Cyclic sieving, skew Macdonald polynomials and Schur positivity

TL;DR

This paper explores the cyclic sieving phenomenon in specialized Macdonald polynomials, introduces skew versions, and demonstrates their Schur positivity and connections to LLT polynomials, advancing combinatorial representation theory.

Contribution

It introduces skew Macdonald polynomials, proves their Schur positivity, and links them to LLT polynomials, expanding the combinatorial understanding of these objects.

Findings

Cyclic sieving phenomenon occurs for partitions with parts multiple of n.

Skew Macdonald polynomials are symmetric and Schur-positive.

Connections established between specialized Macdonald polynomials and LLT polynomials.

Abstract

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ is a multiple of $n$, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under a cyclic shift of the columns. The corresponding CSP polynomial is given by $E_{\lambda}(x;q;0)$. In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B.~Rhoades. We also introduce a skew version of $E_{\lambda}(x;q;0)$. We show that these are symmetric and Schur-positive via a variant of the Robinson--Schenstedt--Knuth correspondence and we also describe crystal raising- and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials.