An Introduction to Buildings

General Information | Syllabus |
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Lectures | Sections in Text | Brief Description |
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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