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Sortable elements in infinite Coxeter groups


Authors: Nathan Reading and David E. Speyer
Journal: Trans. Amer. Math. Soc. 363 (2011), 699-761
MSC (2010): Primary 20F55
DOI: https://doi.org/10.1090/S0002-9947-2010-05050-0
Published electronically: September 24, 2010
MathSciNet review: 2728584
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Abstract: In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group $ W$. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection $ \pi_\downarrow^c$ mapping each element of $ W$ to the unique maximal $ c$-sortable element below it in the weak order. The fibers of $ \pi_\downarrow^c$ essentially define the $ c$-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of $ \pi_\downarrow^c$. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.


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Additional Information

Nathan Reading
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
Email: nathan_reading@ncsu.edu

David E. Speyer
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: speyer@math.mit.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05050-0
Received by editor(s): February 25, 2009
Received by editor(s) in revised form: March 18, 2009
Published electronically: September 24, 2010
Additional Notes: The second author was supported by a research fellowship from the Clay Mathematics Institute.
Article copyright: © Copyright 2010 Nathan Reading and David E Speyer

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