Date of Award

May 2019

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Jeb F Willenbring

Committee Members

Gabriella A Pinter, Allen D Bell, Yi Ming Zou, Kevin McLeod

Keywords

harmonic polynomials, multiplicity, polyhedron, representation theory, Vinberg

Abstract

We consider a family of examples falling into the following context (first considered by

Vinberg): Let G be a connected reductive algebraic group over the complex numbers. A

subgroup, K, of fixed points of a finite-order automorphism acts on the Lie algebra of G.

Each eigenspace of the automorphism is a representation of K. Let g1 be one of the

eigenspaces. We consider the harmonic polynomials on g1 as a representation of K, which

is graded by homogeneous degree. Given any irreducible representation of K, we will see

how its multiplicity in the harmonic polynomials is distributed among the various graded

components. The results are described geometrically by counting integral points on faces of

a polyhedron. The multiplicity in each graded component is given by intersecting these

faces with an expanding sequence of shells.

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