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Ebenso gilt cl = {q} wegen q ∈ cl und l(q) = l(c) + 1, und also aclx = δxq aclq . Wegen apcx axlf = x∈R apxf aclx x∈R folgt apcb ablf = apqf aclq . Weil f minimal in F gew¨ahlt ist, gilt apcb = 1 = aclq (denn wegen p = 1 gilt l(p) ≤ l(f ) − 1 [Skizze]), so dass ablf = apqf ≥ 2. h. b = f im Widerspruch zu l(f ) = l(b) + 1. 2 Austauschbedingung Coxetergruppen sind Gruppen, welche von Involutionen w1 , . . , wm erzeugt werden und der sogenannten Austauschbedingung gen¨ ugen. Zur Formulierung der Austauschbedingung ben¨ otigen wir die L¨ ange von Elementen in W := w1 , .

Beweis. Wir setzen zuerst U ∩S = {1} voraus. F¨ ur jedes Element t ∈ U S existieren u ∈ U und s ∈ S mit t ∈ us. Sind u′ ∈ U und s′ ∈ S mit t ∈ u′ s′ gegeben, so gilt us ∩ u′ s′ = ∅. Daraus folgt u∗ u′ ∩s(s′ )∗ = ∅ (denn es gelten f¨ ur Teilmengen M, O, P, Q ⊆ R ¨ folgende Aquivalenzen: OP ∩ QM = ∅ ⇔ Q ∩ OP M ∗ = ∅ ⇔ O∗ Q ∩ P M ∗ = ∅). Weil U und S abgeschlossen sind, folgt daraus u∗ u′ ∩ s′ s∗ ⊆ U ∩ S = {1}. h. u = u′ und s = s′ . Umgekehrt sei t ∈ U ∩ S ⊆ U S. Dann gilt 1t = t = t1 und somit wegen der Eindeutigkeit der Faktorisierung t = 1.

6 Schursche Schemata 39 Nun sei H eine Untergruppe von G. 3 sei f¨ ur alle g ∈ G auf der ˜ Menge G/H die Relation ˜ ˜ y H) ˜ | y ∈ xH ˜ g˜H} ˜ = {(xH, ˜ y H) ˜ | x−1 y ∈ HgH} g˜H = {(xH, ˜ ˜ ˜ G/ ˜ /H) ˜ mit G/ ˜ /H ˜ = {˜ gegeben. Dann ist T(G)H = (G/H, gH | g ∈ G} ein Assoziationsschema. Ein Assoziationsschema heißt Schursch, wenn es isomorph zu dem Quotientenschema einer Gruppe bez¨ uglich einer Untergruppe ist. Unser Ziel ist ein Kriterium f¨ ur Assoziationsschemata, Schursch zu sein. Es sei (X, R) ein Assoziationsschema.

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Algebraische Kombinatorik by Harm Pralle

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