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Temperley-Lieb Crystals

Son Nguyen, Pavlo Pylyavskyy

Abstract

Elements of Lusztig's dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis - called Temperley-Lieb immanants - we give a generalization of Littlewood-Richardson rule: we provide a combinatorial interpretation for the coefficient of a particular Schur function in the evaluation of a particular Temperley-Lieb immanant on a particular Jacobi-Trudi matrix. For this we introduce shuffle tableaux, and apply Stembridge's axioms to show that certain graphs on shuffle tableaux are type $A$ Kashiwara crystals.

Temperley-Lieb Crystals

Abstract

Elements of Lusztig's dual canonical bases are Schur-positive when evaluated on (generalized) Jacobi-Trudi matrices. This deep property was proved by Rhoades and Skandera, relying on a result of Haiman, and ultimately on the (proof of) Kazhdan-Lusztig conjecture. For a particularly tractable part of the dual canonical basis - called Temperley-Lieb immanants - we give a generalization of Littlewood-Richardson rule: we provide a combinatorial interpretation for the coefficient of a particular Schur function in the evaluation of a particular Temperley-Lieb immanant on a particular Jacobi-Trudi matrix. For this we introduce shuffle tableaux, and apply Stembridge's axioms to show that certain graphs on shuffle tableaux are type Kashiwara crystals.
Paper Structure (10 sections, 18 theorems, 65 equations, 15 figures)

This paper contains 10 sections, 18 theorems, 65 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Generalized Jacobi-Trudi matrix and wiring
  4. Temperley-Lieb immanant
  5. Colored paths
  6. Products of minors
  7. Shuffle tableaux
  8. Crystal operators
  9. Stembridge's axioms
  10. Generalized Littlewood-Richardson rule

Key Result

Theorem 2.6

Let $G$ be a planar network of order $n$ and $A$ be its path matrix. For any basis element $\tau$ of $TL_n(2)$, where the sum is over all wiring $H$ of $G$ such that $\psi(H) = \tau$.

Figures (15)

  • Figure 1: Generalized wiring
  • Figure 2: Two colored covers of a wiring
  • Figure 3: A colored cover
  • Figure 4: A canonical colored cover
  • Figure 5: Another colored cover
  • ...and 10 more figures

Theorems & Definitions (54)

  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6: rhoades2005temperley
  • Definition 2.7
  • Example 2.8
  • Proposition 2.9: skandera2004inequalities
  • proof : Proof sketch
  • ...and 44 more