A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient H of H and show that it has a basis parametrized by a certain subset Wc of the Coxeter group w. Specifically, Wc consists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in 5. We determine which Coxeter groups have finite Wc and compute the cardinality of Wc when W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w E Wc) of an exponential formula of Lusztig which utilizes a specialization of a subalgebra of H.