Artin's theorem on induced characters explained

In representation theory, a branch of mathematics, Artin's theorem, introduced by E. Artin, states that a character on a finite group is a rational linear combination of characters induced from all cyclic subgroups of the group.

There is a similar but somehow more precise theorem due to Brauer, which says that the theorem remains true if "rational" and "cyclic subgroup" are replaced with "integer" and "elementary subgroup".

Statement

In Linear Representation of Finite Groups Serre states in Chapter 9.2, 17 ^[1] the theorem in the following, more general way:

Let

finite group,

family of subgroups.

Then the following are equivalent:

G=cup_g\ing^-1Hg

\forall\chicharacterofG\existsH\inX,\chi_H\inH,d\in\N:d\chi=\sum_H\in

	G(\chi
Ind
	H)

This in turn implies the general statement, by choosing

as all cyclic subgroups of

Proof

Let

be a finite groupe and

(\chi_i)

its irreducible characters. Let us denote, like Serre did in his book,

l{R}(G)

the

-module

oplus_i\chi_iZ

. Since all of

's characters are a linear combination of

(\chi_i)

with positive integer coefficient, the elements of

l{R}(G)

are the difference of 2 characters of

. Moreover, because the product of 2 characters is also a character,

l{R}(G)

is even a ring, a sub-ring of the

-algebra of the class function over

(of which

(\chi_i)

forms a basis), which, by tensor product, is isomorphic to

C ⊗ l{R}(G)

.Both the restriction

Res

of the representation of

to one of its subgroup and its dual operator

Ind

of induction of a representation can be extended to an homomorphisme :

Res:l{R}(G)\tol{R}(H),\sumk_i\chi_i\mapsto\sumk_iRes

	G

	H

\chi_i

Ind:l{R}(H)\tol{R}(G),\sumk_i\chi_i\mapsto\sumk_iInd

	G

	H

\chi_i

With those notations, the theorem can be equivalently re-write as follow :If

is a family of subgroup of

, the following properties are equivalents :

is the reunions of the conjugate of the subgroups of

The cokernel of

Ind:oplus_H\inl{R}(H)\tol{R}(G)

is finite.

This result from the fact that

l{R}(G)

is of finite type. Before getting to the proof of it, understand that the morphisme

Ind:oplus_H\inl{R}(H)\tol{R}(G)

, naturally defined by

(\sum

	H)
k
	H\inX

\mapsto\sum_H\in

	H
k
	i

	G
Res
	H

\chi_i

is well defined because

is finite (because

is) and its cokernel is

l{R}(G)/Im(Ind)

Let’s begin the proof with the implication 2.

\implies

1. Starting with the following lemma :

Let

be an element of

. Then for every

f_H\inl{R}(H)

	G(f
Ind
	H)

is null on

x\inG

isn’t conjugate to any

. It is enough to prove this assertion for the character

\phi_H

of a representation

\theta:H\toGL(W)

of H (as

f_H

is a difference of some). Let

\rho:G\toGL(V)

be the induced representation of

(W,\theta)

. Let now

(r_i)

be a system of representative of

G/H

,by definition, V is the direct sum of the transformed

\rho
	r_i

of which

\rho_x

is a permutation. Indeed

\rho_x\circ\rho


	r_i

W=\rho
	xr_i

W=\rho

r
	i_x

where

xr_i=r


	i_x

for some

t\inH

. To evaluate

\chi(x):=Ind(\phi)(x)=Tr_V(\rho_x)

, we can now choose a basis of

reunion of basis of the

\rho
	r_i

. In such a basis, the diagonal of the matrix of

\rho_x

is null at every

r_{i ≠}

r
	i_x

, and because

r_i=r


	i_x

would imply

r_ixr

	-1

	i

=t\inH

(which is ruled out by hypothesis), it is fully null, we thus have

\chi(x)=Tr_V(\rho_x)=0

which conclude the proof of the lemma.

This particularly insure that, for every element

not in

S:=cup_g\ing^-1Hg

, the elements in the image of

Ind:oplus_H\inl{R}(H)\tol{R}(G)

, which are the

\sum_H\in

	G(f
Ind
	H)

evaluate to zero on

. The prolonged morphisme

C ⊗ Ind:C ⊗ oplus_H\inl{R}(H)\toC ⊗ l{R}(G)

has to be surjective. Indeed if not, its cokernel would contain a

Vect_C(f)

for some

l{R}(G)

, which in turn means the multiples of

are distinct elements of the cokernel of

Ind

contradicting its finitude.Particularly, every element of

C ⊗ l{R}(G)

are thus null on the complementary of

, insuring

S=G

, thereby concluding the implication.

Let’s now prove 1.

\implies

2.To do so, it is enough to prove that the

-linear application

C ⊗ Ind:C ⊗ oplus_H\inl{R}(H)\toC ⊗ l{R}(G)

is surjective (indeed, in that case,

C ⊗ l{R}(G)

would admit a basis

(e_i)

composed of element of the image

of the

Ind

. It would thus have the same cardinality

, than

(\chi_i)

, insuring that the quotient

l{R}(G)/A

is isomorphic to some

	n
Z
	i

H_i\cong

	nZ/H
\prod
	i

which is finite - where the

H_i

are non-trivial ideals of

), which, through duality, is equivalent to prove the injectivity of

C ⊗ Res:C ⊗ l{R}(G)\toC ⊗ oplus_H\inl{R}(H)

Which is obvious : indeed this is equivalent to say that if a class function is null on (at least) one element of each class of conjugation of

, it is null (but class function are constant on conjugation class).

This conclude the proof of the theorem.

References

Book: Serre, Jean-Pierre . Linear Representations of Finite Groups . 1977 . Springer New York . 978-1-4684-9458-7 . New York, NY . 853264255.

Artin's theorem on induced characters explained

Statement

Proof

References

Further reading