Coxeter system

A Coxeter system $(W,I,M)$ records a Coxeter group $W$, its set $I$ of simple generators, and its Coxeter matrix $M=(m_{ij})$.

The Coxeter matrix of a Coxeter system is the symmetric matrix whose entry $m_{ij}$ is the order of $x_i{x}_j$.

The Coxeter graph has vertex set $I$, joins $i$ and $j$ when $m_{ij}\ge 3$, and labels the edge by $m_{ij}$ when $m_{ij}>3$. 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 $E(p,q,k)$ graph is a simply-laced tree with one trivalent vertex and arms containing $p$, $q$, and $k$ vertices beyond that vertex.

A reduced expression for $w\in W$ is a product of the fewest possible simple generators representing $w$. The number of factors is its Coxeter length $\ell (w)$.

If $w=s_1\cdots s_n$ is a reduced expression and $s$ is simple with $\ell (ws)<\ell (w)$, thenfor some index $j$. 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 $ss\mapsto 1$. Consequently, a word is reduced exactly when no sequence of braid moves can make a cancellation possible.

For a finite-rank Coxeter system, let $V$ have basis $(e_i{)}_{i\in I}$ and symmetric bilinear formIts geometric representation sends the generator $x_i$ to the reflection

The Coxeter Gram matrix is the Gram matrix $G=(-2\cos (\pi /m_{ij}){)}_{i,j\in I}$ of the distinguished basis in the Geometric representation of a Coxeter group.

The dual geometric representation acts on $V^{\ast }$ byThe open fundamental chamber consists of the functionals positive on every simple basis vector.

For the closed fundamental chamber $D\subseteq V^{\ast }$, the Tits cone isIt equals all of $V^{\ast }$ exactly when the finite-rank Coxeter group $W$ is finite.

The Coxeter complex has faces represented by cosets $w{W}_J$ of standard parabolic subgroups, ordered by reverse inclusion. Geometrically, its maximal simplices are the chambers of the Coxeter arrangement intersected with a sphere.

For $J\subseteq I$, the standard parabolic subgroup is $W_J=⟨s_j:j\in J⟩$. 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 $s_{i}w$ and $w{s}_j$ have length $\ell (w)+1$, then either $\ell (s_{i}w{s}_j)=\ell (w)+2$ or $w=s_{i}w{s}_j$.

For generators $s_i,s_j$ with $m_{ij}<\infty$, the braid relation equates the two alternating words of length $m_{ij}$ beginning with $s_i$ and $s_j$, respectively. When $m_{ij}=2$, 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.

The Coxeter number is the order of a Coxeter element in a finite irreducible Coxeter group.