Havi Ellers

Thesis Advisor: Dr. Michael Orrison (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Hadi Salmasian (University of Ottawa - Department of Mathematics)

Contact: hellers@hmc.edu

On the Mysteries of Interpolation Jack Polynomials

Interpolation Jack polynomials are certain symmetric polynomials in N variables with coefficients that are rational functions in another parameter k, indexed by partitions of length at most N. Introduced first in 1996 by F. Knop and S. Sahi, and later studied extensively by Sahi, Knop-Sahi, and Okounkov-Olshanski, they have interesting connections to the representation theory of Lie algebras. Given an interpolation Jack polynomial we would like to differentiate it with respect to the variable k and write the result as a linear combination of other interpolation Jack polynomials where the coefficients are again rational functions in k. In this thesis we present proofs of expressions for a few special cases of these coefficients, and develop a general matrix formula which does not provide a concrete formula but can act as a starting point for computing such a formula.