On the Topology of the Cambrian Semilattices
Copyright policy )On the Topology of the Cambrian Semilattices
For an arbitrary Coxeter group $W$, Reading and Speyer defined Cambrian semilattices $\mathcal{C}_{\gamma}$ as sub-semilattices of the weak order on $W$ induced by so-called $\gamma$-sortable elements. In this article, we define an edge-labeling of $\mathcal{C}_{\gamma}$, and show that this is an EL-labeling for every closed interval of $\mathcal{C}_{\gamma}$. In addition, we use our labeling to show that every finite open interval in a Cambrian semilattice is either contractible or spherical, and we characterize the spherical intervals, generalizing a result by Reading.
- University of Vienna Austria
POSETS, Cambrian semilattices, EL-shellability, cambrian semilattices, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], Weak Order, Coxeter groups, Combinatorics of partially ordered sets, SHELLABLE NONPURE COMPLEXES, Reflection and Coxeter groups (group-theoretic aspects), Möbius function, QA1-939, FOS: Mathematics, Mathematics - Combinatorics, 20F55 (Primary) 06A07, 05E15 (Secondary), 101012 Kombinatorik, LATTICES, NONCROSSING PARTITIONS, GENERALIZED ASSOCIAHEDRA, EL-Shellability, 101012 Combinatorics, coxeter groups, el-shellability, Combinatorial aspects of groups and algebras, Coxeter Groups, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Cambrian Semilattices, weak order, Combinatorics (math.CO), FANS, SORTABLE ELEMENTS, Mathematics
POSETS, Cambrian semilattices, EL-shellability, cambrian semilattices, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], Weak Order, Coxeter groups, Combinatorics of partially ordered sets, SHELLABLE NONPURE COMPLEXES, Reflection and Coxeter groups (group-theoretic aspects), Möbius function, QA1-939, FOS: Mathematics, Mathematics - Combinatorics, 20F55 (Primary) 06A07, 05E15 (Secondary), 101012 Kombinatorik, LATTICES, NONCROSSING PARTITIONS, GENERALIZED ASSOCIAHEDRA, EL-Shellability, 101012 Combinatorics, coxeter groups, el-shellability, Combinatorial aspects of groups and algebras, Coxeter Groups, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Cambrian Semilattices, weak order, Combinatorics (math.CO), FANS, SORTABLE ELEMENTS, Mathematics
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