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On cosets in Coxeter groups

In this paper the notion of Coxeter length for a subset of a Coxeter group, as introduced in, is investigated for various subsets of a Coxeter group. Mostly cosets of various subgroups are examined as well as the associated idea of X-posets, which is a vast generalization of the Bruhat order. | Turk J Math 36 (2012) , 77 – 93. ¨ ITAK ˙ c TUB doi:10.3906/mat-0909-65 On cosets in Coxeter groups ∗ Sarah B. Hart, Peter J. Rowley Abstract In this paper the notion of Coxeter length for a subset of a Coxeter group, as introduced in [9], is investigated for various subsets of a Coxeter group. Mostly cosets of various subgroups are examined as well as the associated idea of X -posets, which is a vast generalization of the Bruhat order. Key Words: Coxeter group, Bruhat order, cosets, partially ordered sets 1. Introduction In [9] the authors introduced the notion of length for a subset of a Coxeter group which generalizes the well known length function on elements of a Coxeter group. A number of properties of this generalized length function were obtained there. For a survey of results in this area see [6]. The purpose of the present paper is to investigate the lengths of cosets of various subgroups of the Coxeter group W with particular emphasis on certain partial orders. These partial orders amount to an extensive generalization of the Bruhat order [1], [3], [8] and, indeed, of the Bruhat order defined by Deodhar on the cosets of a standard parabolic subgroup of W [5]. For W a Coxeter group and X a subset of W , we define N (X) = {α ∈ Φ+ |w · α ∈ Φ− for some w ∈ X }, where Φ+ and Φ− are, respectively, the positive and negative roots of the root system Φ of W . So N (X) consists of all the positive roots taken negative by some element of X . Now, from [9], the Coxeter length of X , l(X), is defined to be the cardinality of N (X). If X = {w} , then l(X) is just the length of w in the traditional sense. Let Ref(W ) be the set of reflections of W . For w, w ∈ W write w −→ w if w = w t for some t ∈ Ref(W ) and l(w ) l(w) if w · αs ∈ Φ+ and l(ws) < l(w) if w · αs ∈ Φ− . (iv) The Exchange Condition: Let w = r1 · · · rk , ri ∈ R , be a not necessarily reduced expression for w . Suppose that s ∈ Ref(W ) satisfies l(ws) < l(w). Then there is an index i for which .