Temperley-Lieb Immanants, Key Positivity, and Demazure Crystals

TL;DR

This paper extends Schur positivity results to key positivity using Demazure crystals and shuffle tableaux, providing new combinatorial rules and a characterization of Demazure crystals.

Contribution

It introduces a new Demazure crystal characterization and proves key positivity of Temperley-Lieb immanants, expanding the understanding of key polynomials and their combinatorial structures.

Findings

Temperley-Lieb immanants of flagged Jacobi-Trudi matrices are key positive.

Provides a combinatorial rule for key expansion of flagged skew Schur products.

Establishes a new Demazure crystal characterization.

Abstract

The main goal of this paper is to extend three important Schur positivity results to key positivity, replacing all Schur polynomials in relevant expressions with flagged Schur polynomials. Namely, we first show that the Temperley-Lieb immanants of (many) flagged Jacobi-Trudi matrices are key positive. Using this result, we give a combinatorial rule for the key expansion of (most) products of flagged skew Schur polynomials, and also give a log concavity result inspired by that of Lam-Postnikov-Pylyavskyy. The main tools in our proofs are Demazure crystals, and the recently defined shuffle tableaux of Nguyen and Pylyavskyy. In order to prove our main results, we must develop a new characterization of Demazure crystals, which builds off of prior work of Assaf and Gonzalez. This characterization may be useful in other contexts.