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

Jason Bandlow, Gregg Musiker

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1 Scopus citations

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 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 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, [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.

Original languageEnglish (US)
Pages (from-to)1333-1357
Number of pages25
JournalJournal of Combinatorial Theory. Series A
Volume115
Issue number8
DOIs
StatePublished - 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.

Keywords

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

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