Hall-Littlewood polynomials and a Hecke action on ordered set partitions

TL;DR

This paper constructs a Hecke algebra action on a quotient of polynomial rings, linking combinatorics of ordered set partitions with quantum algebra, generalizing previous results at specific parameter values.

Contribution

It introduces a new Hecke algebra action on polynomial quotients related to ordered set partitions, providing a quantum interpolation of prior combinatorial constructions.

Findings

Dimension matches the number of k-block ordered set partitions

Provides a quantum analog interpolating known results at q=0 and q=1

Establishes a new algebraic framework connecting polynomials and set partitions

Abstract

We construct an action of the Hecke algebra $H_n(q)$ on a quotient of the polynomial ring $F[x_1, \dots, x_n]$, where $F = \mathbb{Q}(q)$. The dimension of our quotient ring is the number of $k$-block ordered set partitions of $\{1, 2, \dots, n \}$. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at $q = 1$ and work of Huang-Rhoades at $q = 0$.