decomposition matrices for the spin characters of modulo...

Basrah Journal of Science Vol.37(2), 253-261,2019

253

Decomposition matrices for the spin characters of Modulo 11

Ahmed H. Jassim and Marwa M. Jawad

Department of Mathematics, College of Science, University of Basrah, Iraq

E-mail: [email protected]

Doi 10.29072/basjs.20190208

Abstract

We found in this paper the 11-decomposition matrices for the spin characters of , which

are a relationship between modular and ordinary characters

.

1. Introduction

For any group there are three kinds of characters ordinary, modular (for a given prime

), and projective (for called spin). The decomposition matrix is the relation between the

ordinary and modular characters for a given prime (see[1], [2]). Spin characters of can

be written as a liner combination with non-negative coefficients of the irreducible spin

characters [2]. The case in found by M. M. Jawad [3]. In this paper we use the

techniques as given in [4]. The matrix for this spin from degree (247,207)[2], [5]. There are

50 blocks, the blocks are of defect two, the blocks ,…, are of defect one the

others of defect zero. The notations used in this section can be found in [6].

Lemma (1). Brauer trees of blocks [7] are:

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

mailto:[email protected]

A.H. Jassim & M.M. Jawad Decomposition matrices for the spin ….

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩

respectively.

Proof. For , we use the ( ̅)-inducing p.i.s. of , , , ,…, , of to

we get , and since it's associate(see [2]), so ⟨ ⟩ ⟨ ⟩ it

follows that splits or there are two columns:

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ,

so that * +[7]. Let . Since ⟨ ⟩ ⟨ ⟩ has no

i.m.s, so . The same way, we find . But deg only when

. It follows that . Since of defect one. So

[3]. In the same way , we use ( ̅)-inducing and to get . Since it's

associate[5]. So divided or there are columns:

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ , * +. Let , and ⟨ ⟩ ⟨ ⟩ has


255

no i.m.s. So . In the same way we get . Then deg only

when so .In the same we divided to

respectively. But of defect one so splits to . In the way we

prove the blocks , . For , we use the ( ̅)-inducing to get on

. Since it's associate so divided or there such that:

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ,

* +. In the same above deg only when

so . But it's of defect one so . In the same way mentional

above we discuss the block and . The other we found it by use the ( ̅)-inducing

from to directly. Finally we find the Brauer trees for the blocks of

the decomposition that we found [7].

Lemma (2). Decomposition matrix for the block of a double [7] is a table (1).

⟨ ⟩ 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 2 1 1 2 2

⟨ ⟩ 1 1 1 2 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1

table (1)

Proof. Using ( ̅) inducing of p.i.s.

of to , we got respectively. Since

( ) ( ) is not p.s., so , consequently we get the table (1).


Theorem (3). Decomposition matrix for spin character of the block is table (2).

⟨ ⟩ 1

⟨ ⟩ 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1

⟨ ⟩ 1

table (2)

Proof. By using ( ̅)-inducing of to , we get approximation matrix in table (3).


257

⟨ ⟩ 1

⟨ ⟩ 1

⟨ ⟩ 2 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 2 1 1 1 1 ⟨ ⟩ 2 1 1 1 1

⟨ ⟩ 2 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 2 1 1

⟨ ⟩ 1 2 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 1 1 1 1

⟨ ⟩ 2 1 1 1 1 1 1

⟨ ⟩ 2 1 1 1 1 1 1

⟨ ⟩ 1 1 2 1 1 2 ⟨ ⟩ 2 2 1 1 2 1 1 ⟨ ⟩ 2 2 1 1 1 1 ⟨ ⟩ 2 1 1 ⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1 1 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1 1

⟨ ⟩ 1

⟨ ⟩ 1


table(3)

Since( ) ( ) is not p.s., so . However we prove that by the way

of contradiction. Let (⟨ ⟩ ⟨ ⟩) is m.s. for but ⟨ ⟩ ⟨ ⟩ ( ) is

not m.s., so .

From [3] must split to , . Also ⟨ ⟩ ⟨ ⟩ on ( )-regular classes

(see[4]). So or there are two columns. If there are two columns then we

have the approximation matrix as above, to describe it's such that ⟨ ⟩ has 4 of

i.m.s. and, from the table(3) we get * +, in the same way we get

* +, * +, * +

, * +, * +. Take * + (if then we

have contradiction) and, since ⟨ ⟩ ⟨ ⟩ has no i.m.s, so we have

by counting the intersections we get on are equal to zero,

and since inducing m.s. is m.s. [8] we have:

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (1)

(⟨ ⟩ ⟨ ⟩) ( ) (2)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (3)

(⟨ ⟩ ⟨ ⟩ ) ( ) (4)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (5)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (6)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (7)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (8)

then deg only when , * +

* + or * + * +. So .

Also ⟨ ⟩ ⟨ ⟩ on ( ) regular classes . It follows that or splits or there

are . Let , and use the same way above, we get:

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩,

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

Since

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (9)

(⟨ ⟩ ⟨ ⟩ ) ( ) (10)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (11)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (12)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (13)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( )

(14)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (15)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( )

, (16)

then deg only when . So .


259

Also ⟨ ⟩ ⟨ ⟩ on ( ) regular classes. Suppose that . Using the

same technic then we get deg only when . So are split to

respectively.

For , we have ⟨ ⟩ ⟨ ⟩ . So it's splits or there are other . So let * +.

By restricting and inducing, we get ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩, ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ and since

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (17)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) , (18)

then deg only when , * +. So

.

For , we have ⟨ ⟩ ⟨ ⟩ . Let * +. By restricting and inducing, we

get: ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩, ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ and, since

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( )

(19)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (20)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (21)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (22)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) , (23)

then deg only when , such that * +

* +, so .

Also ⟨ ⟩ ⟨ ⟩ . Let * +. By restricting and inducing we get:

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ , ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ and since

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (24)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (25)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (26)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (27)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( )

(28)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (29)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) , (30)

then deg only when , * + or

* + * +. So .

Since ⟨ ⟩ ⟨ ⟩ , let . By restricting and inducing we get:


⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ , ⟨ ⟩

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ and, since

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (31)

We set deg only when . It follows that

.

For let * +. By restricting and inducing we get on and, since

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (32)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (33)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (34)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) , (35)

then deg only when , * +, so .

For let * +. By restricting and inducing, we get

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (36)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (37)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (38)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( )

. (39)

Then . For let * +. By restricting and inducing we get on:

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (40)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (41)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( ) (42)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ) ( ) (43)

(⟨ ⟩ ⟨ ⟩ ⟨ ⟩) ( )

, (44)

then deg only when , or

* +. So .

For , let . By restricting and inducing then it splits to and . For let

* + and since

(⟨ ⟩ ⟨ ⟩ ) ( ) . (45)

So it's splits to and . Finally we have 206 columns, and then must divided into

and . The decomposition matrix is shown in Table (3).

References

[1] G. D. James, A. Kerber, The representation theory of the symmetric group. Mass,

Addison-Wesley(1981)

[2] A.O. Morris, A. K. Yaseen, Decomposition matrices for spin characters of symmetric

group. Proc. of Royal society of Edinburgh 108A (1988)145-164.

[3] M. M. Jawad, On Brauer Trees and Decomposition Matrices of Spin Characters of Sn

Modulo 13,11(2018).


261

[4] A.O. Morris, The spin representation of the symmetric group. proc. London Math. Soc

(3)12 (1962) 55-76

[5] B.M. Puttaswamaiah, J.D. Dixon, Modular representation of finite groups. Academic

Press, J.london Math. Soc 15 (1977) 445-455.

[6] A. H. Jassim: 7-Modular Character of The Covering group ̅ . Journal of Basrah

Researches ((Sciences)) V 43. N. 1 A (2017) 108-129.

[7] I. Schur, Uber die Darstellung der symmetrischen und der alternierenden gruppe durch

gebrochene lineare subtituttionen. j.Reine ang.Math 139(1911) 155-250.

[8] J. F. Humphreys, Projective modular representations of finite groups. J. London Math.

Society2.16(1977)51-66.

هصفوفاث التجزئت للوشخصاث األسقاطيت للزهزة التناظزيت

مروى دمحم جواد ,أحمد حسين جاسم

العزاق البصزة، البصزة،جاهعت العلوم،كليت الزياضياث،قسن

الوستخلص

والتي هي عالقت في هذا البحث لقذ وجذنا هصفوفاث التجزئت للوشخصاث األسقاطيت للزهزة التناظزيت

بين الوشخصاث االعتياديت والوعياريت

decomposition matrices for the spin characters of modulo...

Documents