A conjecture on descents, inversions and the weak order on Coxeter groups

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 𝒫 ⊆ W such that the left inversion set of w is the disjoint union of the left inversion set of the elements in 𝒫 . Partitions of elements of W arises in the study of the Belkale-Kumar product on the cohomology H * ( X , ℤ ) , where X is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group 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 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 ).