Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups

Shaul Zemel

TL;DR

The paper develops a unified, representation-theoretic framework for decomposing the $S_{n}$-action on homogeneous polynomials and related quotients, lifting two classical multiplicity formulae to explicit direct-sum decompositions into Specht modules. It introduces generalized higher Specht polynomials and their homogeneous variants, providing explicit bases $F_{M,T}$ and $F_{C,T}^{I}$ (and $F_{C,T}^{I,\\mathrm{hom}}$) that realize irreducible components and yield liftings of the Kostka-number and cocharge-formula structures. By indexing representations with weak compositions and multi-sets, the paper builds the quotients $R_{n,k,s}$ and their homogeneous lifts $R_{n,I}$, $R_{n,I}^{\\mathrm{hom}}$, and proves how these decompose compatibly with orbit decompositions of ordered partitions, the usual induction/restriction rules, and stability phenomena. The results provide explicit, stable decompositions across $n$, $k$, and $s$, and establish a robust framework for analyzing $S_{n}$-representations on polynomial rings and their homogeneous quotients, with connections to FI-modules and central stability.

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.

Generalized Higher Specht Polynomials and Homogeneous Representations of Symmetric Groups

TL;DR

The paper develops a unified, representation-theoretic framework for decomposing the -action on homogeneous polynomials and related quotients, lifting two classical multiplicity formulae to explicit direct-sum decompositions into Specht modules. It introduces generalized higher Specht polynomials and their homogeneous variants, providing explicit bases and (and ) that realize irreducible components and yield liftings of the Kostka-number and cocharge-formula structures. By indexing representations with weak compositions and multi-sets, the paper builds the quotients and their homogeneous lifts , , and proves how these decompose compatibly with orbit decompositions of ordered partitions, the usual induction/restriction rules, and stability phenomena. The results provide explicit, stable decompositions across , , and , and establish a robust framework for analyzing -representations on polynomial rings and their homogeneous quotients, with connections to FI-modules and central stability.

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 on homogeneous polynomials of degree 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.
Paper Structure (3 sections, 30 theorems, 39 equations)

This paper contains 3 sections, 30 theorems, 39 equations.

Table of Contents

  1. Descents, Multi-Sets, and Representations
  2. Generalized Higher Specht Polynomials
  3. Operations Between Representations

Key Result

Lemma 1.1

Fix $n\geq1$, and consider a multi-set $J$ as above, of size $k-1$, which we write in a non-decreasing as $\{j_{h}\}_{h=1}^{k-1}$, and extend with $j_{0}=0$ and $j_{k}=n$. Then the sequence of integers $\{j_{h+1}-j_{h}\}_{h=0}^{k-1}$ is a weak composition $\operatorname{comp}_{n}J\vDash_{w}n$ of len

Theorems & Definitions (103)

  • Lemma 1.1
  • Remark 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Example 1.7
  • Definition 1.8
  • Lemma 1.9
  • ...and 93 more