Reduced expressions in semidirect products of Coxeter groups

Any normal reflection subgroup f W of a Coxeter system (W, S) is a factor in a semidirect product decomposition of W as described by Bonnafe and Dyer. Namely, S is the union of two subsets I and J such that no element of I is conjugate to an element of J , f W is the subgroup generated by WI conjugates of elements of J , and W is the semidirect product of WI by f W . This note describes the reduced expressions of elements of the form wxw−1 with w ∈ WI and x ∈ WJ in terms of reduced expressions of x and a suitable element of WI . Introduction and Preliminaries Following Bonnafe and Dyer, we consider a Coxeter system (W,S) in its internal semidirect product decomposition with respect to two subsets I, J ⊂ S such that I ∩J = ∅ and I ∪J = S and with the requirement that no element of I is conjugate to an element of J . The latter statement says that the vertex set of the Coxeter graph of (W,S) can be divided into two subsets with only edges with even label connecting the two. Let T represent the set of reflections of W . In [4] (see also [1]), it is shown that W is the semidirect product of WI and the normal subgroup W generated by WI -conjugates of elements of J . We observe that any normal reflection subgroup W of W arises from this type of decomposition of S. Then, we describe the reduced expressions of elements of the form wxw−1 where w ∈ WI and x ∈ WJ in terms of reduced expressions of w and x. From this, we get a slightly stronger result describing reduced expressions of reflections of the form wtw−1 with t ∈ WJ ∩ T and w ∈ WI . For general results about Coxeter systems consult [5]. 1. Statement of Main Result We begin with the same setup as [1]. We let (W,S) be a Coxeter system with ` : W → N the corresponding length function. Furthermore, assume that S = I ∪J with I ∩J = ∅ and that no element of I is conjugate to an element of J . We denote by WI and WJ the standard parabolic subgroups generated by I and J respectively. For any K, L ⊂ S, we let W := W/WK (respectively W := WL\W ) be the set of shortest right coset representatives of WK in W (respectively the set of shortest left coset representatives of WL in W ). Then we define W := W ∩ W to be the set of shortest representatives for the double cosets WL\W/WK in W . For brevity, we will use the following notation W K := WK ∩ W. Additionally, let T = ∪w∈W wSw−1 be the set of reflections of W . If P(T ) denotes the power set of Date: February 20, 2009. 1Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556; tedgar@nd.edu