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Highest weight crystals for Schur Q-functions

Eric Marberg, Kam Hung Tong

Abstract

Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur $P$-polynomials. This article studies a modified form of this category, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur $Q$-polynomials. The crystals in this category have some interesting features not present for ordinary $\mathfrak{q}_n$-crystals. For example, there is an extra crystal operator, a different tensor product, and an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of $\mathfrak{q}_n$-crystal structures on certain families of shifted tableaux and factorized reduced words. We describe extended forms of these structures that give similar examples in our new category.

Highest weight crystals for Schur Q-functions

Abstract

Work of Grantcharov et al. develops a theory of abstract crystals for the queer Lie superalgebra . Such -crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur -polynomials. This article studies a modified form of this category, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur -polynomials. The crystals in this category have some interesting features not present for ordinary -crystals. For example, there is an extra crystal operator, a different tensor product, and an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of -crystal structures on certain families of shifted tableaux and factorized reduced words. We describe extended forms of these structures that give similar examples in our new category.
Paper Structure (30 sections, 63 theorems, 80 equations, 2 figures)

This paper contains 30 sections, 63 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Crystals for Schur functions and Schur $P$-functions
  4. Crystals for Schur $Q$-functions
  5. Connections to representation theory
  6. Comparison with Gillespie-Levinson-Purbhoo crystals
  7. Outline
  8. Preliminaries
  9. Shifted tableaux
  10. Symmetric polynomials
  11. Abstract crystals
  12. Crystals for general linear Lie algebras
  13. Crystals for queer Lie superalgebras
  14. Extended queer supercrystals
  15. Crystal operators on words
  16. ...and 15 more sections

Key Result

Theorem 1.1

If $\mathcal{B}$ is a connected normal $\mathfrak{gl}_n$-crystal, then $\mathcal{B}$ has a unique $\mathfrak{gl}_n$-highest weight element, whose weight $\lambda \in \mathbb{Z}^n$ is a partition such that $\mathsf{ch}(\mathcal{B})=s_\lambda(x_1,x_2,\dots,x_n)$. For each partition $\lambda \in \mathb

Figures (2)

  • Figure 1: Crystal graph of $\mathfrak{q}^+_3$-crystal $\mathsf{Incr}^+_3(z)$ for $z=(1,3)(2,4)\in I_\mathbb{Z}$. In this picture we draw styled edges without labels for clarity. Solid blue and red arrows are edges $b \xrightarrow{\ 1\ } c$ and $b \xrightarrow{\ 2\ } c$, respectively. Dotted green and dashed blue arrows are edges $b \xrightarrow{\ 0\ } c$ and $b \xrightarrow{\ \overline 1\ } c$, respectively.
  • Figure 2: Crystal graph of $\mathfrak{q}^+_3$-crystal $\mathsf{ShTab}^+_3(\lambda)$ for $\lambda=(2,1)$. In this picture we draw styled edges without labels for clarity. Solid blue and red arrows are edges $b \xrightarrow{\ 1\ } c$ and $b \xrightarrow{\ 2\ } c$, respectively. Dotted green and dashed blue arrows are edges $b \xrightarrow{\ 0\ } c$ and $b \xrightarrow{\ \overline 1\ } c$, respectively.

Theorems & Definitions (139)

  • Theorem 1.1: See BumpSchilling
  • Corollary 1.2
  • Theorem 1.3: See GJKKK
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 3.1: See BumpSchilling
  • Theorem 3.2: See BumpSchilling
  • Remark 3.3
  • ...and 129 more