A Coxeter group is a group with a presentationwhere and for . The pair consisting of and its distinguished generating reflections is a Coxeter system.
The Coxeter graph has vertex set , joins and when , and labels the edge by when . A Coxeter system is irreducible exactly when this graph is connected.
An irreducible Coxeter system is one whose Coxeter graph is connected. The connected components of a Coxeter graph give the direct-product decomposition of its Coxeter group.
The type graph is a simply-laced tree with one trivalent vertex and arms containing , , and vertices beyond that vertex.
A reduced expression for is a product of the fewest possible simple generators representing . The number of factors is its Coxeter length .
If is a reduced expression and is simple with , thenfor some index . The analogous statement holds for multiplication on the left.
Matsumoto's theorem says that any two reduced expressions for one element of a Coxeter group are connected by a finite sequence of braid moves.
Every word in simple Coxeter generators can be reduced by braid moves and cancellations . Consequently, a word is reduced exactly when no sequence of braid moves can make a cancellation possible.
For a finite-rank Coxeter system, let have basis and symmetric bilinear formIts geometric representation sends the generator to the reflection
The Coxeter Gram matrix is the Gram matrix of the distinguished basis in the Geometric representation of a Coxeter group.
The dual geometric representation acts on byThe open fundamental chamber consists of the functionals positive on every simple basis vector.
For the closed fundamental chamber , the Tits cone isIt equals all of exactly when the finite-rank Coxeter group is finite.
The Coxeter complex has faces represented by cosets of standard parabolic subgroups, ordered by reverse inclusion. Geometrically, its maximal simplices are the chambers of the Coxeter arrangement intersected with a sphere.
For , the standard parabolic subgroup is . Its cosets index faces of a fixed type in the Coxeter complex.
For involutory generators, the folding condition requires that simple multiplication never preserve length and that simultaneous left and right ascents either combine to a two-step ascent or fold: if both and have length , then either or .
For generators with , the braid relation equates the two alternating words of length beginning with and , respectively. When , it says that the generators commute.
A Coxeter element is a product of all simple generators in some order. For a finite irreducible Coxeter group, all Coxeter elements are conjugate.
A finite Coxeter group is a Coxeter group with finitely many elements. Its Coxeter Gram matrix is positive definite.
The longest element is the unique element of maximal Coxeter length in a finite Coxeter group. It is an involution and satisfies .
The Coxeter graph is the simply-laced seven-vertex tree whose three arms have lengths , , and . Its Coxeter Gram matrix is positive definite and has determinant .
The signed symmetric group consists of permutations of commuting with sign change. It is the Coxeter group of type .
The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type .
An irreducible affine Coxeter group has a positive-semidefinite Coxeter Gram matrix with a one-dimensional radical.
A hyperbolic Coxeter group has a nondegenerate Coxeter Gram matrix of Lorentzian signature in its standard geometric realization.
A BN-pair in a group consists of subgroups such that , is normal in , and has distinguished involutory generators satisfying the Bruhat multiplication axioms. The quotient is its Weyl group.
A Hecke algebra is a deformation of the group algebra of a Coxeter group, obtained by deforming the quadratic relations for its simple generators while retaining the braid relations.
Over the polynomial parameter ring, the generic Hecke algebra has basis and multiplicationThe parameters must agree for conjugate simple generators; equivalently, together with the braid relations presents the algebra.
The 0-Hecke algebra is the specialization of a Generic Hecke algebra of a Coxeter system. Its generators obey .
For a finite group with a BN-pair, the Iwahori-Hecke algebra over is the endomorphism algebra of the permutation module , up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.
For a representative , the Hecke parameter isThe Iwahori-Hecke algebra of a BN-pair is obtained from the generic algebra by .
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A Coxeter group is a special type of group that can be defined geometrically using reflections in Euclidean spaces. These groups are named after H.S.M. Coxeter, who studied their properties and relationships to various geometrical structures. ### Basic Definition: A Coxeter group is defined by a set of generators subjected to specific relations. These relations are based on the angles between the reflections corresponding to the generators.