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Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions

Ilse Fischer, Moritz Gangl

Abstract

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_λ$ arise as partition functions of certain path configurations in the $\mathfrak{sl}_2$ higher spin six vertex models. They are multiparameter generalizations of the classical Hall-Littlewood symmetric polynomials. We establish two new generalizations of the classical Littlewood identity, where we express a weighted sum of $F_λ$'s over all partitions $λ$ as a product of the Littlewood kernel and another simple product in one case, and a product of the Littlewood kernel and a Pfaffian in the other case. As a corollary we obtain a novel Littlewood identity for Hall-Littlewood symmetric polynomials. We also elaborate on the newly established connection between the fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_λ$ and the modified Robbins polynomials, the latter being multivariate generating functions for alternating sign matrices. This connection allowed us to discover the two generalizations of the Littlewood identity and we provide a bijection between the underlying combinatorial models in the case where $λ$ is strictly decreasing.

Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions

Abstract

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions arise as partition functions of certain path configurations in the higher spin six vertex models. They are multiparameter generalizations of the classical Hall-Littlewood symmetric polynomials. We establish two new generalizations of the classical Littlewood identity, where we express a weighted sum of 's over all partitions as a product of the Littlewood kernel and another simple product in one case, and a product of the Littlewood kernel and a Pfaffian in the other case. As a corollary we obtain a novel Littlewood identity for Hall-Littlewood symmetric polynomials. We also elaborate on the newly established connection between the fully inhomogeneous spin Hall-Littlewood symmetric rational functions and the modified Robbins polynomials, the latter being multivariate generating functions for alternating sign matrices. This connection allowed us to discover the two generalizations of the Littlewood identity and we provide a bijection between the underlying combinatorial models in the case where is strictly decreasing.

Paper Structure

This paper contains 11 sections, 13 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The combinatorics of the relation between fully inhomogeneous spin Hall-Littlewood symmetric rational functions and alternating sign matrices
  3. The $\mathfrak{sl}_2$ higher spin six vertex model
  4. The modified Robbins polynomials
  5. Modified Robbins polynomials are special cases of fully inhomogeneous spin-Hall Littlewood symmetric rational functions combinatorially
  6. Recurrences for $\mathsf{F}_{\lambda}(u_1,\ldots,u_n | s_0,s_1,\ldots)$ and the left-hand sides of the identities
  7. Proof of Theorem \ref{['main1']}
  8. Proof of Theorem \ref{['main2']}
  9. Reducing \ref{['to be eased by notation']} to Lemma \ref{['lem: Key lemma for main2']}
  10. Reducing \ref{['eq:to show for main 2']} to Lemma \ref{['lem: Key lemma for main2']}
  11. Concluding remarks

Key Result

Theorem 1.1

Let $n,p$ be non-negative integers, then where we set $s_j=s$ for all $j \geqslant p$ and $u_i = \frac{s+x_i}{1 + s x_i}$ for all $i$ and a variable $s$, and consider both sides as power series in $x_1,x_2,\ldots,x_n$.

Figures (5)

  • Figure 1: An ensemble of paths in the $\mathfrak{sl}_2$ higher spin six vertex model for $\lambda=(5,5,2,0)$ with additional horizontal steps at the start and additional vertical steps at the top.
  • Figure 2: $\mathfrak{sl}_2$ higher spin six vertex model weights
  • Figure 3: All possible configurations and their weights in the case $s_j=-q^{-1/2}$.
  • Figure 4: An ensemble of paths and the corresponding monotone triangle
  • Figure 5: The normalized weights $\tilde{w}_{x,-q^{-1/2}}$

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 15 more