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A conjecture on descents, inversions and the weak order

Christophe Hohlweg, Viviane Pons

TL;DR

This work investigates partitions and bipartitions of elements in Coxeter systems via inversion sets and the right weak order, centering on the conjecture that for a bipartition $\{u,v\}$ of $w$, the right-descent count satisfies $d_R(w)=d_R(u)+d_R(v)$. The authors develop a comprehensive framework linking inversion sets to biclosed root subsets and to diameters of intervals in the right weak order, enabling reformulations of the conjecture in terms of rectangled intervals and atoms/coatoms. They provide direct proofs of the conjecture for types $A$ (symmetric groups) and $B$ (hyperoctahedral groups) by constructing recursive decompositions of permutations along the position of the maximal letter $n$ (or its signed analogue) and proving descent-additivity via induction, supported by careful analysis of left/right/forgotten letters and their inversions. The paper also explores long-element bipartitions, establishes parallels between bipartitions and rectangled intervals, and reports extensive enumerative data, highlighting Catalan-like phenomena and posing questions about possible rational generating functions for partition-irreducible elements across types. Overall, the results solidify a structural link between descent statistics, inversion-decomposition, and weak-order geometry, with implications for the Belkale-Kumar product and related combinatorial models in algebraic geometry.

Abstract

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 {\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 $B$).

A conjecture on descents, inversions and the weak order

TL;DR

This work investigates partitions and bipartitions of elements in Coxeter systems via inversion sets and the right weak order, centering on the conjecture that for a bipartition of , the right-descent count satisfies . The authors develop a comprehensive framework linking inversion sets to biclosed root subsets and to diameters of intervals in the right weak order, enabling reformulations of the conjecture in terms of rectangled intervals and atoms/coatoms. They provide direct proofs of the conjecture for types (symmetric groups) and (hyperoctahedral groups) by constructing recursive decompositions of permutations along the position of the maximal letter (or its signed analogue) and proving descent-additivity via induction, supported by careful analysis of left/right/forgotten letters and their inversions. The paper also explores long-element bipartitions, establishes parallels between bipartitions and rectangled intervals, and reports extensive enumerative data, highlighting Catalan-like phenomena and posing questions about possible rational generating functions for partition-irreducible elements across types. Overall, the results solidify a structural link between descent statistics, inversion-decomposition, and weak-order geometry, with implications for the Belkale-Kumar product and related combinatorial models in algebraic geometry.

Abstract

In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system : a partition of an element is a subset such that the left inversion set of is the disjoint union of the left inversion set of the elements in . Partitions of elements of arises in the study of the Belkale-Kumar product on the cohomology , where is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group 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 is the sum of the number of right descents of the elements of and prove that this conjecture holds in the cases of symmetric groups (type ) and hyperoctahedral groups (type ).

Paper Structure

This paper contains 25 sections, 33 theorems, 81 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Partitions of elements in a Coxeter system
  3. First properties of partitions of elements of $W$
  4. Roots and reflections
  5. Bipartitions of the longest element
  6. Proof of Proposition \ref{['prop:Conj']}
  7. Conjecture \ref{['conj:1']}, bipartitions and rectangled intervals in the weak order
  8. The case of symmetric groups (Type A)
  9. Standardization of words, inversions and descents
  10. Bipartition, recursive decomposition of a permutation and proof of Theorem \ref{['thm:Main']} in the case of symmetric groups
  11. Case $k \neq 1$.
  12. Case $k=1$.
  13. Proof of Proposition \ref{['prop:decomposition']}
  14. Proof of Proposition \ref{['prop:decreasing']}
  15. Suppose that the descent is kept in both $\tilde{u}$ and $\tilde{v}$.
  16. ...and 10 more sections

Key Result

Theorem 1.5

Let $n\in\mathbb N^*$ and assume that $W=\mathcal{S}_n$ or $W=W_n$. Then for any $w\in W$ and $\{u,v\}$ a bipartition of $w$, we have: Moreover, if $\{u_1,\dots,u_k\}$ is a $k$-partition of $w$, then $d_R(w)=\sum_{i=1}^k d_R(u_i)$.

Figures (3)

  • Figure 1: The rectangled interval $[e,42131]_R$ for $(W,S)$ of type $D_4$ with Coxeter graph on the upper left side. This rectangled interval has only two diameters: $\{e,42131\}$ and $\{1,4231\}$. The coatoms of $[e,42131]_R$ are $4213,4231$, the coatom of $[e,1]_R$ are $e$ and the coatoms of $[e,4231]_R$ is $423$, so Conjecture \ref{['conj:2']}, and therefore Conjecture \ref{['conj:1']}, hold in this example. The atoms of $[e,42131]_R$ are $1,2,4$ (the left descents), the atoms of $[1,42131]_R$ are $41,21$ and the atom of $[4231,42131]_R$ is $42131$, which illustrates the statement of Conjecture \ref{['conj:3']}.
  • Figure 1: Generating functions of elements admitting a bipartition up to length $k$
  • Figure 2: Generating functions of partition-irreducible elements up to length $k$

Theorems & Definitions (97)

  • Definition 1.1: Partitions of elements of $W$
  • Conjecture 1
  • Example 1.2
  • Example 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Definition 1.7
  • Proposition 1.8
  • Conjecture 2
  • ...and 87 more