Date of Award
Doctor of Philosophy
Jeb F Willenbring
Gabriella A Pinter, Allen D Bell, Yi Ming Zou, Kevin McLeod
harmonic polynomials, multiplicity, polyhedron, representation theory, Vinberg
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.
Heaton, Alexander, "Graded Multiplicity in Harmonic Polynomials from the Vinberg Setting" (2019). Theses and Dissertations. 2074.