October 2018 – Without Loss of Universality

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.

Month: October 2018

An Introduction to Character Theory