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

In 2002, Feigin and Veselov [4] defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of [2]. While many properties of those spaces were proven in [3, 4, 5, 7] from this definition, an explicit computation of a basis was only done in certain cases. In particular, in [4], bases for m-quasiinvariants were computed for dihedral groups, including S3, and Felder and Veselov [5] also computed the non-symmetric m-quasiinvariants of lowest degree for general Sn. In this paper, we provide a new characterization of the m-quasiinvariants of Sn, and use this to provide a basis for the isotypic component indexed by the partition [n − 1, 1]. This builds on a previous paper, [1], in which we computed a basis for S3 via combinatorial methods.