SORTABLE ELEMENTS IN INFINITE COXETER GROUPS

作者：Reading Nathan^*; Spoyer David E

来源：Transactions of the American Mathematical Society, 2011, 363(2): 699-761.

DOI：10.1090/s0002-9947-2010-05050-0

In a series of previous papers. we studied sortable elements infinite Coxeter groups and the related Cambian 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 groyp 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 Fuler form of quiven theory and the projection pi(down arrow) mapping each element of W to the unique maximal sortable element bekwart in the weak order. The fibers of pi(down arrow) essentially define the c Cambrian fan. The most fundamental results are first a precise statement of how sortable elements transform under (BGI) reflection functions and section a precise description of the fibers of pi(down arrow). These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

• 出版日期2011-2