Tensors are interesting in Representation Theory

Initially, I was baffled by the emphasis on tensor products in Representation Theory. I thought of tensors as a way to construct new from the old, in the sense that we can easily obtain new and potentially interesting representations of a group just by “tensoring.” This was definitely important in representations of finite groups, since it gave an easy way to deduce an exhaustive list of irreducible representations for some group G. Yet there was still some motivation missing. This post is more or less how I remedied my uneasiness.

The not-so-well hidden secret is that we can actually discuss representation theory with the language of modules. In particular, there is this really nice object known as the group ring, ( but for our purposes the group algebra over \mathbb C is good enough, and we we will denote it \mathbb C[G].) There are many ways to think about the group ring, but there is a nice category theoretic way so that we don’t have to actually have to do any math:

given a ring R (commutative with unity,) there is a functor

    \[F:\mathrm{R.Alg} \to \mathrm{Grp}\]

that just assigns to each algebra its group of units under multiplication. It turns out that the functor

    \[G: \mathrm{Grp} \to \mathrm{R.Alg}\]

which forms the group ring R[G] is left adjoint to F, so we have the natural isomorphism

    \[\mathrm{Hom}_{\mathrm{Grp}}(A,F(B)) \cong \mathrm{Hom}_{\mathrm{R.Alg}}(G(A),B)\]

fully characterizing our group ring (and actually giving an adjunction between rings and groups if R=\mathbb Z.) This is very cool, but maybe something more concrete would be helpful (Fie! I am a plebian) in the context of representation theory. Consider the group algebra, where \{e_g\} are elements of G, and let them be a basis that defines the underlying vector space of \mathbb C[G]. The algebraic structure is inherited in the most natural way, setting e_ge_h=e_{gh}. But then any representation \rho: G \to \mathrm{Aut}(V) will extend linearly to a map \tilde{\rho}:\mathbb{C}[G] \to \mathrm{End}(V), giving the vector space V a module structure over \mathbb C[G]. It is worth remarking that group homomorphisms in general G \to H will extend linearly to homomorphisms of their corresponding group algebras.

Now, with this in mind, we can just talk about left \mathbb C[G] modules. The tensor, is of immediate importance now, since we know that the functor T(M)= M \otimes N is left adjoint to U(P)=\mathrm{Hom}(N,P)— morally true because A-linear maps f: M \times N \to P are in one to one correspondence with \mathrm{Hom}(M,\mathrm{Hom}(N,P)) and also with maps \mathrm{Hom}(M \otimes N, P), giving an adjunction. From this, we can mutter many things and feign a great deal of sophistication.

Anyhow, from this point of view, it’s possible to start imagining situations ¬†where it would be nice to know some basic facts about representations obtained by tensoring, but I think that there is an even better reason to tensor.

Suppose we have have a group homomorphism \phi: G \to H and we want to think about induced representations. Personally, I first became interested in this by just thinking about projections and inclusions of subgroups– in fact working on representations of the dihedral group via a cyclic subroup. Actually, this led nowhere, but if you’ve read this far, there is a good chance that “going somewhere” wasn’t exactly on your itinerary for the day.

So, we trudge on: there is one straightforward “contravariant” way to get a representation of G from H from \phi, which is to just take the representation \rho: H \to \mathrm{Aut} V and compose, obtaining the map \phi^{*}:=\phi \circ \rho:G \to \mathrm{Aut}(V)which is an induced representation of G. Yet, say we want a “covariant” induced representation. From our discussion of the group ring, we want to see how we can get a representation of H by the homomorphism \phi.

So, maybe something like

    \[\phi_{*}:\mathbb C[G] \to \mathbb C[H].\]

Well, there is actually a really nice way to do this, and yes, it is done exactly by tensoring! In fact, we can discuss this fact in even greater generality:

Given a ring homomorphism \phi: R \to S, with M an R-module. We can consider the induced module S \otimes_{R} M. This makes sense, since we have already assumed that M was an R-module, and S becomes one via \phi. But here is the slight of hand: S \otimes_{R} M is actually an S-module where the action of S is just s \cdot(a \otimes m):=sa \otimes m.

So, tensoring is not just a way to construct “new from the old” arbitrarily, but can be used in way that agrees with homomorphisms of groups, giving rise to homomorphisms of group rings, and hence induced representations by the above discussion.

Here is a final note, only because it is almost funny by now how far this damned category theory goes :

Given a ring morphism f: R \to S, we can actually consider both (seemingly unrelated) processes as functors:

    \[f^{*}: \mathrm{Mod}_S \to \mathrm{Mod}_R\]

 

    \[f_{*} :\mathrm{Mod}_R \to \mathrm{Mod}_S\]

and yes, they are indeed an adjoint pair. *sigh*