Low dimensional Euclidean buildings: II Thibaut Dumont University - PowerPoint PPT Presentation

Low dimensional Euclidean buildings: II Thibaut Dumont University of Jyv¨ askyl¨ a June 2019 – UNCG Thibaut Dumont University of Jyväskylä June 2019 1 / 13

Table of Contents Classification Thibaut Dumont University of Jyväskylä June 2019 2 / 13

Classification: Irreducibility Let ( W, S ) be a Coxeter system. (Draw random Coxeter diagram with at least 2 connected components) ◮ Connected components: I = J 1 ⊔ · · · ∪ J n and S = S 1 ⊔ · · · ⊔ S n . ◮ The subgroups W J k = � S k � pairwise commute. ◮ W ∼ = W 1 × · · · × W n as groups. ◮ ( W, S ) ∼ = ( W 1 , S 1 ) × · · · × ( W n , S n ) as Coxeter systems. Thibaut Dumont University of Jyväskylä June 2019 3 / 13

Classification: Irreducibility Let ∆ be a building of type ( W, S ) and fix a chamber c ∈ C (∆) . ◮ Connected components: I = J 1 ⊔ · · · ∪ J n and S = S 1 ⊔ · · · ⊔ S n . ◮ Let ∆ k denote the J k -residue containing c , a building of type ( W k , S k ) . ◮ The product chamber system ∆ 1 × · · · × ∆ n is a chamber system over I where, for i ∈ J k , the incidence is given by ( c 1 , . . . , c n ) ∼ i ( d 1 , . . . , d n ) if and only if c k ∼ i d k and c ℓ = d ℓ for all ℓ � = k . Thibaut Dumont University of Jyväskylä June 2019 4 / 13

Classification: Irreducibility Let ∆ be a building of type ( W, S ) and fix a chamber c ∈ C (∆) . ◮ Connected components: I = J 1 ⊔ · · · ∪ J n and S = S 1 ⊔ · · · ⊔ S n . ◮ Let ∆ k denote the J k -residue containing c , a building of type ( W k , S k ) . ◮ The product chamber system ∆ 1 × · · · × ∆ n is a chamber system over I where, for i ∈ J k , the incidence is given by ( c 1 , . . . , c n ) ∼ i ( d 1 , . . . , d n ) if and only if c k ∼ i d k and c ℓ = d ℓ for all ℓ � = k . Theorem The product ∆ 1 × · · · × ∆ n is a building isomorphic to ∆ (with σ = id ). Thibaut Dumont University of Jyväskylä June 2019 4 / 13

Classification: Spherical diagrams Coxter (1934): all irreducible spherical Coxeter groups (wikipedia) Thibaut Dumont University of Jyväskylä June 2019 5 / 13

Classification: Spherical diagrams Ronan-Tits: no building thick building of type H 3 or H 4 . (wikipedia) Thibaut Dumont University of Jyväskylä June 2019 6 / 13

Classification: Spherical diagrams Tits spherical classification: all buildings with diagram of rank ≥ 4 . (wikipedia) Thibaut Dumont University of Jyväskylä June 2019 7 / 13

Classification: Spherical diagrams No root system associated, hence no Euclidean extension. (wikipedia) Thibaut Dumont University of Jyväskylä June 2019 8 / 13

Classification: Euclidean diagrams One edges added when Euclidean reflection groups exist. (wikipedia) Thibaut Dumont University of Jyväskylä June 2019 9 / 13

Classification: Euclidean diagrams Classification with regularity parameters. (Parkinson’s thesis) Thibaut Dumont University of Jyväskylä June 2019 10 / 13

Classification: Euclidean diagrams (Parkinson’s thesis) Thibaut Dumont University of Jyväskylä June 2019 11 / 13

Classification: Euclidean diagrams (Parkinson’s thesis) Thibaut Dumont University of Jyväskylä June 2019 12 / 13

Classification Tits’ classification : ◮ Euclidean buildings of rank at least 4 are Bruhat-Tits buildings associated with an algebraic group G ( F ) over a local field F and the automorphism group “is” Aut(∆) = G ( F ) Thibaut Dumont University of Jyväskylä June 2019 13 / 13

Classification Tits’ classification : ◮ Euclidean buildings of rank at least 4 are Bruhat-Tits buildings associated with an algebraic group G ( F ) over a local field F and the automorphism group “is” Aut(∆) = G ( F ) ◮ SL n ( Q p ) Thibaut Dumont University of Jyväskylä June 2019 13 / 13