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# Dissertation Defense: Alexander Heaton

## April 2, 2019 @ 2:00 pm - 4:00 pm

## Graded Multiplicity in Harmonic Polynomials from the Vinberg Setting

**Alexander Heaton**

**University of Wisconsin-Milwaukee**

*PhD Graduate Student – Dissertator*

“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 V be one of the eigenspaces. We consider the harmonic polynomials on V 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.”

**Committee Members:**

Profs. Jeb Willenbring (Advisor); Allen Bell, Kevin McLeod, Gabriella Pinter & Yi Ming Zou