Co-quasi-invariant spaces for finite complex reflection groups

2011/07/04 by Jean-Christophe Aval, François Bergeron, Aval, Jean-Christophe +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1107.0537

Abstract

We study, in a global uniform manner, the quotient of the ring of polynomials in l sets of n variables, by the ideal generated by diagonal quasi-invariant polynomials for general permutation groups W=G(r,n). We show that, for each such group W, there is an explicit universal symmetric function that gives the N^l-graded Hilbert series for these spaces. This function is universal in that its dependance on l only involves the number of variables it is calculated with. We also discuss the combinatorial implications of the observed fact that it affords an expansion as a positive coefficient polynomial in the complete homogeneous symmetric functions.

Related

• Toward explicit Hilbert series of quasi-invariant polynomials in characteristic p and q-deformed quasi-invariants
• Multivariate Diagonal Coinvariant Spaces for Complex Reflection Groups
• Invariant theory for coincidental complex reflection groups
• Hilbert Series of S₃-Quasi-Invariant Polynomials in Characteristics 2, 3
• Invariant rings and representations of the symmetric groups