Odd length for even hyperoctahedral groups and signed generating functions

Abstract

We define a new statistic on the even hyperoctahedral groups which is a natural analogue of the odd length statistic recently defined and studied on Coxeter groups of types A and B. We compute the signed (by length) generating function of this statistic over the whole group and over its maximal and some other quotients and show that it always factors nicely. We also present some conjectures.

Sign-twisted Poincaré series and odd inversions in Weyl groups

“…We have confirmed the above conjecture for all root systems of rank at most 8, so the only open cases involve C n and D n . Note that Brenti and Carnevale have explicit evaluations of the series F J (D n ; q) for the cases with |J| = 1, two cases with |J| = 2, and some conjectured evaluations in a few cases with |J| 3 (see Sections 5-6 of [2]); all are consistent with the above conjecture.…”

Odd length in Weyl groups

“…In this paper we define a new statistic on any Weyl group. This statistic depends on the root system underlying the Weyl group and we compute it combinatorially for the classical root systems of types A, B, C, and D. As a consequence we verify that this statistic coincides, in types A, B, and D, with the odd length statistics defined and studied in [10], [13], [14], [3], [5], and [11] in these types. Our combinatorial computation of the statistic in the classical types shows that it is the sum of some more fundamental statistics and we compute the signed (by length) multivariate generating function of these statistics in types B and D. These results reduce to results in [10], [14], and [2] when all the variables are equal.…”

“…While this statistics depends on the choice of a simple system ∆ ⊆ Φ, where Φ is the root system of W , we show that its generating function over the corresponding Weyl group does not. We then compute combinatorially this new statistics for the classical Weyl groups, for a natural choice of simple system, and show that it coincides with the statistics by the same name that have already been defined and studied in [2], [3], [10], [13], and [14]. Let Φ be a root system and W be the corresponding Weyl group.…”

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We define a new statistic on any Weyl group which we call the odd length and which reduces, for Weyl groups of types A, B, and D, the the statistics by the same name that have already been defined and studied in [10], [13], [14], and [3]. We show that the signed (by length) generating function of the odd length always factors nicely except possibly in type E 8 , and we obtain multivariate analogues of these factorizations in types B and D.

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Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups

“…This statistic combines combinatorial and parity conditions and is now known as the odd inversion number, or odd length [10,12]. Analogous statistics have later been defined and studied for the hyperoctahedral and even hyperoctahedral groups [11,30,31], and more recently for all Weyl groups [12]. A crucial property of this new statistic is that its signed (by length) generating function over the corresponding Weyl group always factors explicitly [12,33].…”