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Cambrian lattices. (English) Zbl 1106.20033
Let $$(W,S)$$ be a Coxeter system. The elements of $$S$$ are called simple reflections and conjugates of simple reflections are called reflections. The left inversion set of $$w$$, denoted $$I(w)$$, is the set of all (not necessarily simple) reflections $$t$$ such that $$l(tw)<l(w)$$. The length $$l(w)$$ is equal to $$|I(w)|$$. The right weak order on $$W$$ is the partial order which sets $$v\leq w$$ if and only if $$I(v)\subset I(w)$$. The term “weak order” always refers to the right weak order.
For an arbitrary finite Coxeter group $$W$$, the author defines the family of Cambrian lattices for $$W$$ as quotients of the weak order on $$W$$ with respect to certain lattice congruences. The author associates to each Cambrian lattice a complete fan. In types $$A$$ and $$B$$ the author obtains, by means of a fiber-polytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, the author proves in types $$A$$ and $$B$$ that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky’s construction of the normal fan as a “cluster fan”. The author also shows that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 06A07 Combinatorics of partially ordered sets 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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