Endo-trivial modules for finite groups with dihedral Sylow 2-subgruops [2-subgroups] (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics)

(1)

Endo‐trivial modules for finite

groups

with dihedral

Sylow

2‐subgruops

Department

of Mathematics and

_Informatics,

Graduate School of

_Science,

Chiba

_University

e‐mail

_{[email protected]‐u.ac.jp}

千葉大学大学院理学研究科

Shigeo

Koshitani

_越谷重夫

1. ENDO‐TRIVIAL MODULES

This is

_joint

work with Caroline Lassueur

_{(see [8], [7]).}

We assume that

k is afield with characteristic

p>0

_, and G is afinite _group with

p||G|.

A

_{finitely generated right}

kG‐module M is called endo‐trivial if

\mathrm{E}\mathrm{n}\mathrm{d}_{k}(M)\cong k_{G}\oplus(

projective)

as

right

kG‐modules where

k_{G}

is the trivial kG‐module. Then in the

stable module

_category

stmod

_(kG)

of

_{finitely generated}

kG

_‐modules,

the set

T(G)

:=

{ [M]\in

stmod

(kG)|M

is

endo‐trivial}

has an abelian _{group structure}

by making

use of the tensor

product

over k.

Namely,

we define an addition + in

T(G)

by

[M]+[N]:=[M\otimes_{k}N]

(note

that

_{M\otimes_{k}N}

is

_again

an endo‐trivial kG‐module ifso are M and

N, and recall also that

\mathrm{E}\mathrm{n}\mathrm{d}_{k}(M)\cong M^{*}\otimes_{k}M

as kG‐modules where

M^{*}

_{:=\mathrm{H}\mathrm{o}\mathrm{m}_{k}(M, k)}

. Remind also that for

right

kG‐modules M and

N,

M\otimes_{k}N

is considered as left kG‐module

by

the

diagonal

action,

namely

(m\otimes n)g

:=mg\otimes ng for

_{g\in G,}

m\in M and n\in N, and

that M^{*} is

_again

a left kG‐modlue via

( $\phi$ g)(m)

:= $\phi$(mg^{-1})

for

g\in G,

m\in M and

_{$\phi$\in M^{*}}

Then it is _{easy to} know that the zero element in

T(G)

is

_{[k_{G}]}

and the inverse element

_-[M]

of

_[M]

in

_T(G)

is

_[M^{*}].

The endo‐trivial modules _go back to Dade in 1978

_[6].

Since then endo‐trivial modules show_{up in many}

_places

inthe modular

_{representa‐}

tion

_theory

of finite_groupsand

_they

do have

_played

_very

_important

role

(2)

in this area.

Actually,

Dade classifies all the endo‐trivial kG‐modules

for the case where G are finite abelian _p_‐groups

(see

[6]).

Since then

the determination of the structure of

_T(G)

has been done for the case

where G are finite _p_‐groups

by

Carlson and Thévenaz

(see [13], [2]).

Then,

we

might

be,interested

in

T(G)

for

artibrary

finite _groups G.

This

_problem

is still _open as far as the author knows. It is known that‐

T(G)

is

_{finitely generated}

_by

a result of

[3] (depending

on initiated

work of

_Puig).

From now on. let

TT(G)

and

TF(G)

be the torsion

part

and the torsion free

_part

of

_T(G)

_{respectively,}

and let P be a

Sylow

p\leftrightarrow

‐subgroup

of G.

First,

if P is

_cyclic,

then

_T(G)

is considered in

_[9].

This is of course

exactly

the case where the _group

algebra

kG is of

finite‐representation

type

as is well‐known since _{many years ago.}

Further,

if

_p=2

and P

is

_generalized

_quaternion

or

semidihedral,

then

T(G)

is treated

by

[4].

Recall that these cases cover almost of all the cases where the _group

algebras

kGareof

finite‐representation

_type

andof

tame‐representation

type.

Then,

whats

_missing?

_Yes,

the case where P is dihedral

(possibly

the Klein‐four _group of order

₄₎

is

_missing.

This is

_actually

our motivaion for the work

(see [7]

and

[8]).

Our main result is the

_folllowing:

Theorem 1.1.

_Suppose

that G is a

finite

_group with a dihedral

Sylow

2‐subgroup

P

_of

order at least

_8,

and assume that

T(G)

is the abelian

group

of

endo‐trivialkG‐modules over an

algebraically

closed

field

k

of

characteristic 2. Set

\overline{G}

_{:=G/O_{2'}(G)}

. Then we have the

following:

(i)

If

\overline{G}\not\cong \mathfrak{A}_{6}

(here

\mathfrak{A}_{6}

is the

_alternating

_group on 6

letters),

then

TT(G)=X(G)

, where

X(G)

:=\{[M]\in T(G)|\dim_{k}(M)=

1\}.

(ii)

If

\overline{G}\cong \mathfrak{A}_{6}

, then either

(a)

TT(G)=X(G)

, or

(b) TT(G)/X(G)

is an

elementary

abelian

3‐group

and each

indecomposable

endo‐trivial kG ‐module M with

_[M]

\in

TT(G)\backslash X(G)

is a 9‐dimensional

simple

module.

Proof.

See

_[8,

Theorem

_1.2].

\square

Remark 1.2. For the case where the

Sylow 2‐subgroup

P of G is the

(3)

Actually

we do have a

pretty

much

general

result to

compute

T(G)

.

It is stated as follows:

Theorem 1.3.

_Suppose

that k is an

algebraically

closed

field of

charac‐

teristic

_p>0

, and that G is a

finite

group

ofp‐rank

at least

2,

namely

G contains a

subgroup

C_{p}\times C_{p}

where

C_{p}

is the

cyclic

_group

of

order

p. Further assume thatG has no

strongly

p‐embedded

subgroups. Now_{f}

we _suppose thatH\triangleleft G with

p\displaystyle \int|H|

. Set

\overline{G}:=G/H

, and assume more

over that

\mathrm{H}^{2}(\overline{G}, k^{\times})=1

.

Then,

we have

k(G)\cong X(G)+k(\overline{G})

where

_K(G)

is the kernel

_of

the restriction map

{\rm Res}_{P}^{G}:T(G)\rightarrow T(P)

given

by

[1\mathrm{J}4i]\mapsto[M\downarrow_{P}^{G}],

and

_further

k(G) :=\{[M]\in T(G)|M=f^{-1}(L)

for

a 1‐dimensional

kN_{G}(P)

‐module L

}

where

_f

is the Green

_{correspondence}

with

_respect

to

_{(G, P, N_{G}(P))}

.

Proof.

See

_[8,

Theorem

_1.1].

\square

Remark 1.4. The

_point

in Theorem 1.3 is the

_following.

We

_actually

would like to

_compute

_TT(G)

, and it is most

likely

thesame as

K(G)

.

Precisely speaking,

K(G)=TT(G)

unless P is

_cyclic

or

generalized

quaternion.

Soat least forour_purpose this is thecase. Soourfinal aim

is recuded from the

_computation

of

_TT(G)

to that of

_K(G)

.

Now,

let

us lookat the

right

hand side.

Then,

first of

all,

X(G)

is

computable

(it

is

_nothing

but the _p

_‐part

of

_{G/[G, G]}

where

_{[G, G]}

is the commutator

subgroup

of G. Then what about the second term

K(\overline{G})

. Since we

can assume that

H\neq 1

, we are able to use inductive

argument.

So,

it

works!

2. OKUYAM\mathrm{A}^{}\mathrm{S} THEOREM

The author would like to introduce a theorem of

Okuyama

in 1981

with a

proof,

which showed _up

only

in

Japanese

[12,

Theorem

1],

be‐

cause the author

hopes/believes

that

Okuyamas

theorem

may/should

be useful even for

understanding

endo‐trivial modules and also endo‐

permutation

modules and so on. Who knows?

Okuyamas

theorem

(4)

dimentional k

_‐algebra

A and

_{I\subseteq A}

we denote

by

\mathrm{A}\mathrm{n}\mathrm{n}_{A}(I)

the

(right)

annihilator of I in A,

namelyì

\mathrm{A}\mathrm{n}\mathrm{n}_{A}(I) :=\{a\in A|Ia=\{0\}\}

, and we

denote

_by

_Z(A)

the center of A.

Theorem 2.1

_{(Okuyama (1981)}

_[12]).

Let B be a block

algebra

of

kG

of

a

finite

_group G over an

algebraically

closed

field

k

of

character‐

istic

_p>0

_(actually,

this theorem holds even

for

an

arbitrary finite

dimensional

_symmetric

k

_‐algebra

B

_).

_Further,

let

_\ell(B)

be the number

of

non‐isomorphic simple

right

B

_‐modules,

and let

_{S_{1},}

\cdots

ì

S_{\ell(B)}

be the

all

_{non‐isomorphic simple}

_right

B ‐modules.

_Then,

it holds

\displaystyle \dim_{k}[\mathrm{A}\mathrm{n}\mathrm{n}_{B}(J(B)^{2})\cap Z(B)]=\ell(B)+\sum_{i=1}^{\ell(B)}\dim_{k}[\mathrm{E}\mathrm{x}\mathrm{t}_{B}^{1}(S_{i}, Si)].

Proof.

(Okuyama

in

_[12,

Theorem

_1]).

Set \ell

_:=\ell(B)

, and let e_{1}, \cdots, e_{\ell}

be the set of all

_primitive

_idempotents

of B such that

_{P_{i}}

_{:=P(S_{i})}

:=

e_{i}B

is the

_projective

cover of

S_{i}

for

i=1,

_\cdots,\ell. Set e :=e_{1}+\cdots+e_{\ell},

and A :=eBe, and hence A is the basic

ring

(algebra)

of B, and A

is a finite dimensional

symmetric

k

‐algebra

(recall

that A and B are

Morita

_equivalent).

_Further,

set J

_:=J(B)

the Jacobson radical of B.

Step

1.

_{\dim_{k}[\mathrm{E}\mathrm{x}\mathrm{t}_{B}^{1}(S_{i}, S_{i})]=\dim_{k}(e_{i}Je_{i}/e_{i}J^{2}e_{i})}

.

Step

2. IfC is a finite dimensional k

‐algebra

such that

C/J(C)\cong k,

then

_{\mathrm{A}\mathrm{n}\mathrm{n}_{C}(J(C)^{2})\subseteq Z(C)}

.

Proof of

_Step

2. Follows

_by

_[10,

Lines 7‐8 of the

_proof

of Lemma

_2],

Step

3. For _any two‐sided ideal I of B, we have that

\dim_{k}[\mathrm{A}\mathrm{n}\mathrm{n}_{B}(I)\cap Z(B)]=\dim_{k}[\mathrm{A}\mathrm{n}\mathrm{n}_{A}(eJe)\cap Z(A)].

Proof of

_Step

3. Recall that B and A are Morita

equivalent.

Step

4. If

_{0\neq f=f^{2}\in C}

for a finite‐dimensional

symmetric

k‐

algebra

C, then

fCf

is

again symmetric.

Proof of

_Step

4. Take a similar _{way to prove} that a Morita

equiva‐

lence _preserves

_being

_symmetric,

_though

_fCf

isnot

_necessarily

Morita

equivalent

to C, of course.

Step

5. Set

T:=\displaystyle \sum_{i}e_{i}J^{2}e_{i}+\sum_{i\neq j}e_{i}Je_{j}.

Then T is a two‐sided ideal of A.

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Acknowledgements.

The author would like to thank Professor Hiroki Shimakura for

_giving

him an

opportunity

to

give

a talk in the

meeting

held in the RIMS of the

_University

of

_Kyoto,

_January

2016. REFERENCES

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