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:
Accepted:
Published online:
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/articles/10.5802/alco.69/}
}
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/
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