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Balance constants for Coxeter groups

Christian Gaetz, Yibo Gao

Abstract

The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/3$-$2/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it.

Balance constants for Coxeter groups

Abstract

The - Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least . By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets of any Coxeter group. Remarkably, we conjecture that the lower bound of still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the - Conjecture to this new setting: we prove our conjecture when is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the - Conjecture, and therefore on which methods are likely to be successful in resolving it.

Paper Structure

This paper contains 20 sections, 21 theorems, 92 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. The $1/3$-$2/3$ Conjecture
  3. Posets as convex subsets of the symmetric group
  4. Balance constants for Coxeter groups
  5. Background on Coxeter groups and root systems
  6. Coxeter groups
  7. Finite Weyl groups and crystallographic root systems
  8. The fully commutative case
  9. Width-two posets and $321$-avoiding permutations
  10. Fully commutative elements in Coxeter groups
  11. Heaps and weak order intervals
  12. Balance constants for fully commutative intervals
  13. Equality cases
  14. Generalized semiorders
  15. A uniform bound for finite Weyl groups
  16. ...and 5 more sections

Key Result

Theorem 2.1

A set $C \subseteq W$ is left convex if and only if it is of the form $W_D^A$ for some $D \subseteq A \subseteq T$.

Figures (8)

  • Figure 1: The Coxeter diagram (left) for the Weyl group of type $A_2$ (the symmetric group $S_3$). The fully commutative element $w=s_1s_2=231$ has heap poset $H_w$ shown on the right with $S$-labels; this example is an equality case $b([\mathrm{id},w]_L)=\mathop{\mathrm{\textit{b}^{\text{ideal}}}}\nolimits(H_w)=\frac{1}{3}$ corresponding to the poset $P_3$ in the original formulation of the $1/3$-$2/3$ Conjecture.
  • Figure 2: The Coxeter diagram (left) for the Weyl group of type $D_4$. The fully commutative element $w=s_4s_2s_3s_1$ has heap poset $H_w$ shown on the right with $S$-labels; this example is an equality case $b([\mathrm{id},w]_L)=\mathop{\mathrm{\textit{b}^{\text{ideal}}}}\nolimits(H_w)=\frac{1}{3}$.
  • Figure 3: The Coxeter diagram (left) for the Weyl group of type $B_3$. The fully commutative element $w=s_3s_2s_3s_1$ has heap poset $H_w$ shown on the right with $S$-labels; this example is an equality case $b([\mathrm{id},w]_L)=\mathop{\mathrm{\textit{b}^{\text{ideal}}}}\nolimits(H_w)=\frac{1}{3}$.
  • Figure 4: The Coxeter diagram (left) for the Weyl group of type $E_6$. The fully commutative element $w=s_6s_3s_2s_4s_1s_3s_5$ has heap poset $H_w$ shown on the right with $S$-labels; this example is an equality case $b([\mathrm{id},w]_L)=\mathop{\mathrm{\textit{b}^{\text{ideal}}}}\nolimits(H_w)=\frac{1}{3}$.
  • Figure 5: The Hasse diagram of the root poset of type $A_{6}$ with a dashed line indicating an order ideal.
  • ...and 3 more figures

Theorems & Definitions (48)

  • Conjecture 1.1: The $1/3$-$2/3$ Conjecture
  • Conjecture 1.2
  • Remark
  • Remark
  • Theorem 2.1: Tits Tits
  • Definition 2.2
  • Theorem 3.1: Linial Linial
  • Definition 3.2: Stembridge Stembridge
  • Proposition 3.3: Stembridge Stembridge
  • Theorem 3.4
  • ...and 38 more