Advanced Algebra I
representation of finite groups
Another interesting realization of the tetrahedral group T is done by choose coordinates such that ±ei are those midpoint of 6 edges. Then one can express T as a finite subgroup of GL(3, R). This is an example of a representation.
Definition 0.1. A n-dimensional matrix representation of a group G is a homomorphism
R : G → GL(n, F ),
where F is a field. A representation is faithful if R is injective. And we write Rg for R(g)
It’s essential to work without fixing a basis. Thus we introduce the concept of representation of a group on a finite dimensional vector space V .
Definition 0.2. By a representation of G on V , we mean a homomor- phism ρ : G → GL(V ), where GL(V ) denote the group of invertible linear transformations on V . We write ρg for ρ(g)
Remark 0.3. By fixing a basis β of V , one has β : GL(V ) → GL(n, F )
T 7→ matrix of T.
And one has a matrix representation R := β ◦ ρ.
Furthermore, if a change of basis is given by a matric P , then one has the conjugate representation R0 = P RP−1, that is R0g = P RgP−1 for all g ∈ G.
Remark 0.4. We would like to remark that the concept of a linear representation of G on V is equivalent to G acts on V linearly. More precisely, G acts on the vector space V and the action satisfying
g(v + v0) = gv + gv0, g(cv) = cg(v) for all g ∈ G, v, v0 ∈ V and c ∈ F .
Definition 0.5. Let ρ, ρ0 be two representations of G on V, V0. They are said to be isomorphic if there is an isomorphism of τ : V → V0 which is compatible with ρ and ρ0. That is,
τ ρs(v) = ρ0sτ (v), for all s ∈ G, v ∈ V .
Example 0.6. A representation of degree 1 is a homomorphism R : G → C∗. Since every element has finite order, Rg is a root of unity.
In particular, |Rg| = 1.
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Example 0.7 (Regular representation). Let G be a finite group of order g and let V be a vector space with basis {et}t∈G. For s ∈ G, let Rs be the linear map of V to V which maps et to est. This is called the regular representation of G.
Note that es = Rs(e1) for all s ∈ G. Hence the image of e1 ∈ V form a basis. On the other hand, if τ : G → W is a representation with the property that there is a v ∈ W such that {τs(v)}s∈G forms a basis. Then W is isomorphic to the regular representation. This is the case by considering τ : V → W with τ (es) = ρs(v).
More generally, if G acts on a finite set X, the one can have a representation similarly on the vector space V with basis X. This is called the permutation representation associated to X.
Let ρ, ρ0 be two representations of G on V, V0, then one can define ρ⊕
ρ0, ρ ⊗ ρ0 naturally. Note that if degree of ρ and ρ0 are d, d0 respectively, then degree of ρ ⊕ ρ0 is d + d0 and degree of ρ ⊗ ρ0 is dd0.
Definition 0.8. V is irreducible representation if V is not a direct sum of two representation non-trivially.
One might ask whether a representation is irreducible or not. We threrefore introduce the G-invariant subspace as we did in linear alge- bra.
Definition 0.9. Let ρ : G → GL(V ) be a representation. A vector subspace W of V is said to be a G-invariant subspace if ρs(W ) ⊂ W for all s ∈ G. It’s clear that the restriction of G action on V to W give a representation of G on W , which is called the subrepresentation of V .
Theorem 0.10 (Maschke’s Theorem). Every representation of a finite is a direct sum of irreducible representations.
Proof. It suffices to prove that for any G-invariant subspace W ⊂ V . There is a G-invariant complement of W . By a complement of W , we mean a subspace W0 such that W ∩ W0 = {e}, and W + W0 = V .
We first pick any complement W0. Then V = W ⊕ W0. Let p : V → W be the projection. We are going to modify W0 to get a G-invariant complement.
To this end, we average p over G to get p0 := 1
g X
t∈G
ρtpρ−1t , where g = |G|.
One checks that p0 : V → W and p0(w) = w for all w ∈ W . That is, p0 : V → W is a projection.
Let W0 := ker(p0). We check that W0 is G-invariant since ρsp0ρ−1s = p0
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for all s ∈ G. It follows that if x ∈ W0, p0ρs(x) = ρs(p0(x)) = 0, which shows that ρs(x) ∈ W0.
This proves that the representation on V is isomorphic to W ⊕ W0.
¤ Remark 0.11. A matrix over C of finite order is diagonalizable. Hence every matrix representation over the field C is diagonalizable. We therefore assume the field to be the complex number field.
Moreover, let λ be an eigenvalue of ρs for some s. Then |λ| = 1.
Definition 0.12. Let ρ : G → GL(V ) be a linear representation on the vector space V . We define the character as χρ := T r ◦ ρ : G → C.
Proposition 0.13. Let χ be the character of ρ : G → GL(V ).
(1) χ(1) = n := dimV ,
(2) χ(s−1) = χ(s) for all s ∈ G, (3) χ(tst−1) = χ(s) for all s, t ∈ G.
(4) if χ0 is the character of another representation ρ0, then the char- acter of ρ ⊕ ρ0 is χ + χ0.
One can define a hermitian dot product on characters as
< χ, χ0 >:= 1 g
X
s∈G
χ(s)χ0(s).
The main theorem for character is the following:
Theorem 0.14. Let G be a group of order g, and let ρ1, ... represent the isomorphism classes of irreducible representations of G. Let χi be the character of ρi for each i.
(1) Orthogonality Relations:
<< χi, χj >= 0if i 6= j,
< χi, χi >= 1 for each i.
(2) The number of isomorphism classes of irreducible representa- tions of G is the same as the number of conjugacy classes of G.
(denote it by r).
(3) Let di be the degree of ρi, then di|g and g =
Xr
i=1
d2i.
Example 0.15. Consider G = D4. It’s clear that r = 5. Hence it’s only possible to have d1 = 2, d2 = ... = d5 = 1.
