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Buildings and their applications in geometry and topology
 ASIAN J. MATH
"... Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (BruhatTits buildings), topological buildings, Rbuildings, in particular Rtrees. They are useful for many differ ..."
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Buildings were first introduced by J.Tits in 1950s to give systematic geometric interpretations of exceptional Lie groups and have been generalized in various ways: Euclidean buildings (BruhatTits buildings), topological buildings, Rbuildings, in particular Rtrees. They are useful for many different applications in various subjects: algebraic groups, finite groups, finite geometry, representation theory over local fields, algebraic geometry, Arakelov intersection for arithmetic varieties, algebraic Ktheories, combinatorial group theory, global geometry and algebraic topology, in particular cohomology groups, of arithmetic groups and Sarithmetic groups, rigidity of cofinite subgroups of semisimple Lie groups and nonpositively curved manifolds, classification of isoparametric submanifolds in R n of high codimension, existence of hyperbolic structures on three dimensional manifolds in Thurston’s geometrization program. In this paper, we survey several applications of buildings in differential geometry and geometric topology. There are four underlying themes in these applications: 1. Buildings often describe the geometry at infinity of symmetric spaces and locally symmetric
Matroids and Coxeter groups
"... The paper describes a few ways in which the concept of a Coxeter group (in its most ubiquitous manifestation, the symmetric group) emerges in the theory of ordinary matroids: • Gale’s maximality principle which leads to the Bruhat order on the symmetric group; • Jordan–Hölder permutation which measu ..."
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The paper describes a few ways in which the concept of a Coxeter group (in its most ubiquitous manifestation, the symmetric group) emerges in the theory of ordinary matroids: • Gale’s maximality principle which leads to the Bruhat order on the symmetric group; • Jordan–Hölder permutation which measures distance between two maximal chains in a semimodular lattice and which happens to be closely related to Tits ’ axioms for buildings; • matroid polytopes and associated reflection groups; • Gaussian elimination procedure, BNpairs and their Weyl groups. These observations suggest a very natural generalisation of matroids; the new objects are called Coxeter matroids and are related to other Coxeter groups in the same way as (classical) matroids are related to the symmetric group.
On the Topology of the Combinatorial Flag Varieties  From a New Point of View
, 2000
"... We prove that the simplicial complex# n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of Wmatroids for an arbitrary finite Coxeter group W . The paper gener ..."
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We prove that the simplicial complex# n of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on shellability of the simplicial complex of Wmatroids for an arbitrary finite Coxeter group W . The paper generalises the well known results by SolomonTits and Bjorner on spherical buildings, and improves the results of [6]. # Partially supported by PSCCUNY Research Award 669437 and EPSRC Research Grant GR/M24707. 1 1 Definitions and main results For any positive integer n, let [n] denote the set { 1, . . . , n }, and let M n be the set of all matroids on [n] of rank distinct from 0 and n. Define a partial ordering on M n by M # # M if M # is a quotient of M . Let# n be the simplicial complex of chains in M n ; every simplex # ## n can be written as # = #M 1 , . . . , M r # , with all M i # M n and M 1 # . . . # M r . Following [10, 11], we shall call# n the n th combinatorial flag variety. In contexts where the value of n is not important we shall omit it from the notation and write M and## The works [10, 11] contain detailed discussions of many remarkable properties of the combinatorial flag variety# n . It behaves like a universal, characteristicfree finite geometry. In particular, every matroid on n points is representable in# n in the sense specified in [10, 11], which is the direct reformulation in combinatorial terms of the classical concept of representability of matroids by means of vector configurations. The present work will concentrate on topological and cohomological properties of the combinatorial flag varieties. It is easy to prove that# n is a purely (n2)dimensional simplicial complex, that is, it has dimension n  2 and eve...
On the Topology of the Combinatorial Flag Varieties
, 1999
"... this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on M + n by M # # M if M # is a quotient of M . Let# + n be the simplicial complex of chains in M + n ; every simplex s ## + n can be written as s = ..."
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this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on M + n by M # # M if M # is a quotient of M . Let# + n be the simplicial complex of chains in M + n ; every simplex s ## + n can be written as s = #M 1 , . . . , M r #, with all M i # M + n and M 1 # . . . # M r . 1.1 Convention. A superscript number on a matroid will denote its rank. Subscripts will be used for general indexing purposes. According to this convention, in a context where all matroids are on [n], M 0 and M n will denote the unique matroids of rank 0 and rank n respectively. For another example of the use of this convention, when we write s =# M i 0 , M i 1 , . . . , M i r # in# + n , the notation will imply that M i 0 # M i 1 # . . . # M i r and that rank(M i 0 ) = i 0 , rank(M i 1 ) = i 1 etc. 1.2 For any subset N # M + n let #(N ) be the subcomplex of# + n generated by N : #(N ) consists of all simplices of# + n all of whose vertices belong to N . In particular let M n = {M # M + n : 1 #rank(M) # n  1}, and let# n = #(M n )