Sortable elements for quivers with cycles.

Nathan Reading ; David E Speyer

The electronic journal of combinatorics, Tome 17 (2010)

Résumé

Each Coxeter element $c$ of a Coxeter group $W$ defines a subset of $W$ called the $c$-sortable elements. The choice of a Coxeter element of $W$ is equivalent to the choice of an acyclic orientation of the Coxeter diagram of $W$. In this paper, we define a more general notion of $\Omega$-sortable elements, where $\Omega$ is an arbitrary orientation of the diagram, and show that the key properties of $c$-sortable elements carry over to the $\Omega$-sortable elements. The proofs of these properties rely on reduction to the acyclic case, but the reductions are nontrivial; in particular, the proofs rely on a subtle combinatorial property of the weak order, as it relates to orientations of the Coxeter diagram. The $c$-sortable elements are closely tied to the combinatorics of cluster algebras with an acyclic seed; the ultimate motivation behind this paper is to extend this connection beyond the acyclic case.

Détail

DOI : 10.37236/362

Classification : 20F55, 06A07, 13F60
Mots-clés : Coxeter elements, Coxeter groups, sortable elements, orientations of Coxeter diagrams, weak order, combinatorics of cluster algebras