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: 2019-01-17

Accepted: 2019-01-21

Published online: 2019-08-01

DOI: https://doi.org/10.5802/alco.62

Classification: 05E15, 05A19, 20F55

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

@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}, publisher = {MathOA foundation}, volume = {2}, number = {4}, year = {2019}, pages = {621-644}, doi = {10.5802/alco.62}, mrnumber = {3997515}, zbl = {1417.05246}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2019__2_4_621_0/} }

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. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_621_0/

[1] Groupes et Algèbres de Lie, Chp. IV–VI, Masson, Paris, 1981 | Zbl 0483.22001

[2] Odd length for even hyperoctahedral groups and signed generating functions, Discrete Math., Volume 340 (2017) no. 12, pp. 2822-2833 | Article | MR 3698070 | Zbl 1370.05010

[3] Odd length in Weyl groups (2017) (https://arxiv.org/abs/1709.03320 ) | Zbl 07140427

[4] Proof of a conjecture of Klopsch–Voll on Weyl groups of type $A$, Trans. Amer. Math. Soc., Volume 369 (2017), pp. 7531-7547 | Article | MR 3683117 | Zbl 1368.05007

[5] Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge, 1990 | Article | Zbl 0725.20028

[6] Igusa-type functions associated to finite formed spaces and their functional equations, Trans. Amer. Math. Soc., Volume 361 (2009) no. 8, pp. 4405-4436 | Article | MR 2500892 | Zbl 1229.05288

[7] Lie groups beyond an introduction, Progress in Mathematics, Volume 140, Birkhäuser, Boston, MA, 1996 | MR 1399083 | Zbl 0862.22006

[8] Proof of Stasinski and Voll’s hyperoctahedral group conjecture, Australas. J. Combin., Volume 71 (2018) no. 2, pp. 196-240 | MR 3786907 | Zbl 1406.05010

[9] The Poincaré series of a Coxeter group, Math. Ann., Volume 199 (1972) no. 2, pp. 161-174 | Article | MR 322069 | Zbl 0286.20062

[10] A new statistic on the hyperoctahedral groups, Electron. J. Combin., Volume 20 (2013) no. 3, 23 pages | Article | MR 3118958 | Zbl 1295.05038

[11] Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $B$, Amer. J. Math., Volume 136 (2014) no. 2, pp. 501-550 | Article | MR 3188068 | Zbl 1286.11140