Coxeter group

A Coxeter group is a group with a presentationwhere $m_{ii}=1$ and $m_{ij}=m_{ji}\in \{2,3,\dots ,\infty \}$ for $i\ne j$. The pair consisting of $W$ and its distinguished generating reflections is a 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.

In the crystallographic convention, a Coxeter graph is admissible when it is realized by linearly independent unit vectors $v_i$ in a Euclidean space, with nonpositive mutual inner products and $4(v_i,v_j{)}^2=n_{ij}\in \{0,1,2,3\}$ for distinct vertices. The number $n_{ij}$ records a single, double or triple connection; equivalently, the displayed Gram matrix is positive-definite. In edge-label notation the nonzero connections have labels $3,4,6$. Forgetting multiplicity, the underlying graph is a forest: for any cycle of $k\ge 3$ distinct vertices, the sum of the corresponding unit vectors would have squared norm at most $k-k=0$, contradicting their linear independence. Multiple lines on one connection are labels and are not length-two cycles. This is the convention of Wehler's Lie algebra notes, section 6.3.

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 in a Coxeter group has the least possible Coxeter length among expressions in the generators of its Coxeter system.

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 $\ell (w)$.

For a group generated by involutions, the deletion condition says that every nonreduced word can be shortened without changing its value by deleting two of its letters.

Arrange the lengths of all consecutive subwords of a word in a triangular grid. Under the folding condition, the boundary between length ascents and descents must contain a folding square. Its equal opposite vertices identify two letters that can be deleted. Induction proves the deletion condition for involutory generators.

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.

A reflection of a Coxeter system $(W,S)$ is a conjugate $ws{w}^{-1}$ of a simple generator $s\in S$.

The wall associated with a reflection $r$ consists of the Cayley-graph edges fixed setwise and reversed by left multiplication by $r$. Its complement has two connected components.

The two connected components of the complement of a wall of a Coxeter group are its half-spaces. A wall separates two group elements when they lie in opposite half-spaces.

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.

A subset $T\subseteq S$ is spherical when its standard parabolic subgroup $W_T$ is finite.

The Davis complex is the geometric realization of the poset of cosets $w{W}_T$ for spherical subsets $T\subseteq S$. Equivalently, it is the basic construction of a Coxeter group obtained by gluing copies of the Davis chamber along mirrors. It is a contractible proper $W$-CW complex.

For a mirrored chamber $K$, the basic construction iswhere $(w,x)\sim (w^{\prime },x)$ when $w^{-1}{w}^{\prime }$ belongs to the subgroup generated by the mirrors containing $x$.

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.

A finite Coxeter group is a Coxeter group with finitely many elements. Its Coxeter Gram matrix is positive definite.

The longest element $w_0$ is the unique element of maximal Coxeter length in a finite Coxeter group. It is an involution and satisfies $\ell (w_{0}w)=\ell (w_0)-\ell (w)$.

The $E_7$ Coxeter graph is the simply-laced seven-vertex tree whose three arms have lengths $1$, $2$, and $3$. Its Coxeter Gram matrix is positive definite and has determinant $2$.

The signed symmetric group consists of permutations of $\{\pm 1,\dots ,\pm n\}$ commuting with sign change. It is the Coxeter group of type $B_n$.

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 $D_n$.

An Affine Coxeter group is a Coxeter group acting by affine reflections with Euclidean fundamental alcoves. 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 $G$ consists of subgroups $B,N$ such that $G=⟨B,N⟩$, $H=B\cap N$ is normal in $N$, and $W=N/H$ has distinguished involutory generators satisfying the Bruhat multiplication axioms. The quotient $W$ 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 $(T_w{)}_{w\in W}$ and multiplicationThe parameters must agree for conjugate simple generators; equivalently, $(T_i-a_i)(T_i+1)=0$ together with the braid relations presents the algebra.

The 0-Hecke algebra is the specialization $a_i=0$ of a Generic Hecke algebra of a Coxeter system. Its generators obey $T_i^2=-T_i$.

This is a finite-group realization of an Iwahori-Hecke algebra. For a finite group $G$ with a BN-pair, the Iwahori-Hecke algebra over $k$ is the endomorphism algebra of the permutation module $k[G/B]$, up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.

For a representative ${\dot{x}}_i\in N$, the Hecke parameter isThe Iwahori-Hecke algebra of a BN-pair is obtained from the generic algebra by $a_i\mapsto q_i\cdot 1_k$.