
Remark 4.1. If either r / 3 or d / 1, then d G 2. Thus Theorem 4.1Ži. is the only case that a s 1.
DECOMPOSITION NUMBERS OF
SpŽ4, q .
107
5. PROOF OF THEOREM 4.1Ži. We fix the prime p s 2 in this section. Since an r-Sylow subgroup of P is a cycle of order r d, it’s easy to determine the r-decomposition matrix of P. We can check the next two lemmas from the character table of SpŽ4, q . in Enomoto w1x. LEMMA 5.1. There are irreducible characters u 3 Ž0., u 3 Ž1., x6 Ž k . of P which make an r-block of P where k s s, 2 s, . . . , ŽŽ r d y 1.r2. s. Moreo¨ er there are 2 irreducible Brauer characters w 1 and w 2 and the decomposition matrix of this block is as follows. Degrees
Chars.
w1
q Ž q y 1 r2 q Ž q y 1. 2r2 q Ž q y 1. 2 .. .
u 3 Ž0. u 3 Ž1. x6 Ž s . .. .
1
.2
x6
q Ž q y 1. 2
ž
rd y 1 2
s
/
w2
1 .. .
1 1 .. .
1
1
LEMMA 5.2. The restriction of the character x 5 of G to P is equal to the character u 3 Ž1. of P. Proof of Theorem 4.1Ži.. From Table I 1
Ž u 3 Ž 1 . , x6 x P . s < P <
ž
q Ž q y 1.
2
= q4 y
2
q 2
= q 3 Ž q y 1. = Ž q y 1. = q
s0 1
Ž x6 Ž k . , x6 x P . s < P <
ž
2
q Ž q y 1. = q 4 y q 3 Ž q y 1. = Ž q y 1. qr2
ž Ý // ž Ý // bi k
=q
is1
s
1 < P<
qr2
ž
2
q Ž q y 1. y q Ž q y 1. 5
4
2
j i k q jyi k
is1
/
a of classes
ik
qj
yi k
1 q y1 qy1 Ž q y 1.Ž q 2 y 1. q Ž q 2 y 1. q Ž q y 1.Ž q 2 y 1. q 2 Ž q y 1.Ž q 2 y 1.r2 q 2 Ž q y 1.Ž q 2 y 1.r2 q 4 Ž q q 1. q 4 Ž q y 1. q 3 Ž q q 1. q 3 Ž q q 1. q3 q 3 Ž q y 1. q 3 Ž q 2 y 1. q 3 Ž q 2 y 1. q 3 Ž q 2 y 1. q 3 Ž q y 1. 2 2
a of elements in this class
where j [ expŽ2p'y 1 rŽ q q 1...
1 1 1 1 1 1 1 1 Ž q y 2.Ž q y 4.r2 q Ž q y 2.r2 qy2 Ž q y 2.r2 qy2 qr2 qy2 Ž q y 2.r2 qy2 qr2
Note. bi k s j
A1 A2 A 31 A 32 A 31 A 32 A 41 A 42 B1Ž i, j . B2 Ž i . C1Ž i . C2 Ž i . C2 Ž i . C4 Ž i . D 1Ž i . D2 Ž i. D2 Ž i. D4 Ž i .
Class of G
u 3 Ž1.
x6 Ž k .
qy1
y1
qy1
y1
ybi k
Ž q y 1. bi k
q Ž q y 1. 2r2 q Ž q y 1. 2r2 q Ž q y 1. 2 yq Ž q y 1.r2 yq Ž q y 1.r2 yq Ž q y 1. yq Ž q y 1.r2 yq Ž q y 1.r2 yq Ž q y 1. qr2 qr2 q q Ž q y 1.r2 yq Ž q y 1.r2 yqr2 qr2 yqr2 qr2 qr2 yqr2
u 3 Ž0.
TABLE I The Restriction to P in Case p s 2
1 y1 q q q yq
q4
x6
108 KATSUSHI WAKI
SpŽ4, q .
DECOMPOSITION NUMBERS OF
s s s
109
qr2
1 < P< 1 < P< 1 < P<
ž ž
2
q 5 Ž q y 1. y q 4 Ž q y 1.
2
žÝ
Žj k.
i
q Žj k.
qq1yi
is1 q
2
q Ž q y 1. y q Ž q y 1. 5
4
2
//
i
ž Ý Ž . // jk
is1
Ž q 5 Ž q y 1. 2 q q 4 Ž q y 1. 2 .
s1 since j k / 1 for each k s 2, 2 s, . . . , ŽŽ r d y 1.r2. s. These scalar products show that the multiplicity of w 2 in the restricted character x6 as a Brauer character is Ž r d y 1.r2 from Lemma 5.1. By the way, the number a is the multiplicity of x 5 in x6 as a Brauer character. Thus Theorem 4.1Ži. follows from Lemma 5.2.
6. PROOF OF THEOREM 4.1Žii. We fix an odd prime p. Let H s Ha = Hb ( SLŽ2, q . = SLŽ2, q .. We can find the character table of SLŽ2, q . in Table II. There is an r-block ˜ b1 in SLŽ2, q . such that this block has the following decomposition matrix, Degrees
Chars.
w1
Ž q y 1.r2 Ž q y 1.r2
x ˜5 x ˜6
1
qy1
x ˜8 Ž k .
1
w2
No. of chars.
1
1 1 rd y 1
1
2
where k g I [ sr2, sr2 q s, . . . , sr2 q ŽŽ r d y 3.r2. s4 . There is an r-block b1 which is constructed by the irreducible characters of H
xŽ5, 5. , xŽ5 , 6. , xŽ5 , 8. Ž k . , xŽ6 , 5. , xŽ6 , 6. , xŽ6 , 8. Ž k . , xŽ8, 5. Ž k . , xŽ8 , 6. Ž k . , xŽ8 , 8. Ž k , l . 4 , where xŽ i, j. s x ˜i Ž h a . x˜j Ž h b . for each h a g Ha and h b g Hb and k, l g I. We can get the following lemma from Tables II and III.
y 2 2 Ž q q 1. Žy1. k Ž q q 1. Ž q y 1. Žy1. k Ž q y 1.
'
2 1 y1
'
2 1 y1
'
2 y1 q eq
'
2 y1 y eq
'
'
2 1 q eq
1 y eq
1 0
P2 1 Ž q 2 y 1.r2
2 y1 y eq
2 y1 q eq
'
'
2 1 y eq
1 q eq
1 0
P1 1 Ž q 2 y 1.r2 1 0
P2 Z 1 Ž q 2 y 1.r2
2 2 2
2 Žy1. k yŽy1. k
2 Žy1. k yŽy1. k
Ž 1 q 'eq . e Ž 1 y 'eq . e
2
Ž 1 y 'eq . e Ž 1 q 'eq . e
2
Ž 1 y 'eq . e Ž 1 q 'eq . e
2
Ž 1 q 'eq . e Ž 1 y 'eq . e
1 0
P1 Z 1 Ž q 2 y 1.r2
Note. e [ Žy1.Ž qy1.r2 ; z [ expŽ2p'y 1 rŽ q y 1..; j [ expŽ2p'y 1 rŽ q q 1...
x ˜7 Ž k . x ˜8 Ž k .
x ˜6
x ˜5 2 Ž q y 1. e
2 Ž q y 1. e
2 qy1
x ˜4 y
2 Ž q q 1. e
2 qq1
2 qy1
Ž q q 1. e
qq1
x˜3
1 q
1 q
x ˜1 x˜2
Z 1 1
I 1 1
SL 2 Ž q . a of classes a of elem.
TABLE II The Character Table of SLŽ2, q . in Case q Is Odd
yŽy1. i
yŽy1. i
0
0
1 y1
z i k q zyi k 0 0 yj i k y jyi k
0
0
Žy1. i
Žy1. i
1 1
Ci Di Ž q y 3.r2 Ž q y 1.r2 q Ž q q 1. q Ž q y 1.
110 KATSUSHI WAKI
DECOMPOSITION NUMBERS OF
SpŽ4, q .
111
TABLE III The Fusion Map between G and H in Case q Is Odd I
Z
P1
P2
I A1 D1 A 21 A 22 X Z D1 A1 D 23 D 24 P1 A 21 D 23 A 31 A 32 P2 A 22 D 24 A 32 A 31 X P1 Z D 21 A 21 D 31 D 33 X P2 Z D 22 A 22 D 32 D 34 X Ci C3 Ž i . C3 Ž i . C41Ž i . C42 Ž i . X Dj C1Ž j . C1Ž j . C21Ž j . C22 Ž j .
P1 Z
P2 Z
Ck
Dl
D 21 X A 21 D 31 D 33 X A 31 X A 32 X C41Ž i . X C21Ž j .
D 22 X A 22 D 32 D 34 X A 32 X A 31 X C42 Ž i . X C22 Ž j .
C3 Ž k . X C3 Ž k . C41Ž k . C42 Ž k . X C41Ž k . X C42 Ž k . B3 Ž i, k . B5 Ž k, j .
C1Ž l . X C1Ž l . C21Ž l . C22 Ž l . X C21Ž l . X C22 Ž l . B5 Ž i, l . B4 Ž j, l .
Note. These entires of the table are names of the conjugacy classes of G in case q ' 1 mod 4. Note that B3 Ž i, i . means B8 Ž i . and B4 Ž j, j . means X X B6 Ž j .. We should exchange A 31 to A 32 and A 31 to A 32 in case q ' 3 mod 4.
LEMMA 6.1.
Ž xŽ5, 6. , u 10 x H . s 1
Ž xŽ5, 8. Ž k . , u 10 x H . s 0 Ž xŽ8, 6. Ž k . , u 10 x H . s 0 Ž xŽ8, 8. Ž k , l . , u 10 x H . s
½
1, 0,
ksl k/l
1, 0,
ksl k/l
Ž xŽ5, 6. , u 11 x H . s 0
Ž xŽ5, 8. Ž k . , u 11 x H . s 0 Ž xŽ8, 6. Ž k . , u 11 x H . s 0 Ž xŽ8, 8. Ž k , l . , u 11 x H . s
½
Ž xŽ5, 6. , u 12 x H . s 0
Ž xŽ5 , 8. Ž k . , u 12 x H . s 0 Ž xŽ8 , 6. Ž k . , u 12 x H . s 0 Ž xŽ8, 8. Ž k, l . , u 12 x H . s 0 Ž xŽ5, 6. , u 13 x H . s 1
Ž xŽ5, 8. Ž k . , u 13 x H . s 1
112
KATSUSHI WAKI
Ž xŽ8, 6. Ž k . , u 13 x H . s 1 Ž xŽ8, 8. Ž k , l . , u 13 x H . s
½
1, 2,
ksl k / l,
where k, l g I. Proof of Theorem 4.1Žii.. It is easy to define the decomposition matrix of b1 by ˜ b1. From the above lemma, the multiplicities of the Brauer character wŽ1, 2. [ xŽ5, 6. of H in u 10 x H , u 11 x H , u 12 x H , and u 13 x H are Ž r d q 1.r2, Ž r d y 1.r2, 0, and r d Ž r d y 1.r2, respectively. Thus we can get r d Ž r d y 1.r2 G a ŽŽ r d q 1.r2. q Ž r d y 1.r2.
REFERENCES 1. H. Enomoto, The characters of the finite symplectic group SpŽ4, q ., q s 2 f , Osaka J. Math. 9, Ž1972., 75]94. 2. B. Srinivasan, The characters of the finite symplectic group SpŽ4, q ., Trans. Amer. Math. Soc. 131 Ž1968., 488]525. 3. S. Wolfram, ‘‘Mathematica: A System for Doing Mathematics by Computer,’’ Addison]Wesley, Reading, MA, 1988. 4. D. L. White, ‘‘The 2-Blocks and Decomposition Numbers of SpŽ4, q ., q Odd,’’ Ph.D. Thesis, Yale University, 1987. 5. D. L. White, Decomposition numbers of SpŽ4, q . for primes dividing q " 1, J. Algebra 132 Ž1990., 488]500. 6. D. L. White, Decomposition numbers of Sp4 Ž2 a . in odd characteristics, J. Algebra 177 Ž1995., 264]276.
186, 105]112 Ž1996.
0364
A Note on the Decomposition Numbers of SpŽ 4, q . Katsushi Waki Department of Information Science, Hirosaki Uni¨ ersity, 3, Bunkyo-cho, Hirosaki, 036 Japan Communicated by Walter Feit Received December 18, 1995
1. INTRODUCTION Let p be a prime and G the symplectic group SpŽ4, q . where q s p n. The decomposition numbers of G in characteristics other than p are almost determined by White w5, 6x. But in case the characteristic divides q q 1, one position in the decomposition matrix of the principal block remains as a variable. We will determine this variable a under some conditions. All calculations of the scalar products in Section 6 are done by ‘‘Mathematica’’ w3x.
2. NOTATION An odd prime r which divides q q 1 is fixed in this paper. Thus there are numbers d and s such that q q 1 s r d s and s is not divisible by r. Let P be the parabolic subgroup of G. The order of P is q 4 Ž q y 1. 2 Ž q q 1.. The subgroup H of G which is denoted by K in Srinivasan w2x is isomorphic to SLŽ2, q . = SLŽ2, q .. The order of H is q 2 Ž q 2 y 1. 2 . We can see the fusion map between G and H in Table III. We use the same notation of White w5, 6x for the ordinary characters of G. The irreducible ordinary characters of P can be found in Enomoto w1x. The ordinary characters and conjugacy classes of G are given in Srinivasan w2x and Enomoto w1x. The group Cn denotes the cyclic group of order n. 105 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
106
KATSUSHI WAKI
3. A RESULT OF WHITE THEOREM 3.1 ŽWhite w5, 6x.. The decomposition matrix for the principal r-block is as follows. The unknown entry a is an integer satisfying 1 F a F Ž q y 1.r2 Ž or qr2 if p s 2.. Degrees
Chars.
No. of chars.
1 q Ž q y 1. 2r2 q Ž q 2 q 1.r2 q Ž q 2 q 1.r2 q4 Ž q 2 q 1.Ž q y 1. 2 Ž q 2 q 1.Ž q y 1. q Ž q 2 q 1.Ž q y 1. Ž q 2 q 1.Ž q y 1. q Ž q 2 q 1.Ž q y 1.
1G Ž x 1 . u 10 Ž x 5 . u 11 Ž x 3 . u 12 Ž x4 . u 13 Ž x6 . x4 Ž x 19 . x6 Ž x 9 . x 7 Ž x 10 . j 1 Ž x 13 . j 1X Ž x 14 .
1 1 1 1 1
1
a ay2 1 ay1 1 ay1
1
1 1 1 1
1 1
1 1 1 1
1 1 1 1 1 Ž r d y 1.Ž r d y 3.r8 Ž r d y 1.r2 Ž r d y 1.r2 Ž r d y 1.r2 Ž r d y 1.r2
The parentheses in the table mean the name of characters in case p s 2.
4. A MAIN RESULT THEOREM 4.1. Ži. Žii.
In Theorem 3.1, we ha¨ e the following.
If p is e¨ en than a F Ž r d y 1.r2. If p is odd then a F Ž r d y 1. 2rŽ r d q 1..
COROLLARY 4.1. From Theorem 4.1, we can determine the ¨ alue of a in the two special cases. Ži. If a 3-Sylow subgroup of SpŽ4, q . is isomorphic to the elementary abelian group of order 3 2 , then a s 1 for r s 3. Žii. If a 5-Sylow subgroup of SpŽ4, q . is isomorphic to the elementary abelian group of order 5 2 , then a s 2 for r s 5. Proof. It is easy to check that an r-Sylow to Cr d = Cr d . Thus the assumption means r d a F 1 if r d s 3. Since a G 1, a must be 1. If a F 2 and Theorem 4.1Žii. shows a F 8r3. a G 2. Thus corollary is proved.
subgroup of G is isomorphic is 3 or 5. From Theorem 4.1, r d s 5, Theorem 4.1Ži. shows In case r d is bigger than 3,
Remark 4.1. If either r / 3 or d / 1, then d G 2. Thus Theorem 4.1Ži. is the only case that a s 1.
DECOMPOSITION NUMBERS OF
SpŽ4, q .
107
5. PROOF OF THEOREM 4.1Ži. We fix the prime p s 2 in this section. Since an r-Sylow subgroup of P is a cycle of order r d, it’s easy to determine the r-decomposition matrix of P. We can check the next two lemmas from the character table of SpŽ4, q . in Enomoto w1x. LEMMA 5.1. There are irreducible characters u 3 Ž0., u 3 Ž1., x6 Ž k . of P which make an r-block of P where k s s, 2 s, . . . , ŽŽ r d y 1.r2. s. Moreo¨ er there are 2 irreducible Brauer characters w 1 and w 2 and the decomposition matrix of this block is as follows. Degrees
Chars.
w1
q Ž q y 1 r2 q Ž q y 1. 2r2 q Ž q y 1. 2 .. .
u 3 Ž0. u 3 Ž1. x6 Ž s . .. .
1
.2
x6
q Ž q y 1. 2
ž
rd y 1 2
s
/
w2
1 .. .
1 1 .. .
1
1
LEMMA 5.2. The restriction of the character x 5 of G to P is equal to the character u 3 Ž1. of P. Proof of Theorem 4.1Ži.. From Table I 1
Ž u 3 Ž 1 . , x6 x P . s < P <
ž
q Ž q y 1.
2
= q4 y
2
q 2
= q 3 Ž q y 1. = Ž q y 1. = q
s0 1
Ž x6 Ž k . , x6 x P . s < P <
ž
2
q Ž q y 1. = q 4 y q 3 Ž q y 1. = Ž q y 1. qr2
ž Ý // ž Ý // bi k
=q
is1
s
1 < P<
qr2
ž
2
q Ž q y 1. y q Ž q y 1. 5
4
2
j i k q jyi k
is1
/
a of classes
ik
qj
yi k
1 q y1 qy1 Ž q y 1.Ž q 2 y 1. q Ž q 2 y 1. q Ž q y 1.Ž q 2 y 1. q 2 Ž q y 1.Ž q 2 y 1.r2 q 2 Ž q y 1.Ž q 2 y 1.r2 q 4 Ž q q 1. q 4 Ž q y 1. q 3 Ž q q 1. q 3 Ž q q 1. q3 q 3 Ž q y 1. q 3 Ž q 2 y 1. q 3 Ž q 2 y 1. q 3 Ž q 2 y 1. q 3 Ž q y 1. 2 2
a of elements in this class
where j [ expŽ2p'y 1 rŽ q q 1...
1 1 1 1 1 1 1 1 Ž q y 2.Ž q y 4.r2 q Ž q y 2.r2 qy2 Ž q y 2.r2 qy2 qr2 qy2 Ž q y 2.r2 qy2 qr2
Note. bi k s j
A1 A2 A 31 A 32 A 31 A 32 A 41 A 42 B1Ž i, j . B2 Ž i . C1Ž i . C2 Ž i . C2 Ž i . C4 Ž i . D 1Ž i . D2 Ž i. D2 Ž i. D4 Ž i .
Class of G
u 3 Ž1.
x6 Ž k .
qy1
y1
qy1
y1
ybi k
Ž q y 1. bi k
q Ž q y 1. 2r2 q Ž q y 1. 2r2 q Ž q y 1. 2 yq Ž q y 1.r2 yq Ž q y 1.r2 yq Ž q y 1. yq Ž q y 1.r2 yq Ž q y 1.r2 yq Ž q y 1. qr2 qr2 q q Ž q y 1.r2 yq Ž q y 1.r2 yqr2 qr2 yqr2 qr2 qr2 yqr2
u 3 Ž0.
TABLE I The Restriction to P in Case p s 2
1 y1 q q q yq
q4
x6
108 KATSUSHI WAKI
SpŽ4, q .
DECOMPOSITION NUMBERS OF
s s s
109
qr2
1 < P< 1 < P< 1 < P<
ž ž
2
q 5 Ž q y 1. y q 4 Ž q y 1.
2
žÝ
Žj k.
i
q Žj k.
qq1yi
is1 q
2
q Ž q y 1. y q Ž q y 1. 5
4
2
//
i
ž Ý Ž . // jk
is1
Ž q 5 Ž q y 1. 2 q q 4 Ž q y 1. 2 .
s1 since j k / 1 for each k s 2, 2 s, . . . , ŽŽ r d y 1.r2. s. These scalar products show that the multiplicity of w 2 in the restricted character x6 as a Brauer character is Ž r d y 1.r2 from Lemma 5.1. By the way, the number a is the multiplicity of x 5 in x6 as a Brauer character. Thus Theorem 4.1Ži. follows from Lemma 5.2.
6. PROOF OF THEOREM 4.1Žii. We fix an odd prime p. Let H s Ha = Hb ( SLŽ2, q . = SLŽ2, q .. We can find the character table of SLŽ2, q . in Table II. There is an r-block ˜ b1 in SLŽ2, q . such that this block has the following decomposition matrix, Degrees
Chars.
w1
Ž q y 1.r2 Ž q y 1.r2
x ˜5 x ˜6
1
qy1
x ˜8 Ž k .
1
w2
No. of chars.
1
1 1 rd y 1
1
2
where k g I [ sr2, sr2 q s, . . . , sr2 q ŽŽ r d y 3.r2. s4 . There is an r-block b1 which is constructed by the irreducible characters of H
xŽ5, 5. , xŽ5 , 6. , xŽ5 , 8. Ž k . , xŽ6 , 5. , xŽ6 , 6. , xŽ6 , 8. Ž k . , xŽ8, 5. Ž k . , xŽ8 , 6. Ž k . , xŽ8 , 8. Ž k , l . 4 , where xŽ i, j. s x ˜i Ž h a . x˜j Ž h b . for each h a g Ha and h b g Hb and k, l g I. We can get the following lemma from Tables II and III.
y 2 2 Ž q q 1. Žy1. k Ž q q 1. Ž q y 1. Žy1. k Ž q y 1.
'
2 1 y1
'
2 1 y1
'
2 y1 q eq
'
2 y1 y eq
'
'
2 1 q eq
1 y eq
1 0
P2 1 Ž q 2 y 1.r2
2 y1 y eq
2 y1 q eq
'
'
2 1 y eq
1 q eq
1 0
P1 1 Ž q 2 y 1.r2 1 0
P2 Z 1 Ž q 2 y 1.r2
2 2 2
2 Žy1. k yŽy1. k
2 Žy1. k yŽy1. k
Ž 1 q 'eq . e Ž 1 y 'eq . e
2
Ž 1 y 'eq . e Ž 1 q 'eq . e
2
Ž 1 y 'eq . e Ž 1 q 'eq . e
2
Ž 1 q 'eq . e Ž 1 y 'eq . e
1 0
P1 Z 1 Ž q 2 y 1.r2
Note. e [ Žy1.Ž qy1.r2 ; z [ expŽ2p'y 1 rŽ q y 1..; j [ expŽ2p'y 1 rŽ q q 1...
x ˜7 Ž k . x ˜8 Ž k .
x ˜6
x ˜5 2 Ž q y 1. e
2 Ž q y 1. e
2 qy1
x ˜4 y
2 Ž q q 1. e
2 qq1
2 qy1
Ž q q 1. e
qq1
x˜3
1 q
1 q
x ˜1 x˜2
Z 1 1
I 1 1
SL 2 Ž q . a of classes a of elem.
TABLE II The Character Table of SLŽ2, q . in Case q Is Odd
yŽy1. i
yŽy1. i
0
0
1 y1
z i k q zyi k 0 0 yj i k y jyi k
0
0
Žy1. i
Žy1. i
1 1
Ci Di Ž q y 3.r2 Ž q y 1.r2 q Ž q q 1. q Ž q y 1.
110 KATSUSHI WAKI
DECOMPOSITION NUMBERS OF
SpŽ4, q .
111
TABLE III The Fusion Map between G and H in Case q Is Odd I
Z
P1
P2
I A1 D1 A 21 A 22 X Z D1 A1 D 23 D 24 P1 A 21 D 23 A 31 A 32 P2 A 22 D 24 A 32 A 31 X P1 Z D 21 A 21 D 31 D 33 X P2 Z D 22 A 22 D 32 D 34 X Ci C3 Ž i . C3 Ž i . C41Ž i . C42 Ž i . X Dj C1Ž j . C1Ž j . C21Ž j . C22 Ž j .
P1 Z
P2 Z
Ck
Dl
D 21 X A 21 D 31 D 33 X A 31 X A 32 X C41Ž i . X C21Ž j .
D 22 X A 22 D 32 D 34 X A 32 X A 31 X C42 Ž i . X C22 Ž j .
C3 Ž k . X C3 Ž k . C41Ž k . C42 Ž k . X C41Ž k . X C42 Ž k . B3 Ž i, k . B5 Ž k, j .
C1Ž l . X C1Ž l . C21Ž l . C22 Ž l . X C21Ž l . X C22 Ž l . B5 Ž i, l . B4 Ž j, l .
Note. These entires of the table are names of the conjugacy classes of G in case q ' 1 mod 4. Note that B3 Ž i, i . means B8 Ž i . and B4 Ž j, j . means X X B6 Ž j .. We should exchange A 31 to A 32 and A 31 to A 32 in case q ' 3 mod 4.
LEMMA 6.1.
Ž xŽ5, 6. , u 10 x H . s 1
Ž xŽ5, 8. Ž k . , u 10 x H . s 0 Ž xŽ8, 6. Ž k . , u 10 x H . s 0 Ž xŽ8, 8. Ž k , l . , u 10 x H . s
½
1, 0,
ksl k/l
1, 0,
ksl k/l
Ž xŽ5, 6. , u 11 x H . s 0
Ž xŽ5, 8. Ž k . , u 11 x H . s 0 Ž xŽ8, 6. Ž k . , u 11 x H . s 0 Ž xŽ8, 8. Ž k , l . , u 11 x H . s
½
Ž xŽ5, 6. , u 12 x H . s 0
Ž xŽ5 , 8. Ž k . , u 12 x H . s 0 Ž xŽ8 , 6. Ž k . , u 12 x H . s 0 Ž xŽ8, 8. Ž k, l . , u 12 x H . s 0 Ž xŽ5, 6. , u 13 x H . s 1
Ž xŽ5, 8. Ž k . , u 13 x H . s 1
112
KATSUSHI WAKI
Ž xŽ8, 6. Ž k . , u 13 x H . s 1 Ž xŽ8, 8. Ž k , l . , u 13 x H . s
½
1, 2,
ksl k / l,
where k, l g I. Proof of Theorem 4.1Žii.. It is easy to define the decomposition matrix of b1 by ˜ b1. From the above lemma, the multiplicities of the Brauer character wŽ1, 2. [ xŽ5, 6. of H in u 10 x H , u 11 x H , u 12 x H , and u 13 x H are Ž r d q 1.r2, Ž r d y 1.r2, 0, and r d Ž r d y 1.r2, respectively. Thus we can get r d Ž r d y 1.r2 G a ŽŽ r d q 1.r2. q Ž r d y 1.r2.
REFERENCES 1. H. Enomoto, The characters of the finite symplectic group SpŽ4, q ., q s 2 f , Osaka J. Math. 9, Ž1972., 75]94. 2. B. Srinivasan, The characters of the finite symplectic group SpŽ4, q ., Trans. Amer. Math. Soc. 131 Ž1968., 488]525. 3. S. Wolfram, ‘‘Mathematica: A System for Doing Mathematics by Computer,’’ Addison]Wesley, Reading, MA, 1988. 4. D. L. White, ‘‘The 2-Blocks and Decomposition Numbers of SpŽ4, q ., q Odd,’’ Ph.D. Thesis, Yale University, 1987. 5. D. L. White, Decomposition numbers of SpŽ4, q . for primes dividing q " 1, J. Algebra 132 Ž1990., 488]500. 6. D. L. White, Decomposition numbers of Sp4 Ž2 a . in odd characteristics, J. Algebra 177 Ž1995., 264]276.