Following recent work of Brenti and Carnevale, we investigate a sign-twisted Poincaré series for finite Weyl groups $W$ that tracks “odd inversions”; i.e. the number of odd-height positive roots transformed into negative roots by each member of $W$. We prove that the series is divisible by the corresponding series for any parabolic subgroup ${W}_{J}$, and provide sufficient conditions for when the quotient of the two series equals the restriction of the first series to coset representatives for $W/{W}_{J}$. We also show that the series has an explicit factorization involving the degrees of the free generators of the polynomial invariants of a canonically associated reflection group.

Revised:

Accepted:

Published online:

DOI: 10.5802/alco.62

Keywords: Weyl group, root system, Poincaré series, inversion

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@article{ALCO_2019__2_4_621_0, author = {Stembridge, John R.}, title = {Sign-twisted {Poincar\'e} series and odd inversions in {Weyl} groups}, journal = {Algebraic Combinatorics}, pages = {621--644}, publisher = {MathOA foundation}, volume = {2}, number = {4}, year = {2019}, doi = {10.5802/alco.62}, mrnumber = {3997515}, zbl = {1417.05246}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.62/} }

TY - JOUR AU - Stembridge, John R. TI - Sign-twisted Poincaré series and odd inversions in Weyl groups JO - Algebraic Combinatorics PY - 2019 SP - 621 EP - 644 VL - 2 IS - 4 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.62/ DO - 10.5802/alco.62 LA - en ID - ALCO_2019__2_4_621_0 ER -

Stembridge, John R. Sign-twisted Poincaré series and odd inversions in Weyl groups. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 621-644. doi : 10.5802/alco.62. http://www.numdam.org/articles/10.5802/alco.62/

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