Adding \pm 1 to the argument of an Hall-Littlewood polynomial

TL;DR

This paper explores how shifting the power sums by b1 1 affects Hall-Littlewood polynomials, leading to new descriptions involving plane partitions and connections to Rogers-Ramanujan identities.

Contribution

It introduces a transformation of symmetric functions via b1 1 shifts and derives new representations and identities for Hall-Littlewood polynomials.

Findings

Describes the effect of b1 1 shifts on Hall-Littlewood polynomials

Provides a plane partition-based description of these polynomials

Recovers an identity related to Rogers-Ramanujan identities

Abstract

Shifting by \pm 1 powers sums: p_i \to p_i \pm 1 induces a transformation on symmetric functions that we detail in the case of Hall-Littlewood polynomials. By iteration, this gives a description of these polynomials in terms of plane partitions, as well as some generating functions. We recover in particular an identity of Warnaar related to Rogers-Ramanujan identities.