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...
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
|
| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001592/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832589946650624000 |
|---|---|
| author | Grant Barkley Colin Defant Eliot Hodges Noah Kravitz Mitchell Lee |
| author_facet | Grant Barkley Colin Defant Eliot Hodges Noah Kravitz Mitchell Lee |
| author_sort | Grant Barkley |
| collection | DOAJ |
| description | 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
$\widetilde G_2$
. |
| format | Article |
| id | doaj-art-e649fa0bcf2c471a8bce24081f8fcc0f |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-e649fa0bcf2c471a8bce24081f8fcc0f2025-01-24T05:20:12ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.159Bender–Knuth Billiards in Coxeter GroupsGrant Barkley0Colin Defant1Eliot Hodges2Noah Kravitz3Mitchell Lee4Harvard University, 1 Oxford Street, 02138, U.S.A.; E-mail:Harvard University, 1 Oxford Street, 02138, U.S.A.; E-mail:Harvard University, 1 Oxford Street, 02138, U.S.A.; E-mail:Princeton University, Washington Road, 08540, U.S.A.;Harvard University, 1 Oxford Street, 02138, U.S.A.; E-mail: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 $\widetilde G_2$ .https://www.cambridge.org/core/product/identifier/S2050509424001592/type/journal_article05E1820F5537B20 |
| spellingShingle | Grant Barkley Colin Defant Eliot Hodges Noah Kravitz Mitchell Lee Bender–Knuth Billiards in Coxeter Groups Forum of Mathematics, Sigma 05E18 20F55 37B20 |
| title | Bender–Knuth Billiards in Coxeter Groups |
| title_full | Bender–Knuth Billiards in Coxeter Groups |
| title_fullStr | Bender–Knuth Billiards in Coxeter Groups |
| title_full_unstemmed | Bender–Knuth Billiards in Coxeter Groups |
| title_short | Bender–Knuth Billiards in Coxeter Groups |
| title_sort | bender knuth billiards in coxeter groups |
| topic | 05E18 20F55 37B20 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001592/type/journal_article |
| work_keys_str_mv | AT grantbarkley benderknuthbilliardsincoxetergroups AT colindefant benderknuthbilliardsincoxetergroups AT eliothodges benderknuthbilliardsincoxetergroups AT noahkravitz benderknuthbilliardsincoxetergroups AT mitchelllee benderknuthbilliardsincoxetergroups |