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A combinatorial formula for Interpolation Macdonald polynomials

Houcine Ben Dali, Lauren Williams

TL;DR

The paper develops a comprehensive combinatorial framework for interpolation Macdonald polynomials via signed multiline queues and their two-line/ tableaux incarnations. It proves a central formula expressing both interpolation ASEP and symmetric interpolation Macdonald polynomials as weighted sums over signed MLQs and signed queue tableaux, with a detailed recursive and algebraic structure that mirrors the underlying Hecke action. The work connects to Okounkov’s tableau formula, establishes integrality results for the integral forms, and yields a factorization at $q=1$, thereby deepening the link between combinatorial models (MLQ, two-line queues, tableaux) and the algebraic theory of interpolation Macdonald polynomials. This approach provides new tools for computation, structural understanding, and potential applications in related algebraic combinatorics and representation theory contexts.

Abstract

In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials, defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials. Like the Macdonald polynomials, these interpolation Macdonald polynomials are closely connected to the Hecke algebra, and admit nonsymmetric versions, which generalize the nonsymmetric Macdonald polynomials. In this paper we give a combinatorial formula for interpolation Macdonald polynomials in terms of signed multiline queues; this formula generalizes the combinatorial formula for Macdonald polynomials in terms of multiline queues given by Corteel-Mandelshtam-Williams.

A combinatorial formula for Interpolation Macdonald polynomials

TL;DR

The paper develops a comprehensive combinatorial framework for interpolation Macdonald polynomials via signed multiline queues and their two-line/ tableaux incarnations. It proves a central formula expressing both interpolation ASEP and symmetric interpolation Macdonald polynomials as weighted sums over signed MLQs and signed queue tableaux, with a detailed recursive and algebraic structure that mirrors the underlying Hecke action. The work connects to Okounkov’s tableau formula, establishes integrality results for the integral forms, and yields a factorization at , thereby deepening the link between combinatorial models (MLQ, two-line queues, tableaux) and the algebraic theory of interpolation Macdonald polynomials. This approach provides new tools for computation, structural understanding, and potential applications in related algebraic combinatorics and representation theory contexts.

Abstract

In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials, defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials. Like the Macdonald polynomials, these interpolation Macdonald polynomials are closely connected to the Hecke algebra, and admit nonsymmetric versions, which generalize the nonsymmetric Macdonald polynomials. In this paper we give a combinatorial formula for interpolation Macdonald polynomials in terms of signed multiline queues; this formula generalizes the combinatorial formula for Macdonald polynomials in terms of multiline queues given by Corteel-Mandelshtam-Williams.

Paper Structure

This paper contains 25 sections, 42 theorems, 136 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Interpolation polynomials
  3. Multiline queues and signed multiline queues
  4. Combinatorial formulas for ASEP and interpolation ASEP polynomials
  5. Integrality and Comparison with Okounkov's Formula
  6. Background on interpolation polynomials
  7. Notation
  8. Interpolation Macdonald polynomials
  9. Hecke Operators
  10. ASEP and interpolation ASEP polynomials
  11. Characterization of interpolation ASEP polynomials
  12. An algebraic recursion for f* when the composition is packed
  13. An algebraic recursion for f* indexed by arbitrary compositions
  14. Some preliminaries about Hecke operators
  15. Action of Hecke operators on extended ASEP polynomials
  16. ...and 10 more sections

Key Result

Theorem 1.1

Knop1997bSahi1996 For each partition $\lambda=(\lambda_1,\dots,\lambda_n)$, there is a unique inhomogeneous symmetric polynomial $P^*_{\lambda} =P_{\lambda}^*(\boldsymbol{x};q,t) = P_{\lambda}^*(x_1,\dots,x_n;q,t)$ called the interpolation Macdonald polynomial such that Moreover, the top homogeneous component of $P_{\lambda}^*$ is the usual Macdonald polynomial $P_{\lambda}.$

Figures (8)

  • Figure 1: A multiline queue of type $(2,2,0,0,0,3,2,1)$.
  • Figure 2: \ref{['fig:forbidden_configurations_classic']} illustrates the forbidden configurations for the classic layers, while the three other figures (\ref{['fig:forbidden_configurations_ghost']}) show the forbidden configurations for the signed layers. The three figures on the left show two balls on top of each other, which are not trivially paired, whereas the rightmost figure features a regular ball with an empty position beneath it.
  • Figure 3: A signed multiline queue of type $(2,2,0,0,0,2,3,1)$.
  • Figure 4: The signed multiline queues of type $(0,2)$, with their weights superimposed. Note that a ball labeled $\pm 2$ represents the fact that the corresponding ball can either be a positive or a negative ball. Thus, the six diagrams above actually represent $15$ signed multiline queues.
  • Figure 5: An example of a signed two-line queue in $\mathcal{G}_{(0,0,4,4,0,0,3,2,1)}^{(0,4,-4,0,-2,-3,0,1,0)}.$
  • ...and 3 more figures

Theorems & Definitions (112)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 102 more