Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups

TL;DR

This paper extends the theory of higher Specht polynomials to include generalized versions and explores their role in decomposing symmetric group actions on homogeneous polynomials, providing new formulas and stable representations.

Contribution

It introduces generalized higher Specht polynomials with empty sets and multi-sets, enabling new decompositions of symmetric group actions and lifting classical formulas involving Kostka numbers.

Findings

Decomposition of $S_{n}$ action into irreducible representations using generalized polynomials.

Lifting of Stanley's formula through new polynomial bases.

Development of stable versions of generalized Specht polynomials.

Abstract

We consider actions, similar to those of Haglund, Rhoades, and Shimozono on ordered partitions, and their basis in terms of the higher Specht polynomials of Ariki, Terasoma, and Yamada, as carried out by Gillespie and Rhoades. By allowing empty sets and working with multi-sets and weak partitions as indices, we obtain a decomposition of the action of $S_{n}$ on homogeneous polynomials of degree $d$ into irreducible representations, in a way that lifts a formula of Stanley. By considering generalized higher Specht polynomials, we obtain yet another such decomposition, lifting another formula involving Kostka numbers. We also investigate several operations on both types of representations, which are based on normalizations of the generalized higher Specht polynomials that allow for defining their stable versions.