On quasiinvariants of $S_n$ of hook shape

TL;DR

This paper generalizes the description of bases for quasiinvariants of the symmetric group $S_n$ to all hook-shaped standard tableaux, extending previous work limited to the shape $(n-1,1)$.

Contribution

It provides a new basis description for quasiinvariants of $S_n$ associated with arbitrary hook-shaped tableaux, broadening the understanding of their structure.

Findings

Extended basis descriptions to all hook shapes

Connected quasiinvariants with standard tableaux of arbitrary shape

Generalized previous specific shape results

Abstract

Chalykh, Veselov and Feigin introduced the notions of quasiinvariants for Coxeter groups, which is a generalization of invariants. In [2], Bandlow and Musiker showed that for the symmetric group $S_n$ of order $n$, the space of quasiinvariants has a decomposition indexed by standard tableaux. They gave a description of basis for the components indexed by standard tableaux of shape $(n-1,1)$. In this paper, we generalize their results to a description of basis for the components indexed by standard tableaux of arbitrary hook shape.