Compatibility of Higher Specht Polynomials and Decompositions of Representations
Shaul Zemel
TL;DR
This work develops a stable, combinatorial framework for higher Specht polynomials and their representations. It first normalizes higher Specht polynomials and constructs stable, infinite-variable limits compatible with embedding $S_n \hookrightarrow S_{n+1}$, using evacuation and ct-destandardization tied to RSK. It then decomposes the $S_n$-action on higher Specht polynomials into irreducible components indexed by standard and cocharge tableaux, with multiplicities governed by Kostka numbers, and identifies a natural lift to orbit representations on ordered partitions. Finally, it builds explicit maps between representations via star/bar insertions and Ind/Ext operations, lifting the Branching Rule to polynomial representations and establishing stability results as $n$ grows. The results unify normalization, decomposition, and interlevel maps, enabling a systematic passage to stable, infinite-variable limits and offering a robust framework for representation stability in this combinatorial setting.
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.