ALGEBRAIC COMBINATORICS

no. 6
Odd length in Weyl groups
Brenti, Francesco1; Carnevale, Angela2
1 Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma, Italy
2 Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld, Germany
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.69
Classification: 17B22, 20F55, 05E99
Keywords: Root system, Weyl group, Coxeter group, odd length, enumeration.
Brenti, Francesco 1; Carnevale, Angela 2

1 Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma, Italy
2 Fakultät für Mathematik Universität Bielefeld Postfach 100131 D-33501 Bielefeld, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/articles/10.5802/alco.69/

[1] Björner, A.; Brenti, F. Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[6] Geck, M. PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras, LMS J. Comput. Math., Volume 15 (2012), pp. 231-256 | DOI | MR | Zbl

[7] Humphreys, J. E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | MR | Zbl

[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 | DOI | MR | Zbl

[9] Landesman, A. Proof of Stasinski and Voll’s hyperoctahedral group conjecture, Australas. J. Combin., Volume 71 (2018), pp. 196-240 | MR | Zbl

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

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

[12] Sage Developers SageMath, the Sage Mathematics Software System (Version 8.6) (2019) https://www.sagemath.org

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

[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 | DOI | MR | Zbl

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

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