Hook-Shape Immanant Characters from Stanley-Stembridge Characters
Nathan R. T. Lesnevich
TL;DR
This work advances the understanding of Schur-positivity for monomial immanants of Jacobi-Trudi matrices by focusing on hook-shaped immanant characters. It develops a representation-theoretic framework tying immanants to Hessenberg-pattern data and Kostka numbers, culminating in a explicit decomposition: for hook partitions $\\theta=(N-k,1,...,1)$, the immanant character $\\Gamma^{\\theta}_{\\mu/\\nu}$ is a nonnegative integer sum of Stanley-Stembridge characters with a concrete formula. The results yield new cases where the Stanley-Stembridge conjecture holds (e.g., pre-abelian or small Hessenberg cases) and provide computational reductions that streamline calculations of immanant characters. Collectively, the paper narrows the gap toward the general conjecture by delivering a complete hook-case proof and practical tools for broader applicability in symmetric-function positivity problems.
Abstract
We consider the Schur-positivity of monomial immanants of Jacobi-Trudi matrices, in particular whether a non-negative coefficient of the trivial Schur function implies non-negative coefficients for other Schur functions in said immanants. We prove that this true for hook-shape Schur functions using combinatorial methods in a representation theory setting. Our main theorem proves that hook-shape immanant characters can be written as finite non-negative integer sums of Stanley-Stembridge characters, and provides an explicit combinatorial formula for these sums. This resolves a special case of a longstanding conjecture of Stanley and Stembridge that posits such a sum exists for all immanant characters. We also provide several simplifications for computing immanant characters, and several corollaries applying the main result to cases where the coefficient of the trivial Schur function in monomial immanants of Jacobi-Trudi matrices is known to be non-negative.