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## Posts Tagged ‘Hilbert series’

Let $G$ be a group and let

$\displaystyle V = \bigoplus_{n \ge 0} V_n$

be a graded representation of $G$, i.e. a functor from $G$ to the category of graded vector spaces with each piece finite-dimensional. Thus $G$ acts on each graded piece $V_i$ 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

$\displaystyle \chi_V(g) = \sum_{n \ge 0} \chi_{V_n}(g) t^n$.

In particular, the graded trace of the identity is the graded dimension or Hilbert series of $V$.

Classically a case of particular interest is when $V_n = \text{Sym}^n(W^{*})$ for some fixed representation $W$, since $V = \text{Sym}(W^{*})$ is the symmetric algebra (in particular, commutative ring) of polynomial functions on $W$ invariant under $G$. In the nicest cases (for example when $G$ is finite), $V$ is finitely generated, hence Noetherian, and $\text{Spec } V$ is a variety which describes the quotient $W/G$.

In a previous post we discussed instead the case where $V_n = (W^{*})^{\otimes n}$ for some fixed representation $W$, hence $V$ is the tensor algebra of functions on $W$. I thought it might be interesting to discuss some generalities about these graded representations, so that’s what we’ll be doing today.

## Set-multiset duality and supervector spaces

Recall that the elementary symmetric functions $e_{\lambda}$ generate the ring of symmetric functions $\Lambda$ as a module over any commutative ring $R$. A corollary of this result, although I didn’t state it explicitly, is that the elementary symmetric functions $e_1, e_2, ...$ are algebraically independent, hence any ring homomorphism from the symmetric functions is determined freely by the images of $e_1, e_2, ...$.

A particularly interesting choice of endomorphism $\omega : \Lambda \to \Lambda$ occurs when we set $\omega(e_n) = h_n$. This endomorphism sends the generating function $E(t)$ for the elementary symmetric polynomials to the generating function $H(t) = \frac{1}{E(-t)}$, and this is an involution – in other words, it follows that $\omega(h_n) = e_n$ and $\omega$ itself must also be an involution on the ring of symmetric functions. Thus the elementary symmetric functions and complete homogeneous symmetric functions are dual in a very strong sense. This is closely related to the identity

$\displaystyle \left( {n \choose d} \right) = (-1)^d {-n \choose d}$

and today I’d like to try to explain one interpretation of what’s going on that I learned from Todd Trimble, which is that “the exterior algebra is the symmetric algebra of a purely odd supervector space.”

## Introduction to symmetric functions

The theory of symmetric functions, which generalizes some ideas that came up in the previous discussion of Polya theory, can be motivated by thinking about polynomial functions of the roots of a monic polynomial $P(x) \in \mathbb{Z}[x]$. Problems on high school competitions often ask for the sum of the squares or the cubes of the roots of a given polynomial, and those not familiar with symmetric functions will often be surprised that these quantities end up being integers even though the roots themselves aren’t integers. The key is that the sums being asked for are always symmetric in the roots, i.e. invariant under an arbitrary permutation. The coefficients of a monic polynomial are the elementary symmetric polynomials of its roots, which we are given are integers. It follows that any symmetric polynomial that can be written using integer coefficients in terms of the elementary symmetric polynomials must in fact be an integer, and as we will see, every symmetric polynomial with integer coefficients has this property.

These ideas lead naturally to the use of symmetric polynomials in Galois theory, but combinatorialists are interested in symmetric functions for a very different reason involving the representation theory of $S_n$. This post is a brief overview of the basic results of symmetric function theory relevant to combinatorialists. I’m mostly following Sagan, who recommends Macdonald as a more comprehensive reference.

Often in mathematics we work in an algebra with the property that the “degree” of an element has a multiplicative property. For example, in a polynomial ring in $n$ variables we can define the degree of a monomial to be the vector of its degrees with respect to each variable, and the product of monomials corresponds to the sum of vectors. More typically we can define the degree of a monomial to be its total degree (the sum of the components of the above vector); this degree is also multiplicative.
Algebras with this additional property are called graded algebras, and they show up surprisingly often in mathematics. As Alexandre Borovik notes, when schoolchildren work with units such as “apples” and “people” they are really working in a $\mathbb{Z}^n$-graded algebra, and one could argue that the study of homogeneous elements (that is, elements of the same degree) in $\mathbb{Z}^n$-graded algebras is the entire content of dimensional analysis.