## Abstract

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 S_{3}, 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 S_{3}, J. Combin. Theory Ser. A 109 (2005) 281-298], in which we computed a basis for S_{3} via combinatorial methods.

Original language | English (US) |
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Pages (from-to) | 1333-1357 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 115 |

Issue number | 8 |

DOIs | |

State | Published - Nov 2008 |

### Bibliographical note

Funding Information:The authors are grateful for Adriano Garsia’s guidance in this project as well as the support of the NSF. We also would like to thank Vic Reiner for conversations which helped motivate the analysis of Section 8.

Copyright:

Copyright 2008 Elsevier B.V., All rights reserved.

## Keywords

- Calogero-Moser operator
- Invariants
- Isotypic component
- Quasiinvariants
- Symmetric group