This is a continuation of the ongoing series on representation theory, see Overview of blog posts for previous posts on that subject.
Let be a group and let be a field. We begin with some remarks on dual spaces and tensor products of representations.
Lemma/Definition 5.1 If is a -module, then the dual space is a -module via . 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 can be considered a functor , where we regard as a one-object category. The inversion map is an anti-isomorphism, i.e. an isomorphism , as , which means that inversion gives a functor , now if we compose the functors , we get a covariant functor , since we composed two contravariant and one covariant functors, this is the dual representation defined in the lemma.
Lemma/Definition 5.2 If and are two -modules, then is a -module via the “diagonal action” given on elementary tensors by .
Proof As before, this is a straightforward computation. For a more conceptual proof, one can use that the diagonal map induces a ring homomorphism given on basis elements by (this is called “comultiplication” in the setting of Hopf algebras) and that is canonically a -module (where the first copy of acts on and the other one on ), so one can restrict scalars along the comultiplication .
Lemma/Definition 5.3 If and are two -modules, then is a -module where is defined via
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 and be -modules, then we have .
Proof clear from the definition of the -module structure on .
Lemma 5.5 Let and be two finite-dimensional -modules, then the map given on elementary tensors by is an isomorphism of -modules.
Proof That this is an isomorphism of -vector spaces is known from linear algebra. One checks that it is -equivariant.
For the rest of this post, let be a finite group and let be a field of characteristic . (All representations shall have coefficients in )
By Maschke’s theorem, every representation of with coefficients in can be decomposed as a direct sum of irreducible representations. If is an irreducible representation and is a finite-dimensional representation, then we get by Schur’s lemma that the number of times that appears in the decomposition of is equal to (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 be a -module, then the map , is a linear projection.
Proof To see that is well-defined note that for , we get , since the map is a bijection. That restricts to the identity on and linearity is clear.
Lemma 5.7 Let and be finite-dimensional, then
Proof Since the map , is a projection onto the subspace (by 5.6), we get that . Using the isomorphism (by 5.5), we get that . Using properties of traces and the definition of the dual representation, this equals .
Let us ponder for a moment what we have shown so far. We know that every finite-dimensional representation of may be decomposed as a direct sum of irreducible submodules. For each irreducible representation of , we can compute the multiplicity of in the decomposition of if we know and (cf. the discussion preceeding 5.6). But now 5.7 gives us an expression for these dimension that only involves the traces and . It thus makes sense to give a special name to the function
Definition 5.8 Let be a finite-dimensional representation of , then the function is called the character of and is denoted by . (Note that the trace of a matrix is invariant under conjugation, so the character doesn’t depend on a choice of basis for .)
Using the above discussion, we obtain the following surprising corollary:
Corollary 5.9 A finite-dimensional representation of is uniquely determined by its character: if two finite-dimensional representations of 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.