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A conjecture on descents, inversions and the weak order on Coxeter groups
Hohlweg, Christophe1; Pons, Viviane2
1Université du Québec à Montréal, LaCIM et Département de Mathématiques, CP 8888 Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada
2Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, Orsay, France, and IRL CRM-CNRS - Centre de recherches mathématiques International Research Lab, Montréal, Canada.
Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1037-1066

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal{P}\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the left inversion set of the elements in $\mathcal{P}$. Partitions of elements of $W$ arises in the study of the Belkale-Kumar product on the cohomology $H^*(X,\mathbb{Z})$, where $X$ is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group $\mathcal{S}_n$ are also related to the Babington-Smith model in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone.

Moreover, we state and discuss the conjecture that the number of right descents of $w$ is the sum of the number of right descents of the elements of $\mathcal{P}$. In particular: (1) we state an equivalent conjecture in term of the number of atoms in the interval $[e,w]_R$ of the right weak order, with a word-metric flavour; (2) we give a direct proof that this conjecture holds in the cases of symmetric groups (type $A$) and hyperoctahedral groups (type $B$).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.501
Classification: 20F55, 05A05, 17B22, 05E16
Keywords: Coxeter groups, inversion sets, weak order

Hohlweg, Christophe  1 ; Pons, Viviane  2

1 Université du Québec à Montréal, LaCIM et Département de Mathématiques, CP 8888 Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada
2 Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, Orsay, France, and IRL CRM-CNRS - Centre de recherches mathématiques International Research Lab, Montréal, Canada.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Hohlweg, Christophe; Pons, Viviane. A conjecture on descents, inversions and the weak order on Coxeter groups. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 1037-1066. doi: 10.5802/alco.501
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