Let be a group and let
be a graded representation of , i.e. a functor from to the category of graded vector spaces with each piece finite-dimensional. Thus acts on each graded piece individually, each of which is an ordinary finite-dimensional representation. We want to define a character associated to a graded representation, but if a character is to have any hope of uniquely describing a representation it must contain information about the character on every finite-dimensional piece simultaneously. The natural definition here is the graded trace
In particular, the graded trace of the identity is the graded dimension or Hilbert series of .
Classically a case of particular interest is when for some fixed representation , since is the symmetric algebra (in particular, commutative ring) of polynomial functions on invariant under . In the nicest cases (for example when is finite), is finitely generated, hence Noetherian, and is a variety which describes the quotient .
In a previous post we discussed instead the case where for some fixed representation , hence is the tensor algebra of functions on . I thought it might be interesting to discuss some generalities about these graded representations, so that’s what we’ll be doing today.
Molien’s theorem and pals
Let denote a linear transformation on a finite-dimensional complex vector space of dimension , and let denote its eigenvalues. acts diagonally on the tensor powers of , hence on the symmetric powers of .
Lemma: The trace of acting on is given by where is the complete homogeneous symmetric polynomial of degree .
Proof. First suppose that has a full set of eigenvectors . Then is spanned by the monomials of degree in these eigenvectors. Any such monomial is an eigenvector for the action of on with eigenvalue , and this gives a full set of eigenvectors for the action of on . The sum of these eigenvalues is as desired. Since the set of with a full set of eigenvectors is dense in , the general result follows by continuity.
As a corollary, the graded trace of acting on is . Compare with the following result: since the trace is multiplicative under tensor product, the trace of acting on is , hence the graded trace of acting on is .
Now let be a compact group. The category of graded unitary representations of with each graded piece finite-dimensional has an internal Hom given by the space of grade-preserving linear transformations from to , where the graded piece of degree consists of the direct sum of the linear transformations from to . As in the case of ordinary representations, is canonically isomorphic to , which has graded character . Again as in the case of ordinary representations, the external Hom , given by the space of grade-preserving linear transformations from to which respect the action of , is precisely the direct sum of the copies of the trivial representation in . By inspecting each graded piece, it follows that
which exactly mimics the statement in the ordinary representation case. In particular, the graded dimension of the trivial part of a graded representation is
which corresponds to the case where is the trivial representation in degree zero. As a corollary, we obtain the following results.
Theorem (Molien): The graded dimension of the trivial part of is
In particular, when is finite this is a rational function. Molien’s theorem is, as I understand it, used to help describe explicitly the structure of the ring of polynomial invariants; we’ll give some examples below. Similarly, for the tensor algebra we have the following.
Theorem: The graded dimension of the trivial part of is
Let acting by rotation matrices. Then the graded dimension of the trivial part of , hence the Hilbert series of , is
Using the complex-analytic technique described, for example, here, this integral evaluates to ; in other words, is generated by .
The graded dimension of the trivial part of , on the other hand, is
which, again using the complex-analytic method, evaluates to ; this agrees with the combinatorial answer we found in the previous post by counting walks on the infinite cycle graph.
Let acting by rotation matrices. Then the Hilbert series of is
It’s not hard to see that are always polynomial invariants, so we expect the denominator of the above to have a and a in it. In fact, it’s not hard to show that
by comparing the residue of both sides at an root of unity and at . This will let us describe the structure of as follows. The three invariants satisfy the relation
so the ring they generate is isomorphic to . By always replacing with , every monomial in this ring is either a monomial in or a monomial in , and since has degree and have degree the Hilbert series of this ring agrees with the Hilbert series of ; in other words, the relation we’ve identified is the only relation and
This exhibits the quotient of by the action of as a variety embedded in .
Similarly, the graded dimension of the trivial part of is equal to
and it has the same interpretation as before in terms of counting walks on the cycle graph with vertices.
Since the character of a finite-dimensional unitary representation of a compact group uniquely determines it, it follows that the graded character of a graded representation also uniquely determines it. This implies that it should be possible to interpret identities between certain generating functions in terms of isomorphisms between certain graded representations.
So let with the defining representation yet again. Since we know the symmetric powers of are irreducible, they contain no copy of the trivial representation, so consists only of the constant functions. We know, however, that the graded dimension of the trivial part of is
which is the generating function for the Catalan numbers. This generating function satisfies the Catalan relation , which implies an isomorphism
of graded vector spaces, where denotes the trivial representation in degree zero and denotes the trivial representation in degree one. I am very curious as to whether this isomorphism can be written down explicitly (or, better, canonically); if so, it should be possible to construct a basis for which is in bijection with binary trees. Presumably one can wrestle such a basis out of the description of the graded parts of given in the last post, but it would be interesting if there were a general construction which specialized to this.