Coxeter complex

In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building. In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building. The first ingredient in the construction of the Coxeter complex associated to a Coxeter group W is a certain representation of W, called the canonical representation of W. Let ( W , S ) {displaystyle (W,S)} be a Coxeter system associated to W, with Coxeter matrix M = ( m ( s , t ) ) s , t ∈ S {displaystyle M=(m(s,t))_{s,tin S}} . The canonical representation is given by a vector space V with basis of formal symbols ( e s ) s ∈ S {displaystyle (e_{s})_{sin S}} , which is equipped with the symmetric bilinear form B ( e s , e t ) = − cos ⁡ ( π m ( s , t ) ) {displaystyle B(e_{s},e_{t})=-cos left({frac {pi }{m(s,t)}} ight)} . The action of W on this vector space V is then given by s ( v ) = v − 2 B ( e s , v ) B ( e s , e s ) e s {displaystyle s(v)=v-2{frac {B(e_{s},v)}{B(e_{s},e_{s})}}e_{s}} , as motivated by the expression for reflections in root systems. This representation has several foundational properties in the theory of Coxeter groups; for instance, the bilinear form B is positive definite if and only if W is finite. It is (always) a faithful representation of W. One can think of this representation as expressing W as some sort of reflection group, with the caveat that B might not be positive definite. It becomes important then to distinguish the representation V from its dual V*. The vectors e s {displaystyle e_{s}} lie in V, and have corresponding dual vectors e s ∨ {displaystyle e_{s}^{vee }} in V*, given by: where the angled brackets indicate the natural pairing of a dual vector in V* with a vector of V, and B is the bilinear form as above. Now W acts on V*, and the action satisfies the formula for s ∈ S {displaystyle sin S} and any f in V*. This expresses s as a reflection in the hyperplane H s = { f ∈ V ∗ : ⟨ f , e s ⟩ = 0 } {displaystyle H_{s}={fin V^{*}:langle f,e_{s} angle =0}} . One has the fundamental chamber C = { f ∈ V ∗ : ⟨ f , e s ⟩ > 0   ∀ s ∈ S } {displaystyle {mathcal {C}}={fin V^{*}:langle f,e_{s} angle >0 forall sin S}} , this has faces the so-called walls, H s {displaystyle H_{s}} . The other chambers can be obtained from C {displaystyle {mathcal {C}}} by translation: they are the w C {displaystyle w{mathcal {C}}} for w ∈ W {displaystyle win W} . Given a fundamental chamber C {displaystyle {mathcal {C}}} , the Tits cone is defined to be X = ⋃ w ∈ W w C ¯ {displaystyle X=igcup _{win W}w{overline {mathcal {C}}}} . This need not be the whole of V*. Of major importance is the fact that the Tits cone X is convex. The action of W on the Tits cone X has fundamental domain the fundamental chamber C {displaystyle {mathcal {C}}} .