Bjorner and Brenti, Selected Chapter 2 Notes

References for these notes are Sections 2.7 and 2.8 as well as Appendicies A2.3, A2.4, and A2.5. of Bjorner and Brenti. There are several supplementary sources listed at the end. Section 2.7 explores the combinatorial structure of the interval between two elements of a coxeter group W , or more generally of a quotient W J . First of all, Corollary 2.5.6 assures us that every maximal chain in [w, u] is of the same length. This allows us to parametrize (if I am using that word correctly) the maximal chains by a set of k tuples, λ(m) = (λ1(m), . . . , λk(m)) for every maximal chain m, and where k = `(w, u) = `(w) − `(u). Given a reduced expression of w = s1s2 · · · sq and a maximal chain w = x0 .x1 . · · ·.xk−1 .xk = u, we know by the strong exchange property (and perhaps from the chain property), that x1 = s1 · · · ŝi · · · sq, and in general xj = s1 · · · ŝi1 · · · ŝij · · · sq (and we potentially could have deleted s1 and/or sq at any stage.) Then, λ(m) will record the index of the deleted generators of w in the order they are deleted. That is, if x1 = s1 · · · ŝi · · · sq, then λ(m)1 = i, and if xj = s1 · · · ŝi1 · · · ŝij · · · sq, then the first j positions of λ(m) will have i1, . . . , ij in the order they were deleted. For an example that may make more sense than the definition, consider maximal chains in the full Bruhat order of S3.