2024/12/12 by Christophe Hohlweg, Hohlweg, Christophe, Viviane Pons +1 · 1 voice
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #math.CO #math.GR
paper · pdf · doi:10.48550/arxiv.2412.09227
openalex publication_date 2024/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊆ 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 Sn are also related to the \em Babington-Smith model in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone. We state 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 and prove that this conjecture holds in the cases of symmetric groups (type A) and hyperoctahedral groups (type B).