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. For an abstract simplicial complex Delta, let Gamma(Delta) be the Boolean cell complex whose cells are indexed by all injective words over the sets forming the faces of Delta. The boundary of a cell indexed by a given word w consists of those cells that are indexed by subwords of w. For a partial order P on S, we study the subcomplex Gamma(Delta, P) of Gamma(Delta) consisting of those cells that are indexed by words whose letters are arranged in increasing order with respect to some linear extension of the order P. For a graph G = (S, E) on vertex set S and a word w over S, let [w] be the class of all words that we can obtain from w via a sequence of commutations ss' -> s's s such that {s, s'} g is not an edge in E. We study the Boolean cell complex Gamma/G(Delta) whose cells are indexed by commutation classes [w] of words indexing cells in Gamma(Delta). We prove: If Delta is shellable then so are Gamma(Delta, P) and Gamma/G(Delta). If Delta is Cohen-Macaulay (respectively sequentially Cohen-Macaulay) then so are Gamma(Delta, P) and Gamma/G(Delta). The complex Gamma(Delta) is partitionable. Our work generalizes work by Farmer and by Bjorner and Wachs on the complex of all injective words.

Nyckelord

injective word

Boolean cell complex

simplicial complex

Cohen-Macaulay complex

shellable complex

linear-groups

homology

stability

posets

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