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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.

The Coxeter complex of a Coxeter system $(W,S)$ is the abstract simplicial complex whose faces are the cosets $w{W}_J$ of proper standard parabolic subgroups, ordered by reverse inclusion. Chambers correspond to elements $w\in W$; two chambers sharing a panel of type $s$ are $w$ and $ws$. The face $w{W}_J$ has type $S\setminus J$ and dimension $|S\setminus J|-1$ when $S$ is finite.

For a finite Coxeter group, the Geometric representation of a Coxeter group is a real reflection action with positive-definite Coxeter Gram matrix. Its reflecting hyperplanes cut the unit sphere into the chambers of this simplicial complex. For $S_3$ with its two simple transpositions, the complex is a circle subdivided into six edges. For an infinite Coxeter group, the geometric chamber construction instead uses the Tits cone; a decomposition of the entire unit sphere is not generally available. The Davis complex uses spherical standard parabolic subgroups to give a different, contractible complex.

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

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

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.

John R. Stembridge works on algebraic combinatorics, representation theory, Coxeter groups and symmetric functions. His Michigan profile also identifies computational methods in algebra as a focus.^[ref]

With Richard P. Stanley, he studied immanants of matrices associated with the Jacobi–Trudi identity. An immanant replaces the sign character in a determinant by another symmetric group character. Their paper proves special cases of stronger positivity conjectures and connects the calculation with cycle structure of permutations having restricted positions.^[ref]

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

If a finitely generated Coxeter group $W$ is finite, its integer-valued Coxeter length has a maximum. Conversely, if some $w_0$ has globally maximal length $N$, every group element has a word of length at most $N$. There are only finitely many words of bounded length in the finite set of simple generators, so $W$ is finite.

Realize the finite group as the reflection group of a root system with fundamental system $\Delta$ and positive system $\Pi$. Maximality and the fact that multiplication by a simple generator changes Coxeter length by one giveThe positive-root criterion for Coxeter length therefore gives $w_0\Delta \subseteq -\Pi$. Since $w_0\Delta$ is itself fundamental, it must be the simple system $-\Delta$ of the positive system $-\Pi$.

If $v_0$ is another maximal-length element, the same argument gives $v_0\Delta =-\Delta =w_0\Delta$. Hence $v_0^{-1}{w}_0$ stabilizes $\Delta$, and part c gives $v_0^{-1}{w}_0=1$. The longest element of a finite Coxeter group is therefore unique.

Every Hecke parameter of a BN-pair $q_i$ divides $|B|$ and, because $B$ is a $q$-group, is a power of $q$. If $q\equiv 1(\mathrm{mod}p)$, then $q_i=1$ in the field $k$ of characteristic $p$. The specialized quadratic relation becomeswhile the braid relations are unchanged. These are the defining relations of the Coxeter group $W$, so $x_i\mapsto T_i$ induces a surjective homomorphismBoth algebras have bases indexed by $W$, hence the homomorphism is an isomorphism of group algebras.