Sortable Elements for Quivers with Cycles
Name
Reading-2010-Sortable elements.pdf
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209.49 KB
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Adobe PDF
Checksum (MD5)
ae1f1edf6af045c3f883dd4aa9e4390b
Author(s)
Reading, Nathan
Speyer, David E.
Date Issued
June 2010
Journal
Electronic Journal of Combinatorics
Publisher
Electronic Journal of Combinatorics
Citation
Reading, Nathan, and David E. Speyer. "Sortable Elements for Quivers with Cycles." Electronic Journal of Combinatorics, Volume 17 (2010).
Version
Final published version
Abstract
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 Ω-sortable elements, where Ω is an arbitrary orientation of the diagram, and show that the key properties of c-sortable elements carry over to the Ω-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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.37236/362
