Proof of a conjecture of Klopsch-Voll on Weyl groups of type 𝐴

Abstract

We prove a conjecture of Klopsch-Voll on the signed generating function of a new statistic on the quotients of the symmetric groups. As a consequence of our results we also prove a conjecture of Stasinski-Voll in type B.

Odd length for even hyperoctahedral groups and signed generating functions

“…The proof is analogous to that of [4,Proposition 3.3] noting that , since i > 2, σ ∈ D I n if and only if σ(i + 2j, i + 2k + 2)(−i − 2j, −i − 2k − 2) ∈ D I n .…”

Odd length in Weyl groups

“…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.…”

“…We showed in the previous section that the odd length defined combinatorially for type A coincides with L Φ(∆) for a very natural choice of simple system of type A n−1 . Nice formulae for the signed (by length) distribution of this statistic over all quotients of the symmetric groups were proved in [2]. For later use (see Section 5.2), we prove here that the signed generating function of L Φ(∆) (= oinv) over S n is the same as the one over the set of unimodal permutations, whose definition we now recall.…”

“…Recall that for σ ∈ D n the descent set is Des(σ) = {i ∈ [0, n − 1] | σ(i) > σ(i + 1)}, where we set σ(0) := −σ (2). Also, recall that S n is naturally isomorphic to the parabolic subgroup D n[n−1] of D n , and that D n can be written as…”

The average size of the kernel of a matrix and orbits of linear groups

“…We note that permutation statistics have previously featured in explicit formulae for representation zeta functions ; see also .…”