I’m following some lecture notes on introductory finite representation theory/character theory that present topics like irreductible/indecomposable representations, Maschke’s theorem… and then move on to dual representations, tensorial products of representations, to finally reach the definition of the canonical homomorphism
$$i: W \otimes_K V^* \rightarrow \operatorname{Hom}_K(V,W)$$
which is an isomorphism when $V$ and $W$ are finite-dimentional vector spaces. It is then proved that a finite group’s ($G$) representation over $\operatorname{Hom}_K(V,W)$ is isomorphic to $\pi \otimes \rho^*$, where $\pi$ and $\rho$ are representations of $G$ over $V$ and $W$ respectively, and $\rho^*$ is the dual representation corresponding to $\rho$.
I think I followed the proofs so far, but I’m a bit lost regarding the motivation behind all of this. If I understand correctly, representation theory is for instance used to study groups through their actions on vector spaces, in order to reduce abstract algebra problems into linear algebra ones, which are in general easier. Following this line of reasoning, are dual representations/tensor products of representations especially useful to capture some particular aspects of the abstract groups from where they arise? If so, why? Or is there an altogether different reason why they’re interesting?
P.S.: I included the “characters” tag because my lecture notes later define the character of a dual representation and show that it can be computed by $\chi_{V^*}(g)=\chi_V(g^{-1})$. Since I don’t understand the motivation of this either, I don’t know if they’re relevant to the question/possible answers, but I would be interested to know if that’s the case.
Best Answer
Suppose that $K$ is a field of characteristic $0$ and $G$ is a finite group. I'll write $\mathrm{Hom}_{K[G]}(V,W)$ for the $K$-vector space of morphisms of representations $V \to W$. Note that we have the relation $\mathrm{Hom}_{K}(V,W)^G=\mathrm{Hom}_{K[G]}(V,W)$, where $X^G$ denotes the invariants under the action. Maschke's theorem tells us that any representation can be expressed uniquely as a sum of irreducible representations. A possible proof of Maschke proceeds by describing a projection operator $\pi:\mathrm{Hom}_{K}(V,W) \to \mathrm{Hom}_{K[G]}(V,W)f \mapsto \frac{1}{|G|} \sum_{g \in G} gf$. By a standard linear algebra fact, if we have a projection onto a subspace, the trace of that projection is the dimension of the subspace. Now we can compute the trace of that using the isomorphism $\mathrm{Hom}_{K}(V,W) \cong W \otimes V^*$. We know how the character of tensor product and dual representation is computed. We thus get that $$\mathrm{Trace}(\pi)=\frac{1}{|G|} \sum_{g \in G}\chi_V(g^{-1})\chi_W(g)$$ This is also called the inner product of the characters $\chi_V$ and $\chi_W$. Why is this interesting? Well, it tells us that we can determine the dimension of the space $\mathrm{Hom}_{K[G]}(V,W)$ just from knowing the characters of $V$ and $W$.
We can combine that with Maschke's theorem to show why a representation is determined by its character. Suppose that $V_1, \dots V_n$ are all irreducible representations. Suppoe that $W \cong \bigoplus V_i^{n_i}$, where the $n_i \in \Bbb Z_{\geq 0}$. Then we have by an application of Schur's lemma $\mathrm{Hom}_{K[G]}(V_i,V) \cong \mathrm{Hom}_{K[G]}(V_i,V_i)^{n_i}$, thus $\operatorname{dim} \mathrm{Hom}_{K[G]}(V_i,V)/\operatorname{dim} \mathrm{Hom}_{K[G]}(V_i,V_i)=n_i$. Thus if we know the character of $V$, we can compute all the numbers $n_i$ and thus reconstruct $V$ up to isomorphism from its character.