Download CHARACTERS AS CENTRAL IDEMPOTENTS I have recently

CHARACTERS AS CENTRAL IDEMPOTENTS
CİHAN BAHRAN
I have recently noticed (while thinking about the “skewed orthogonality” business Theo
has mentioned) that the irreducible characters of a finite group G are nothing but the
central idempotents (up to a scalar multiple) of the group algebra CG. The purpose of
this document is to make sense of and explain this. Actually, I will show how characters can be defined in this way and the fact that they are scaled central idempotents
immediately imply the usual and the skew orthogonality relations.
1. Endomorphisms Induced by Central Elements
In this section, I will work with a more general setup than the group algebra. The main
results are Theorem 7, which is stated in a form that doesn’t refer to previous notation;
so the reader may jump to there.
Let k be an algebraically closed field of characteristic 0 (can be taken as C for all intents
and purposes) and let A be a finite-dimensional k-algebra.
Definition 1. For a left A-module M , we write ρM : A → Endk M for the ring
homomorphism which defines the A-action on M as a k-vector space.
Definition 2. Let S be a simple left A-module. For any A-module M , define MS to
be the sum of submodules of M which are isomorphic to S.
If M has no submodule isomorphic to S, then MS = 0. Note that MS can be written
as a direct sum of copies of S.
If M is semisimple, then we have
M
M=
MS
simple S≤M
in which case we may refer to MS as the S-constituent of M . Note that a finitedimensional algebra has finitely many simple (left) modules up to isomorphism. But of
course each individual MS might be a direct sum of infinitely many S’s.
Remark. In the remaining of this section, we assume M to be a fixed finitely generated semisimple left A-module. Therefore each MS consists of finitely many copies
of S, say MS ∼
= S nS . In fact nS is determined by M : it is the number of copies of S
occurring in M , the well-definition of nS can be checked by Jordan-Hölder, for instance.
Proposition 3. Writing ιS : MS ,→ M for the inclusion and πS : M MS for the
projection maps, the map
Y
Ξ : EndA M →
EndA MS
simple S≤M
ϕ 7→ (πS ◦ ϕ ◦ ιS )S
is a k-algebra isomorphism.
1
CHARACTERS AS CENTRAL IDEMPOTENTS
Proof. This is because HomA (MS , MT ) = 0 if S T .
2
Observe that if a ∈ A is central, then ρM (a) ∈ Endk M actually lies inside EndA M .
This is because the subring EndA M of Endk M is nothing but the centralizer of ρM (A).
And in the notation of Proposition 3, since ιS : MS ,→ M is an A-module homomorphism, we have
πS ◦ ρM (a) ◦ ιS = πS ◦ ιS ◦ ρMS (a) = ρMS (a) .
Thus the isomorphism Ξ in Proposition 3 sends ρM (a) to the tuple (ρMS (a))S . To
understand the ρMS (a)’s, we start with a standard fact.
Proposition 4. For every simple S, we have Z(EndA MS ) ∼
= k.
Proof. Writing n = nS , we have k-algebra isomorphisms
EndA MS ∼
= EndA S n ∼
= Mn (EndA S)
but EndA S ∼
= k by Schur’s lemma (here we use the assumption that k is algebraically
closed). Thus
Z(EndA MS ) ∼
= Z(Mn (k)) ∼
= k.
Definition 5. Given a ∈ A, we write trM (a) for the trace of the k-linear operator
ρM (a) : M → M .
Corollary 6. If a ∈ Z(A) and ρM (a) ∈ Z(EndA M ), then for every simple S ≤ M ,
the endomorphism ρMS (a) ∈ EndA MS is the scalar multiplication by
trMS (a)
,
nS
where tr denotes taking the trace of a k-linear operator on a finite-dimensional k-vector
space.
Proof. The isomorphism Ξ in Proposition 3 yields an isomorphism
Y
Z(EndA M ) ∼
Z(EndA MS )
=
simple S≤M
which sends ρM (a) to the tuple (ρMS (a))S , where each coordinate ρMS (a) is scalar
multiplication by Proposition 4. The scalar can be recovered by dividing the trace by
nS . Note that we use the assumption that char k = 0 to be able to divide with nS . Also
nS 6= 0 because we are assuming S ≤ M .
Application: If A is semisimple as a ring, then we can take M to be the regular
left module A A, as this is finitely generated and semisimple. Let us write IS for the
S-constituent of M (it can be shown that IS ⊆ A is not just a left ideal, but a two-sided
ideal). Moreover, it is a standard fact that every simple S occurs in M with multiplicity
dimk S. Moreover, the endomorphism algebra EndA M is isomorphic to the opposite
algebra Aop , via the maps
Aop ∼
= EndA M
a 7→ σa
ϕ(1) ←[ ϕ
CHARACTERS AS CENTRAL IDEMPOTENTS
3
where σa : A → A is the “multiply by a from the right” map. Composing with the
isomorphism in Proposition 3, we get a k-algebra isomorphism
Aop ∼
=
(?)
Y
EndA IS .
S-simple
Moreover if a ∈ Z(A), then σa ∈ Z(EndA M ) is also equal to the “multiply by a from
the left” map, which is exactly ρM (a). Thus the second assumption of Corollary 6
comes for free. In short, we get the following result:
Theorem 7. Let A be a finite-dimensional semisimple algebra over an algebraically
closed field of characteristic zero. Let S be a simple left A-module and write IS for
the S-constituent of the regular left module A A. Then the action of a central element
a ∈ Z(A) on IS is given by its trace divided by dimk S.
Proof. Immediate from above. Really.
We may also deduce the following from the technology we’ve developed:
Theorem 8. Let A be as in Theorem 7. Then for every simple left A-module S, there
exists a unique element eS in A such that the action of eS on S is trivial and on any
other simple T S is zero. Consider the (finite) set B := {eS : S-simple}.
(1) B is the set of central primitive idempotents of A.
(2) B is a k-basis of Z(A).
(3) Let S, T be simple left A-modules. Writing trT (a) for the trace of the action of
an element a ∈ A on T , we have

dim
trT (eS ) = 
k
S
0
if S ∼
=T,
otherwise.
= dimk S · δS,T .
Thus, because taking trace is k-linear, the map
Z(A) × B → k
(a, eT ) 7→ trT (a)
uniquely extends to a non-degenerate bilinear form on Z(A) for which B is an orthogonal
basis and heS , eS i = dimk S.
Proof. No kidding.
The Group Algebra
Next we apply the generalities in the previous section to the group algebra CG where
G is a finite group. CG definitely satisfies the conditions of Theorem 7.
The general convention to write an element of CG is something like G ag g, that is,
formal C-linear combinations of the group elements. This is very useful in calculating
products like (1 + g)(1 + h). But in this document, I consider CG as the set of functions
from G to C. This allows to realize a character χ as a genuine element of the group
P
CHARACTERS AS CENTRAL IDEMPOTENTS
algebra, instead of writing them like
tg ∈ CG for the function
P
g∈G
4
χ(g)g. And for every g ∈ G, we write
tg : G → C
h 7→ δg,h
The set {tg : g ∈ G} forms a C-basis for CG and the multiplication of basis elements is
defined by
tg ∗ th = tgh .
and is extended linearly to all elements. I write ∗ for the product instead of a · or
juxtaposition because for α, β ∈ CG, it is easy to make the mistake that their product
should send g to α(g)β(g). The multiplication is NOT pointwise. In fact, by definition
it is given by
ÑÑ
é
X
(α ∗ β)(g) =
α(h)th
Ñ
∗
h∈G
=
X
éé
X
β(k)tk
(g)
k∈G
α(h)β(k)(th ∗ tk )(g)
h,k∈G
=
X
α(h)β(k)thk (g)
h,k∈G
=
X
α(h)β(k)
hk=g
=
X
α(gk −1 )β(k) .
k∈G
Note the similarity to the convolution product
(f ∗ g)(y) =
Z
f (y − x)g(x)dx
from real analysis. In fact, putting the counting measure on G, the map α∗β is precisely
the convolution of α and β.
By Theorem 8, we already have a non-degenerate C-bilinear form on Z(CG) via traces,
for which we write h−, −itr . By using the convolution product (which is really the
product on CG), we shall define another bilinear form h−, −i∗ . Here it is:
h−, −i∗ : Z(CG) × Z(CG) → C
(α, β) 7→ (α ∗ β)(1) .
This is clearly a bilinear form (it is actually symmetric since elements of the center
commute with respect to ∗, by definition). What is not clear, for now, is that whether
h−, −i∗ is non-degenerate.
We will show h−, −itr and h−, −i∗ are basically the same forms. For this, we make use
of the basis {tg : g ∈ G} of CG and chase the definition of h−, −itr . For every pair S, T
of simple left CG-modules, we have
Ñ
heS , eT itr = trT (eS ) = trT
é
X
g∈G
=
X
g∈G
eS (g) trT (tg ) .
eS (g)tg
CHARACTERS AS CENTRAL IDEMPOTENTS
5
We want to realize this expression as a convolution product of eS with some other
function. The expression suggests to define a function
χM : G → C
g 7→ trM (tg ) .
for every left CG-module M which is of course the character associated to M !
Clearly χT is a class function, hence lies in Z(CG). Then, writing T ∗ for the dual of T ,
which is also a simple CG-module, we get
dimk S · δS,T = heS , eT itr
=
X
eS (g)χT (g)
g∈G
=
X
eS (g)χT (g −1 )
g∈G
=
X
eS (g)χT ∗ (g −1 )
g∈G
= (eS ∗ χT ∗ )(1)
= heS , χT ∗ i∗ .
Thus h−, −i∗ is non-degenerate, {χS : S-simple} is also a basis of Z(CG) which the
bilinear form h−, −i∗ pairs with the basis {eS : S-simple} via eS ↔ χS ∗ . But on the
other hand because eS ’s are central primitive idempotents, we have
heS , eT i∗ = (eS ∗ eT )(1)
= eS (1) · δS,T .
Therefore h−, −i∗ also pairs the basis {eS : S-simple} with itself. So
eS (1)
eS =
χS ∗ .
dimk S
With using extra standard information about the characters, it can be shown that
eS (1) = (dimk S)2 /|G|, but we don’t need this for the skew orthogonality: Given S T
simples, then S ∗ T ∗ are also simples; so eS ∗ ∗ eT ∗ = 0 and χS , χT are (nonzero) scalar
multiples of eS ∗ and eT ∗ , respectively; so χS ∗ χT = 0 as functions from G to C. Thus
for every g ∈ G, we have
0 = (χS ∗ χT )(g) =
X
h∈G
χS (gh−1 )χT (h) .