Bender–Knuth Billiards in Coxeter Groups
Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$ , where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involuti...
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| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001592/type/journal_article |
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| Summary: | Let
$(W,S)$
be a Coxeter system, and write
$S=\{s_i:i\in I\}$
, where I is a finite index set. Fix a nonempty convex subset
$\mathscr {L}$
of W. If W is of type A, then
$\mathscr {L}$
is the set of linear extensions of a poset, and there are important Bender–Knuth involutions
$\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$
indexed by elements of I. For arbitrary W and for each
$i\in I$
, we introduce an operator
$\tau _i\colon W\to W$
(depending on
$\mathscr {L}$
) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on
$\mathscr {L}$
that coincides with
$\mathrm {BK}_i$
in type A. Given a Coxeter element
$c=s_{i_n}\cdots s_{i_1}$
, we consider the operator
$\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$
. We say W is futuristic if for every nonempty finite convex set
$\mathscr {L}$
, every Coxeter element c and every
$u\in W$
, there exists an integer
$K\geq 0$
such that
$\mathrm {Pro}_c^K(u)\in \mathscr {L}$
. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types
$\widetilde A$
and
$\widetilde C$
, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if
$s_{i_N}\cdots s_{i_1}$
is a reduced expression for the long element of W, then
$\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$
; this allows us to determine the smallest integer
$\mathrm {M}(c)$
such that
$\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$
for all
$\mathscr {L}$
. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type
$\widetilde A$
,
$\widetilde C$
, or
$\widetilde G_2$
. |
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| ISSN: | 2050-5094 |