Date of Award
Doctor of Philosophy
Jeb F. Willenbring
Allen Bell, Craig Guilbault, Peter Hinow, Yi Ming Zou
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.
Johnson, Wayne Andrew, "Multivariate Hilbert Series of Lattice Cones and Homogeneous Varieties" (2015). Theses and Dissertations. 1001.