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

In 2002, Feigin and Veselov [M. Feigin, A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002) 521-545] defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of [O.A. Chalykh, A.P. Veselov, Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990) 597-611]. While many properties of those spaces were proven in [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 3 (2002) 555-566; M. Feigin, A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002) 521-545; G. Felder, A.P. Veselov, Action of Coxeter groups on m-harmonic polynomials and Knizhnik-Zamolodchikov equations, Mosc. Math. J. 4 (2003) 1269-1291; A. Garsia, N. Wallach, The non-degeneracy of the bilinear form of m-quasi-invariants, Adv. in Appl. Math. 3 (2006) 309-359. [7]] from this definition, an explicit computation of a basis was only done in certain cases. In particular, in [M. Feigin, A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002) 521-545], bases for m-quasiinvariants were computed for dihedral groups, including S₃, and Felder and Veselov [G. Felder, A.P. Veselov, Action of Coxeter groups on m-harmonic polynomials and Knizhnik-Zamolodchikov equations, Mosc. Math. J. 4 (2003) 1269-1291] also 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, [J. Bandlow, G. Musiker, Quasiinvariants of S₃, J. Combin. Theory Ser. A 109 (2005) 281-298], in which we computed a basis for S₃ via combinatorial methods.