2 00 7 Complexes of Injective Words and Their Commutation Classes

Let S be a finite alphabet. An injective word over S is a word over S such that each letter in S appears at most once in the word. We study Boolean cell complexes of injective words over S and their commutation classes. This generalizes work by Farmer and by Björner and Wachs on the complex of all injective words. Specifically, for an abstract simplicial complex ∆, we consider the Boolean cell complex Γ(∆) whose cells are indexed by all injective words over the sets forming the faces of ∆. ⊲ For a partial order P = (S,≤P ) on S, we study the Boolean cell complex Γ(∆, P ) of all words from Γ(∆) whose sequence of letters comes from a linear extension of P . ⊲ For a graph G = (S,E) on vertex set S, we study the Boolean cell complex Γ/G(∆) whose cells are indexed by commutation classes [w] of words from Γ(∆). More precisely, [w] consists of all words that can be obtained from w by successively applying commutations of neighboring letters not joined by an edge of G. Our main results are as follows: ⊲ If ∆ is shellable then so are Γ(∆, P ) and Γ/G(∆). ⊲ If ∆ is Cohen-Macaulay (resp. sequentially Cohen-Macaulay) then so are Γ(∆, P ) and Γ/G(∆). ⊲ The complex Γ(∆) is partitionable.