Abstract
Motivated by permutation statistics, we define, for any complex reflection group W, a family of bivariate generating functions Wσ(t, q). They are defined either in terms of Hilbert series for W-invariant polynomials when W acts diagonally on two sets of variables or, equivalently, as sums involving the fake degrees of irreducible representations for W. It is shown that Wσ(t, q) satisfies a 'bicyclic sieving phenomenon' which combinatorially interprets its values when t and q are certain roots of unity.
Original language | English (US) |
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Pages (from-to) | 627-646 |
Number of pages | 20 |
Journal | Journal of the London Mathematical Society |
Volume | 77 |
Issue number | 3 |
DOIs | |
State | Published - Jun 2008 |
Bibliographical note
Funding Information:The first author is supported by NSA grant H98230-05-1-0256. The second and third authors are supported by NSF grants DMS-0601010 and DMS-0503660, respectively.