Paper Info
Title
A new characterization for the m-quasiinvariants of Sn and explicit basis for two row hook shapes
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.
Year
DOI
Venue
2008
10.1016/j.jcta.2008.01.011
J. Comb. Theory, Ser. A
Keywords
Field
DocType
explicit basis,isotypic component,invariants,new characterization,knizhnik-zamolodchikov equation,j. combin,j. bandlow,g. musiker,row hook shape,quasiinvariants,non-symmetric m-quasiinvariants,g. felder,a.p. veselov,symmetric group,calogero–moser operator,m-harmonic polynomial,coxeter group,m. feigin,dihedral group,indexation
Combinatorics,Symmetric group,Dihedral group,Artin group,Pure mathematics,Point group,Commutative ring,Coxeter element,Lie algebra,Mathematics,Coxeter group
Journal
Volume
Issue
ISSN
115
8
Journal of Combinatorial Theory, Series A
Citations 
PageRank 
References 
0
0.34
2
Authors
2
Name
Order
Citations
PageRank
Jason Bandlow1254.10
Gregg Musiker2114.86