A new characterization for the m-quasiinvariants of S_n and explicit basis for two row hook shapes

TL;DR

This paper introduces a new way to characterize m-quasiinvariants of the symmetric group S_n and constructs an explicit basis for the isotypic component corresponding to the partition [n-1,1], advancing the understanding of these algebraic objects.

Contribution

It provides a novel characterization of m-quasiinvariants of S_n and explicitly constructs a basis for a specific isotypic component, extending previous computational results.

Findings

New characterization of m-quasiinvariants of S_n

Explicit basis for the [n-1,1] isotypic component

Extension of previous combinatorial basis computations

Abstract

In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of Chalykh and Veselov. While many properties of those spaces were proven from this definition, an explicit computation of a basis was only done in certain cases. In particular, Feigin and Veselov computed bases for the m-quasiinvariants of dihedral groups, including S_3, and Felder and Veselov computed the non-symmetric m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a new characterization of the m-quasiinvariants of S_n, and use this to provide a basis for the isotypic component indexed by the partition [n-1,1]. This builds on a previous paper in which we computed a basis for S_3 via combinatorial methods.