Date of Award

August 2015

Degree Type


Degree Name

Doctor of Philosophy



First Advisor

Jeb F. Willenbring

Committee Members

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.

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