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Temperley-Lieb Immanants, Key Positivity, and Demazure Crystals

Rosa Paten, Dora Woodruff

TL;DR

This work extends classical Schur-positivity results to the realm of key positivity by replacing Schur polynomials with flagged Schur polynomials and leveraging Demazure-crystal machinery. The authors prove that Temperley-Lieb immanants of many flagged Jacobi-Trudi matrices are key-positive and develop a practical Demazure-crystal criterion (extension and gluing) to analyze flagged shuffle tableaux, enabling a flagged Littlewood-Richardson rule and a log-concavity analogue. They also introduce a two-local-property characterization of Demazure crystals, providing new local tools for crystal-theoretic proofs and broad applicability beyond flagged tableaux. The results establish a robust combinatorial framework that links TL-immanants, Demazure crystals, shuffle tableaux, and key polynomials, with implications for positivity phenomena in representation-theoretic and algebraic-combinatorial contexts.

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.

Temperley-Lieb Immanants, Key Positivity, and Demazure Crystals

TL;DR

This work extends classical Schur-positivity results to the realm of key positivity by replacing Schur polynomials with flagged Schur polynomials and leveraging Demazure-crystal machinery. The authors prove that Temperley-Lieb immanants of many flagged Jacobi-Trudi matrices are key-positive and develop a practical Demazure-crystal criterion (extension and gluing) to analyze flagged shuffle tableaux, enabling a flagged Littlewood-Richardson rule and a log-concavity analogue. They also introduce a two-local-property characterization of Demazure crystals, providing new local tools for crystal-theoretic proofs and broad applicability beyond flagged tableaux. The results establish a robust combinatorial framework that links TL-immanants, Demazure crystals, shuffle tableaux, and key polynomials, with implications for positivity phenomena in representation-theoretic and algebraic-combinatorial contexts.

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.
Paper Structure (31 sections, 36 theorems, 112 equations)

This paper contains 31 sections, 36 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Main results and organization
  3. Acknowledgements
  4. Preliminaries
  5. Skew Schur polynomials and tableau conventions
  6. Flagged Schur polynomials
  7. Key polynomials
  8. Jacobi-Trudi matrices
  9. Temperley-Lieb Immanants
  10. Shuffle Tableaux
  11. Permutation conventions
  12. Crystal conventions
  13. Crystal operators and shuffle tableaux
  14. Demazure crystals
  15. Characterizing Demazure Crystals
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\mu$ be a partition, $\lambda$ be a strict partition, and $\vec{b}$ be a nondecreasing flag. Then, all Temperley-Lieb immanants of the flagged Jacobi-Trudi matrix $A_{\lambda, \mu}^{\vec{b}}$ are key nonnegative.

Theorems & Definitions (120)

  • Theorem 1.1
  • Example 1.2: Counterexample for nonstrict $\lambda$
  • Theorem 1.3
  • Corollary 1.4
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 110 more