An Introduction to Character Theory

This is a continuation of the ongoing series on representation theory, see Overview of blog posts for previous posts on that subject.

Let G be a group and let k be a field. We begin with some remarks on dual spaces and tensor products of representations.

Lemma/Definition 5.1 If V is a k[G]-module, then the dual space V^*=\mathrm{Hom}_k(V,k) is a k[G]-module via g \cdot f(v) := f(g^{-1}v). This is called the dual representation.

Proof This can be checked via an explicit (but boring) computation, so let’s do a more conceptual proof (feel free to ignore it if you’re not into categories): The representation V can be considered a functor \rho_V:G \to k\textrm{-}\mathrm{Vect}, where we regard G as a one-object category. The inversion map G \to G, g \mapsto g^{-1} is an anti-isomorphism, i.e. an isomorphism G^{op} \to G, as (gh)^{-1}=h^{-1}g^{-1}, which means that inversion \iota gives a functor \iota:G^{op} \to G, now if we compose the functors \mathrm{Hom}_k(-,k) \circ \rho \circ \iota, we get a covariant functor \rho_V:G \to k\textrm{-}\mathrm{Vect}, since we composed two contravariant and one covariant functors, this is the dual representation defined in the lemma.

Lemma/Definition 5.2 If V and W are two k[G]-modules, then V \otimes_k W is a k[G]-module via the “diagonal action” given on elementary tensors by g\cdot (v \otimes w):=gv \otimes gw.

Proof As before, this is a straightforward computation. For a more conceptual proof, one can use that the diagonal map G \to G \times G, g \mapsto (g,g) induces a ring homomorphism k[G] \to k[G\times G]\cong k[G] \otimes_k k[G] given on basis elements by g \mapsto g \otimes g (this is called “comultiplication” in the setting of Hopf algebras) and that V \otimes_k W is canonically a k[G] \otimes_k k[G]-module (where the first copy of k[G] acts on V and the other one on W), so one can restrict scalars along the comultiplication k[G] \to k[G] \otimes_k k[G].

Lemma/Definition 5.3 If V and W are two k[G]-modules, then \mathrm{Hom}_k(V,W) is a k[G]-module where g \cdot f is defined via (g \cdot f) (v)= gf(g^{-1}v)

Proof this is a straightforward computation. One can also give a conceptual proof similar to those before using the maps from the two previous proofs.

Lemma 5.4 Let V and W be k[G]-modules, then we have \mathrm{Hom}_k(V,W)^G=\mathrm{Hom}_{k[G]}(V,W).

Proof clear from the definition of the k[G]-module structure on \mathrm{Hom}_k(V,W).

Lemma 5.5 Let V and W be two finite-dimensional k[G]-modules, then the map V^* \otimes_k W \to \mathrm{Hom}_k(V,W) given on elementary tensors by \xi\otimes w \mapsto (v \mapsto \xi(v)w) is an isomorphism of k[G]-modules.

Proof That this is an isomorphism of k-vector spaces is known from linear algebra. One checks that it is G-equivariant.

For the rest of this post, let G be a finite group and let k be a field of characteristic 0. (All representations shall have coefficients in k)

By Maschke’s theorem, every representation of G with coefficients in k can be decomposed as a direct sum of irreducible representations. If V is an irreducible representation and W is a finite-dimensional representation, then we get by Schur’s lemma that the number of times that V appears in the decomposition of W is equal to \mathrm{dim}_k\mathrm{Hom}_{k[G]}(V,W)/ \mathrm{dim}_k \mathrm{End}_{k[G]}(V) (cf. 3.19)
Let us revisit the averaging method from proof of Maschke’s theorem to find another expression for this dimension.

Lemma 5.6 Let V be a k[G]-module, then the map V\mapsto V^G, \mathrm{avg}:v \mapsto \frac{1}{|G|} \sum_{g \in G} gv is a linear projection.

Proof To see that \mathrm{avg} is well-defined note that for h \in G, we get h \frac{1}{|G|}\sum_{g \in G}gv=\frac{1}{|G|}\sum_{g \in G} hgv= \frac{1}{|G|}\sum_{g \in G} gv, since the map G \to G, g \mapsto hg is a bijection. That \mathrm{avg} restricts to the identity on V^G and linearity is clear.

Lemma 5.7 Let (V,\rho_V) and (W,\rho_W) be finite-dimensional, then \mathrm{dim}_k \mathrm{Hom}_{k[G]}(V,W)=\frac{1}{|G|}\sum_{g \in G}\mathrm{Tr}(\rho_V(g^{-1})) \mathrm{Tr}(\rho_W(g))

Proof Since the map \mathrm{avg}:\mathrm{Hom}_k(V,W) \to \mathrm{Hom}_k(V,W)^G = \mathrm{Hom}_{k[G]}(V,W), f \mapsto (v \mapsto \frac{1}{|G|}\sum_{g \in G}\rho_W(g)f(\rho_V(g^{-1})v)) is a projection onto the subspace \mathrm{Hom}_k(V,W)^G (by 5.6), we get that \mathrm{dim}_k \mathrm{Hom}_{k[G]}(V,W) = \mathrm{Tr}(\mathrm{avg}). Using the isomorphism V^* \otimes_k W \cong \mathrm{Hom}_k(V,W) (by 5.5), we get that \mathrm{Tr}(\mathrm{avg})=\frac{1}{|G|}\sum_{g\in G}\mathrm{Tr}(\rho_V^*(g) \otimes \rho_W(g)). Using properties of traces and the definition of the dual representation, this equals \frac{1}{|G|}\sum_{g\in G} \rho_V(g^{-1})\rho_W(g).

Let us ponder for a moment what we have shown so far. We know that every finite-dimensional representation (W,\rho_W) of G may be decomposed as a direct sum of irreducible submodules. For each irreducible representation (V,\rho_V) of G, we can compute the multiplicity of (V,\rho_V) in the decomposition of W if we know \mathrm{dim}_k \mathrm{Hom}_{k[G]}(V,W) and \mathrm{dim}_k \mathrm{End}_{k[G]}(V) (cf. the discussion preceeding 5.6). But now 5.7 gives us an expression for these dimension that only involves the traces \mathrm{Tr}(\rho_W(g)) and \mathrm{Tr}(\rho_V(g)). It thus makes sense to give a special name to the function G \to k, g \mapsto \mathrm{Tr}(\rho_W(g))

Definition 5.8 Let (V,\rho_V) be a finite-dimensional representation of G, then the function G \to k, g \mapsto \mathrm{Tr}(\rho_V(g)) is called the character of (V,\rho_V) and is denoted by \chi_V. (Note that the trace of a matrix is invariant under conjugation, so the character doesn’t depend on a choice of basis for V.)

Using the above discussion, we obtain the following surprising corollary:

Corollary 5.9 A finite-dimensional representation of G is uniquely determined by its character: if two finite-dimensional representations of G have the same character, then they are isomorphic.

This concludes a short intro and motivation for character theory, we will continue the study in future posts.

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