Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Andrew Baxter**, Rutgers University, baxter{at} math [dot] rutgers [dot] edu**Lara Pudwell**, Rutgers University, lpudwell {at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Generating functions for quantum character immanants

Speaker: **Mark Skandera**, Lehigh University

Date: Thursday, April 3, 2008 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Immanants, interpreted in a broad sense, are certain polynomials in commuting variables whose coefficients are functions from the symmetric group S_n to the complex numbers. Functions most often used in this context are S_n-characters. Goulden and Jackson expressed irreducible character immanants as coefficients in generating functions given by permanents and determinants. Merris and Watkins gave similar expressions for induced character immanants. Deforming the commutative coordinate ring in n^2 variables into the noncommutative quantum coordinate ring, and S_n into the noncommutative Hecke algebra H_n(q), we define quantum immanants for irreducible and induced H_n(q) characters. We show that these arise as coefficients in natural quantizations of the formulae given by Goulden-Jackson and Merris-Watkins. Moreover, our quantized Goulden-Jackson formula is related to the quantized Master Theorem of Garoufalidis-Le-Zeilberger in much the same way that Goulden-Jackson's original formula is related to MacMahon's classical Master Theorem.

This is joint work with Matjaz Konvalinka.