ALGEBRAIC COMBINATORICS

Odd length in Weyl groups
Algebraic Combinatorics, Volume 2 (2019) no. 6, p. 1125-1147

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 A, B, and D, 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 B and D.

Received : 2018-08-19
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},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     pages = {1125-1147},
     doi = {10.5802/alco.69},
     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] Björner, A.; Brenti, F. Combinatorics of Coxeter groups, Graduate Texts in Mathematics, Volume 231, Springer, New York, 2005, xiv+363 pages | MR 2133266 | Zbl 1110.05001

[2] Brenti, F. q-Eulerian polynomials arising from Coxeter groups, European J. Combin., Volume 15 (1994) no. 5, pp. 417-441 | Article | MR 1292954 | Zbl 0809.05012

[3] Brenti, F.; Carnevale, A. 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] Brenti, F.; Carnevale, A. Proof of a conjecture of Klopsch–Voll on Weyl groups of type A, Trans. Amer. Math. Soc., Volume 369 (2017) no. 10, pp. 7531-7547 | Article | MR 3683117 | Zbl 1368.05007

[5] Carnevale, A.; Shechter, S.; Voll, C. 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] Geck, M. 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] Humphreys, J. E. 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] Klopsch, B.; Voll, C. 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] Landesman, A. Proof of Stasinski and Voll’s hyperoctahedral group conjecture, Australas. J. Combin., Volume 71 (2018), pp. 196-240 | MR 3786907 | Zbl 1406.05010

[10] Reiner, V. Signed permutation statistics, European J. Combin., Volume 14 (1993) no. 6, pp. 553-567 | Article | MR 1248063 | Zbl 0793.05005

[11] Reiner, V. Descents and one-dimensional characters for classical Weyl groups, Discrete Math., Volume 140 (1995) no. 1-3, pp. 129-140 | Article | MR 1333715

[12] Sage Developers SageMath, the Sage Mathematics Software System (Version 8.6) (2019), 50 https://www.sagemath.org | MR 3118958 | Zbl 1295.05038

[13] Stasinski, A.; Voll, C. A new statistic on the hyperoctahedral groups, Electron. J. Combin., Volume 20 (2013) no. 3, 50, 23 pages | Article | MR 3188068 | Zbl 1286.11140

[14] Stasinski, A.; Voll, C. 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 | Zbl 1417.05246

[15] Stembridge, J. Signed-twisted Poincaré series and odd inversions in Weyl groups, Algebraic Comb., Volume 2 (2019) no. 4, pp. 621-644