Multivariate Hilbert Series of Lattice Cones and Homogeneous Varieties

Abstract

We consider the dimensions of irreducible representations whose highest weights lie on a given finitely generated lattice cone. We present a rational function representing the multivariate formal power series whose coefficients encode these dimensions. This result generalizes the formula for the Hilbert series of an equivariant embedding of an homogeneous projective variety. We use the multivariate generating function to compute Hilbert series for the Kostant cones and other affine and projective varieties of interest in representation theory. As a special case, we show how the multivariate series can be used to compute the Hilbert series of the three classical families of determinantal variety.