Subword complexes and 2-truncated cubes (Q2925754)

Let \(W\) be a Coxeter group and \(S\) a set of simple reflections that generate \(W\). A word of size \(m\) is an ordered sequence of length \(m\) of elements of \(S\). A subword is an ordered subsequence of a word. Let \(q\) be a word and \(w \in W\). The subword complex \(\Delta(q, w)\) is the set of subwords \(q^\prime \subseteq q\) such that some subsequence of \(q \setminus q^\prime\) is a reduced expression for \(w\). These were introduced and studied first by \textit{A. Knutson} and \textit{E. Miller} [Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007); Adv. Math. 184, No. 1, 161--176 (2004; Zbl 1069.20026)]. They showed that these complexes are pure and vertex-decomposable.NEWLINENEWLINENEWLINELet \(w_0\) be the longest element in \(W\). Let \(w_0\) be an arbitrary reduced expression for \(w_0\). Let \(c\) be a word. In the paper under review, the author studies the subword complex \(\Delta(cw_0, w_0)\). Among the results proved in this paper, it is shown that the set of vertices of \(\Delta(cw_0, w_0)\) is in bijective correspondence with the set consisting of all the negative simple roots and the \((c,w_0)\)-stable positive roots (theorem 2(i)), and that every generalized associahedron is a 2-truncated cube (corollary 1). The proofs are based on earlier results of the author [Proc. Steklov Inst. Math. 286, 114--127 (2014); translation from Tr. Mat. Inst. Steklova 286, 129--143 (2014; Zbl 1338.20037)].