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

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Low dimensional Euclidean buildings: II Thibaut Dumont University of Jyv askyl a June 2019 UNCG Thibaut Dumont University of Jyvskyl June 2019 1 / 13 Table of Contents Classification Thibaut Dumont University of Jyvskyl

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Thibaut Dumont University of Jyväskylä June 2019

Low dimensional Euclidean buildings: II

Thibaut Dumont

University of Jyv¨ askyl¨ a

June 2019 – UNCG

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Thibaut Dumont University of Jyväskylä June 2019

Classification

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Irreducibility

Let (W, S) be a Coxeter system.

(Draw random Coxeter diagram with at least 2 connected components)

◮ Connected components: I = J1 ⊔ · · · ∪ Jn and S = S1 ⊔ · · · ⊔ Sn. ◮ The subgroups WJk = Sk pairwise commute. ◮ W ∼

= W1 × · · · × Wn as groups.

◮ (W, S) ∼

= (W1, S1) × · · · × (Wn, Sn) as Coxeter systems.

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Irreducibility

Let ∆ be a building of type (W, S) and fix a chamber c ∈ C(∆).

◮ Connected components: I = J1 ⊔ · · · ∪ Jn and S = S1 ⊔ · · · ⊔ Sn. ◮ Let ∆k denote the Jk-residue containing c, a building of type

(Wk, Sk).

◮ The product chamber system ∆1 × · · · × ∆n is a chamber system

ver I where, for i ∈ Jk, the incidence is given by

(c1, . . . , cn) ∼i (d1, . . . , dn) if and only if ck ∼i dk and cℓ = dℓ for all ℓ = k.

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Irreducibility

Let ∆ be a building of type (W, S) and fix a chamber c ∈ C(∆).

◮ Connected components: I = J1 ⊔ · · · ∪ Jn and S = S1 ⊔ · · · ⊔ Sn. ◮ Let ∆k denote the Jk-residue containing c, a building of type

(Wk, Sk).

◮ The product chamber system ∆1 × · · · × ∆n is a chamber system

ver I where, for i ∈ Jk, the incidence is given by

(c1, . . . , cn) ∼i (d1, . . . , dn) if and only if ck ∼i dk and cℓ = dℓ for all ℓ = k.

Theorem

The product ∆1 × · · · × ∆n is a building isomorphic to ∆ (with σ = id).

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Spherical diagrams

Coxter (1934): all irreducible spherical Coxeter groups

(wikipedia) 5 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Spherical diagrams

Ronan-Tits: no building thick building of type H3 or H4.

(wikipedia) 6 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Spherical diagrams

Tits spherical classification: all buildings with diagram of rank ≥ 4.

(wikipedia) 7 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Spherical diagrams

No root system associated, hence no Euclidean extension.

(wikipedia) 8 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Euclidean diagrams

One edges added when Euclidean reflection groups exist.

(wikipedia) 9 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Euclidean diagrams

Classification with regularity parameters.

(Parkinson’s thesis) 10 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Euclidean diagrams

(Parkinson’s thesis) 11 / 13

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Thibaut Dumont University of Jyväskylä June 2019

Classification: Euclidean diagrams

(Parkinson’s thesis) 12 / 13

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Thibaut Dumont University of Jyväskylä June 2019

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)

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Thibaut Dumont University of Jyväskylä June 2019

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)

◮ SLn(Qp)

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