We define for any crystallographic root system a new statistic on the corresponding Weyl group which we call the odd length. This statistic reduces, for Weyl groups of types , , and , to the each of the statistics by the same name that have already been defined and studied in [8], [12], [13], and [3]. We show that the sign-twisted generating function of the odd length always factors completely, and we obtain multivariate analogues of these factorizations in types and .
Revised: 2019-02-28
Accepted: 2019-03-02
Published online: 2019-12-04
DOI: https://doi.org/10.5802/alco.69
Classification: 17B22 20F55 05E99
Keywords: Root system, Weyl group, Coxeter group, odd length, enumeration.
@article{ALCO_2019__2_6_1125_0, author = {Brenti, Francesco and Carnevale, Angela}, title = {Odd length in Weyl groups}, journal = {Algebraic Combinatorics}, pages = {1125--1147}, publisher = {MathOA foundation}, volume = {2}, number = {6}, year = {2019}, doi = {10.5802/alco.69}, mrnumber = {4049840}, zbl = {07140427}, language = {en}, url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1125_0/} }
Brenti, Francesco; Carnevale, Angela. Odd length in Weyl groups. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1125-1147. doi : 10.5802/alco.69. https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1125_0/
[1] Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Volume 231, Springer, New York, 2005, xiv+363 pages | MR 2133266 | Zbl 1110.05001
[2] -Eulerian polynomials arising from Coxeter groups, European J. Combin., Volume 15 (1994) no. 5, pp. 417-441 | Article | MR 1292954 | Zbl 0809.05012
[3] 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
[4] Proof of a conjecture of Klopsch–Voll on Weyl groups of type , Trans. Amer. Math. Soc., Volume 369 (2017) no. 10, pp. 7531-7547 | Article | MR 3683117 | Zbl 1368.05007
[5] Enumerating traceless matrices over compact discrete valuation rings, Israel J. Math., Volume 227 (2018) no. 2, pp. 957-986 | Article | MR 3846349 | Zbl 1414.05024
[6] PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras, LMS J. Comput. Math., Volume 15 (2012), pp. 231-256 | Article | MR 2988815 | Zbl 1296.20009
[7] Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Volume 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | MR 1066460 | Zbl 0725.20028
[8] 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
[9] Proof of Stasinski and Voll’s hyperoctahedral group conjecture, Australas. J. Combin., Volume 71 (2018), pp. 196-240 | MR 3786907 | Zbl 1406.05010
[10] Signed permutation statistics, European J. Combin., Volume 14 (1993) no. 6, pp. 553-567 | Article | MR 1248063 | Zbl 0793.05005
[11] Descents and one-dimensional characters for classical Weyl groups, Discrete Math., Volume 140 (1995) no. 1-3, pp. 129-140 | Article | MR 1333715 | Zbl 0830.20025
[12] SageMath, the Sage Mathematics Software System (Version 8.6) (2019) https://www.sagemath.org
[13] A new statistic on the hyperoctahedral groups, Electron. J. Combin., Volume 20 (2013) no. 3, 23 pages | Article | MR 3118958 | Zbl 1295.05038
[14] Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type , Amer. J. Math., Volume 136 (2014) no. 2, pp. 501-550 | Article | MR 3188068 | Zbl 1286.11140
[15] Signed-twisted Poincaré series and odd inversions in Weyl groups, Algebraic Comb., Volume 2 (2019) no. 4, pp. 621-644 | Article | MR 3997515 | Zbl 1417.05246