Sortable Elements for Quivers with Cycles
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...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Electronic Journal of Combinatorics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89812 |
| _version_ | 1811094885414993920 |
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| author | Reading, Nathan Speyer, David E. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Reading, Nathan Speyer, David E. |
| author_sort | Reading, Nathan |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T16:07:04Z |
| format | Article |
| id | mit-1721.1/89812 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:07:04Z |
| publishDate | 2014 |
| publisher | Electronic Journal of Combinatorics |
| record_format | dspace |
| spelling | mit-1721.1/898122022-09-29T18:18:28Z Sortable Elements for Quivers with Cycles Reading, Nathan Speyer, David E. Massachusetts Institute of Technology. Department of Mathematics Speyer, David E. 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. Clay Mathematics Institute (Research Fellowship) 2014-09-18T17:10:53Z 2014-09-18T17:10:53Z 2010-06 2009-09 Article http://purl.org/eprint/type/JournalArticle 1077-8926 http://hdl.handle.net/1721.1/89812 Reading, Nathan, and David E. Speyer. "Sortable Elements for Quivers with Cycles." Electronic Journal of Combinatorics, Volume 17 (2010). en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r90 Electronic Journal of Combinatorics 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. application/pdf Electronic Journal of Combinatorics Electronic Journal of Combinatorics |
| spellingShingle | Reading, Nathan Speyer, David E. Sortable Elements for Quivers with Cycles |
| title | Sortable Elements for Quivers with Cycles |
| title_full | Sortable Elements for Quivers with Cycles |
| title_fullStr | Sortable Elements for Quivers with Cycles |
| title_full_unstemmed | Sortable Elements for Quivers with Cycles |
| title_short | Sortable Elements for Quivers with Cycles |
| title_sort | sortable elements for quivers with cycles |
| url | http://hdl.handle.net/1721.1/89812 |
| work_keys_str_mv | AT readingnathan sortableelementsforquiverswithcycles AT speyerdavide sortableelementsforquiverswithcycles |