Path tableaux and combinatorial interpretations of immanants for class functions on S n

Abstract

Let χ λ be the irreducible S n-character corresponding to the partition λ of n, equivalently, the preimage of the Schur function s λ under the Frobenius characteristic map. Let φ λ be the function S n → ℂ which is the preimage of the monomial symmetric function m λ under the Frobenius characteristic map. The irreducible character immanant Imm λ (x) =∑ wisin;Sn χ λ(w) x 1,w 1 ⋯ x n,wn evaluates nonnegatively on each totally nonnegative matrix A. We provide a combinatorial interpretation for the value Imm λ (A) in the case that λ is a hook partition. The monomial immanant Imm φλ(x) =∑ w∈Sn φ λ(w)x 1,w1 ⋯ x n,wn is conjectured to evaluate nonnegatively on each totally nonnegative matrix A. We confirm this conjecture in the case that λ is a two-column partition by providing a combinatorial interpretation for the value Imm φλ(A). © 2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.