Show simple item record

dc.contributor.authorHart, Sarah
dc.contributor.authorKelsey, Veronica
dc.contributor.authorRowley, Peter
dc.date.accessioned2022-09-15T09:04:49Z
dc.date.available2022-09-15T09:04:49Z
dc.date.issued2022-09-15
dc.identifier.urihttp://publications.mfo.de/handle/mfo/3973
dc.description.abstractFor an element $w$ of a Coxeter group $W$ there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on $\Phi$, the root system of $W$. This paper investigates the interaction between these two features of $w$, introducing the notion of the crossing number of $w$, $\kappa(w)$. Writing $w = c_1 \cdots c_r$ as a product of disjoint cycles we associate to each cycle $c_i$ a `crossing number' $\kappa(c_i)$, which is the number of positive roots $\alpha$ in $c_i$ for which $w\cdot \alpha$ is negative. Let Seq$_k(w)$ be the sequence of $\kappa(c_i)$ written in increasing order, and let $\kappa(w)$ = max Seq$_k(w)$. The length of $w$ can be retrieved from this sequence, but Seq$_k(w)$ provides much more information. For a conjugacy class $X$ of $W$ let $k_{\min}(X)=\min \{\kappa(w) \;|\;w \in X\}$ and let $\kappa(W)$ be the maximum value of $k_{\min}$ across all conjugacy classes of $W$. We call $\kappa(w)$ and $\kappa(W)$, respectively, the crossing numbers of $w$ and $W$. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups if $u$ and $v$ are two elements of minimal length in the same conjugacy class $X$, then Seq$_k(u)$ = Seq$_k(v)$ and $k_{\min}(X)=\kappa(u)=\kappa(v)$. Also it is shown that the crossing number of an arbitrary Coxeter group is bounded below by the crossing number of a standard parabolic subgroup. Finally, examples are given to show that crossing numbers can be arbitrarily large for finite and infinite irreducible Coxeter groups.en_US
dc.language.isoen_USen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2022-16
dc.subjectCoxeter groupsen_US
dc.subjectRoot systemen_US
dc.subjectRoot cyclesen_US
dc.subjectLength functionen_US
dc.titleRoot Cycles in Coxeter Groupsen_US
dc.typePreprinten_US
dc.rights.licenseDieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.de
dc.rights.licenseThis document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.en
dc.identifier.doi10.14760/OWP-2022-16
local.scientificprogramOWRF 2022en_US
local.series.idOWP-2022-16en_US
dc.identifier.urnurn:nbn:de:101:1-2022112209111135998137
dc.identifier.ppn1823179940


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record