Math 125
An Introduction to Buildings

Last updated June 17, 2015 12:56:58 EDT

General Information Syllabus


The course will use both Garrett's book as well as Abramenko and Brown's book on buildings, primarily Garrett in the first half and Abramenko and Brown once we get past the local (Coxeter) theory.

Lectures Sections in Text Brief Description
Week 1 Introduction An important example: $GL_n(k)$ and flag complex, $BN$-pair, Weyl group, stabilizers, strongly transitive action on the building, loose definition of a building, mentioned chamber system approach; BN-pair (loose definition), Bruhat decompostion, affine reflections, locally finite invariant families of hyperplanes ($A_1$, $\tilde A_1$, $A_2$, $\tilde A_2$, $\tilde C_2$); Coxeter matrix, system, diagram, Tit's linear representation
Week 2 Tit's geometric realization Roots, proof that linear representation of Coxeter group is isomorphic to the group generated by reflections in a real vector space with symmetric bilinear form; The Deletion, Exchange, and Folding conditions; Intro to the Coxeter complex
Week 3 The Coexter Complex and foldings A dizzying day of definitions: chamber complexes and chamber maps, labellings, retracts, Justify the Coxter poset is a thin chamber complex with W-invariant action and type function; Standard uniqueness arguments and foldings
Week 4 The main theorem of Tits Prove that at Coxter complex is a thin chamber complex with reversible foldings corresponding to each panel of adjacent chambers
Week 5 Chamber systems Chamber systems, examples, recovering simplicial structure from chamber system
Week 6 Buildings Definitions, basic properties, $A_n(k)$
Week 7 Buildings Canonical retractons, convexity, BN-pairs, $\Delta(G,B)$
Week 8 Buildings Verifying BN-pair axioms for $GL_n$, $Sp_{2n}$, discrete valuations
Week 9 Buildings BN-pair for $SL_n(K)$, $K$ a non-archimedean local field, Euclidean buildings, Hecke Operators and Euler Products

T. R. Shemanske
Last updated June 17, 2015 12:56:58 EDT