12,334 research outputs found
Cambrian Lattices
For an arbitrary finite Coxeter group W we define the family of Cambrian
lattices for W as quotients of the weak order on W with respect to certain
lattice congruences. We associate to each Cambrian lattice a complete fan,
which we conjecture is the normal fan of a polytope combinatorially isomorphic
to the generalized associahedron for W. In types A and B we obtain, 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, we prove 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." Our
construction does not require a crystallographic Coxeter group and therefore
suggests a definition, at least on the level of cellular spheres, of a
generalized associahedron for any finite Coxeter group. The Tamari lattice is
one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are
identified and characterized in terms of signed pattern avoidance. We also show
that open intervals in Cambrian lattices are either contractible or homotopy
equivalent to spheres.Comment: Revisions in exposition (partly in response to the suggestions of an
anonymous referee) including many new figures. Also, Conjecture 1.4 and
Theorem 1.5 are replaced by slightly more detailed statements. To appear in
Adv. Math. 37 pages, 8 figure
Combing Euclidean buildings
For an arbitrary Euclidean building we define a certain combing, which
satisfies the `fellow traveller property' and admits a recursive definition.
Using this combing we prove that any group acting freely, cocompactly and by
order preserving automorphisms on a Euclidean building of one of the types
A_n,B_n,C_n admits a biautomatic structure.Comment: 32 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper2.abs.htm
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