Kazhdan Lusztig Cells in Infinite Coxeter Groups

Abstract

Groups defined by presentations of the form $\langle s_1,\ldots,s_n | s_i^2 = 1, (s_is_j)^{m_{ij}} = 1(i,j=1,\ldots,n)\rangle$ are called Coxeter groups. The exponents $m_{i,j} ∈ N ∪ {∞}$ form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group $S_n$ is isomorphic to the Coxeter group with presentation $\langle s_1,\ldots,s_n | s_i^2 = 1 (i=1,\ldots,n),(s_is_{i+1})^3=1(i=1,\ldots,n-1)\rangle$, and is also known as the Weyl group of type $A_{n-1}$.