The principle investigator will continue his research on problems involving the interaction between combinatorics and algebra. He will focus on two problems. The first concerns new conjectures arising from the principle investigator's application of Kazhdan-Lusztig theory to a problem on combinatorial immanants which assert that certain Hecke algebra virtual characters are polynomials with with non-negative, even unimodal, integer coefficients. A combinatorial approach to these conjectures will be sought, along with combinatorial interpretations of known non-negativity results in Kazhdan-Lusztig theory which now rest on difficult geometric machinery. The second problem concerns the conjecture that the Shur function expansion of Macdonald's symmetric functions are polynomials with non-negative integer coefficients. This project is on the interface between combinatorics and Lie algebra. The principle investigator is concerned with certain polynomials which are combinatorially defined and have deep connections with structures in geometry and representation theory. This work is important both in mathematics and physics.
