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A Littlewood-type identity for Robbins polynomials

Ilse Fischer, Hans Höngesberg

TL;DR

This work generalizes the classical Littlewood identity to a Littlewood-type identity for modified Robbins polynomials $R^{*}_{\boldsymbol{k}}$, revealing a deep link between symmetric-function theory, down-arrowed monotone triangles (DAMTs), and diagonally symmetric alternating sign matrices (DSASMs). The main result expresses the sum over $R^{*}_{\boldsymbol{k}}$ as a product of a simple prefactor and a Pfaffian, while interpreting the right-hand side as a DSASM partition function from a six-vertex model. A comprehensive combinatorial model (DAMTs) for $R^{*}_{\boldsymbol{k}}$ is developed, together with two proofs of the identity, including an Izergin–Korepin-style outline. The paper further analyzes the leading-term coefficient of the DSASM partition function via Pfaffian transforms, derives recovery routes to the classical Littlewood identity, and proposes conjectural generalizations to fully inhomogeneous spin Hall–Littlewood polynomials, hinting at broader connections to Hall–Littlewood theory and potential bijective correspondences among ASM-related objects.

Abstract

We provide a generalization of the Littlewood identity, both sides of which are related to alternating sign matrices. The classical Littlewood identity establishes a nice product formula for the sum of all Schur polynomials. Compared to the classical identity, Schur polynomials are replaced by so-called modified Robbins polynomials. These polynomials are a generalization of Schur polynomials and enumerate down-arrowed monotone triangles, and thus also alternating sign matrices. As an additional factor on the other side of the identity, we have a Pfaffian formula which we interpret in terms of the partition function of six-vertex model configurations corresponding to diagonally symmetric alternating sign matrices.

A Littlewood-type identity for Robbins polynomials

TL;DR

This work generalizes the classical Littlewood identity to a Littlewood-type identity for modified Robbins polynomials , revealing a deep link between symmetric-function theory, down-arrowed monotone triangles (DAMTs), and diagonally symmetric alternating sign matrices (DSASMs). The main result expresses the sum over as a product of a simple prefactor and a Pfaffian, while interpreting the right-hand side as a DSASM partition function from a six-vertex model. A comprehensive combinatorial model (DAMTs) for is developed, together with two proofs of the identity, including an Izergin–Korepin-style outline. The paper further analyzes the leading-term coefficient of the DSASM partition function via Pfaffian transforms, derives recovery routes to the classical Littlewood identity, and proposes conjectural generalizations to fully inhomogeneous spin Hall–Littlewood polynomials, hinting at broader connections to Hall–Littlewood theory and potential bijective correspondences among ASM-related objects.

Abstract

We provide a generalization of the Littlewood identity, both sides of which are related to alternating sign matrices. The classical Littlewood identity establishes a nice product formula for the sum of all Schur polynomials. Compared to the classical identity, Schur polynomials are replaced by so-called modified Robbins polynomials. These polynomials are a generalization of Schur polynomials and enumerate down-arrowed monotone triangles, and thus also alternating sign matrices. As an additional factor on the other side of the identity, we have a Pfaffian formula which we interpret in terms of the partition function of six-vertex model configurations corresponding to diagonally symmetric alternating sign matrices.
Paper Structure (8 sections, 13 theorems, 133 equations, 1 figure, 4 tables)

This paper contains 8 sections, 13 theorems, 133 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Modified Robbins polynomials
  3. Down-arrowed monotone triangles
  4. Proof of Theorem \ref{['thm:Robbins_Littlewood']}
  5. Six-vertex model configurations
  6. The coefficient of the highest term in the polynomial expansion of the partition function
  7. How to recover the classical Littlewood identity
  8. A conjectural generalization of Theorem \ref{['thm:Robbins_Littlewood']} for fully inhomogeneous spin Hall--Littlewood polynomials

Key Result

Theorem 1.1

Let $n$ be a positive integer. Then where $\mathop{\mathrm{Pf}}\limits$ denotes the Pfaffian of an upper triangular array and $\chi_{\mathrm{even}}(n)$ equals $1$ if $n$ is even and $0$ otherwise.

Figures (1)

  • Figure 1: The partition function $Z_{\mathop{\mathrm{DSASM}}\nolimits}(x_1,\ldots,x_n)$ with respect to the weights in Table \ref{['tab:six-vertex weightsw=0']} is given by the generating function of perfect matchings of the grid graph in \ref{['subfig:graph']} (here for $n=5$), where the red vertices on the left boundary may remain uncovered. The corresponding edge weights are provided in \ref{['subfig:weights']}.

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 3.1
  • Example 3.2
  • Definition 3.3
  • ...and 22 more