Involution Subword Complexes in Coxeter Groups

Let (W,S) be a Coxeter system. An element in W and an ordered list of elements in S give us a subword complex, as defined by Knutson and Miller. We define the “fish product” between an involution in W and a generator in S, which is also discussed by Hultman. This “fish product” always produces an involution. The structure of the involution subword complex ∆̂(Q,w) behaves very similarly to the regular subword complex. In particular, we prove that ∆̂(Q,w) is either a ball or sphere. We then give an explicit description of a special class of involution subword complexes in Sn and demonstrate that they are isomorphic to the dual associahedron.