Compatibility of Higher Specht Polynomials and Decompositions of Representations

TL;DR

This paper develops a stable normalization for higher Specht polynomials and analyzes their decomposition under certain group actions, linking these to representation theory and branching rules.

Contribution

It introduces a stable version of higher Specht polynomials and explores their behavior under non-transitive group actions, connecting to representation theory.

Findings

Defined a compatible normalization for higher Specht polynomials.

Decomposed non-transitive actions into orbits respecting polynomial bases.

Linked orbit structures to representation lifting and branching rules.

Abstract

%We show how to normalize the higher Specht polynomials of Ariki, Terasoma, and Yamada in a compatible way in order to define a stable version of these polynomials. We also decompose the non-transitive actions of Haglund, Rhoades, and Shimozono into orbits, and show how the associated basis of higher Specht polynomials of Gillespie and Rhoades respects that decomposition. For a given $n$, the orbits of the action of $S_{n}$ are associated with subsets of the set of positive integers that are smaller than $n$, and we relate the representation associated with a set $I$ to the ones of $S_{n+1}$ associated with $I$ and with its union with $n$, the latter being a lifting of the Branching Rule.