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Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation

Jean-Christophe Novelli, Jean-Yves Thibon

TL;DR

This work builds a cohesive noncommutative extension of chromatic quasisymmetric functions by embedding Shareshian–Wachs theory into $\mathbf{WQSym}$, formulating noncommutative Macdonald polynomials that align with the Haglund–Wilson framework and its multi-$t$ analogues. It then links these objects to Yang–Baxter elements in Hecke algebras, framing Macdonald polynomials as equivariant traces and exploring q=0 specializations with representation-theoretic interpretations via $U_q(\widehat{\mathfrak{sl}}_n)$. The paper further analyzes Bruhat-interval traces, Lascoux factorizations, and modular relations governed by the Kazhdan–Lusztig basis, shedding light on positivity and structural connections between noncommutative combinatorics and Hecke-algebra theory. Altogether, it proposes a unifying perspective that bridges noncommutative symmetric functions, LLT polynomials, and Macdonald theory through algebraic and combinatorial constructions.

Abstract

As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-$t$ version of these noncommutative analogues. For rectangular partitions, their commutative images at $q=0$ appear to coincide with the multi-$t$ Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi-$t$ Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.

Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation

TL;DR

This work builds a cohesive noncommutative extension of chromatic quasisymmetric functions by embedding Shareshian–Wachs theory into , formulating noncommutative Macdonald polynomials that align with the Haglund–Wilson framework and its multi- analogues. It then links these objects to Yang–Baxter elements in Hecke algebras, framing Macdonald polynomials as equivariant traces and exploring q=0 specializations with representation-theoretic interpretations via . The paper further analyzes Bruhat-interval traces, Lascoux factorizations, and modular relations governed by the Kazhdan–Lusztig basis, shedding light on positivity and structural connections between noncommutative combinatorics and Hecke-algebra theory. Altogether, it proposes a unifying perspective that bridges noncommutative symmetric functions, LLT polynomials, and Macdonald theory through algebraic and combinatorial constructions.

Abstract

As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to , the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi- version of these noncommutative analogues. For rectangular partitions, their commutative images at appear to coincide with the multi- Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi- Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis.

Paper Structure

This paper contains 20 sections, 13 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. General setup
  3. Examples
  4. The $\omega$-involution
  5. The $(q-1)$-transform of unicellular LLT polynomials
  6. Noncommutative Macdonald polynomials
  7. Multi-$t$-analogues and $R$-matrices
  8. Multi $t$ analogues of the Macdonald polynomials
  9. The $q=0$ specialization
  10. Connection with the irreducible representations of $U_q(\widehat{\mathfrak{sl}}_n)$
  11. Computing the $c_{\lambda\mu}$
  12. Characters of $H_N(q)$
  13. Yang-Baxter elements in the Hecke algebra
  14. Yang-Baxter elements and Macdonald polynomials
  15. Generic traces of Bruhat intervals in the Hecke algebra
  16. ...and 5 more sections

Key Result

Proposition 2.1

$G\mapsto {\bf X}_G(A)$ is a morphism of Hopf algebras from ${\mathcal{G}}$ to ${\bf WQSym}$.

Theorems & Definitions (19)

  • Proposition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Theorem 4.1
  • ...and 9 more