On the distance between the expressions of a permutation

We prove that the combinatorial distance between any two reduced expressions of a given permutation of {1, ..., n} in terms of transpositions lies in O(n4), a sharp bound. Using a connection with the intersection numbers of certain curves in van Kampen diagrams, we prove that this bound is sharp, and give a practical criterion for proving that the derivations provided by the reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also show the existence of length ` expressions whose reversing requires C`4 elementary steps. This paper is about the various ways of expressing a permutation as a product of transpositions and the complexity of transforming one such expression into another. We consider both the absolute complexity (“combinatorial distance”), which deals with the minimal possible number of steps, and the more specific complexity (“reversing complexity”), which arises when one uses subword reversing, a certain prescribed strategy for transforming expressions. Throughout the paper, we denote by [[1, n]] the set {1, 2, ..., n}, and by si the transposition that exchanges i and i + 1. A well known result—see for instance [8]—states that, if π is any permutation of [[1, n]] and u, v are any two reduced (i.e., minimal length) expressions of π in terms of s1, ..., sn−1, then one can transform u into v only using the braid relations sisjsi = sjsisj with |i− j| = 1, (I) sisj = sjsi with |i− j| > 2. (II) In this context, we define the combinatorial distance dist(u, v) of u and v to be the minimal number of braid relations needed to transform u into v. The standard proof for the finiteness of dist(u, v) relies on the so-called Exchange Lemma of Coxeter groups, and it leads to an exponential upper bound for dist(u, v) in terms of n. The first aim of this paper is to establish a polynomial upper bound, namely a sharp degree 4 one. Using “n-expression” as a shorthand for “expression representing a permutation of [[1, n]]”, i.e., involving letters about s1, ..., sn−1 only, we prove Proposition 1. There exist positive constants C1, C2 such that, for each n, • all equivalent reduced n-expressions u, v satisfy dist(u, v) 6 C1 n, • there exist equivalent n-expressions u, v satisfying dist(u, v) > C2 n. (The values C1 = 1/2 and C2 = 1/8 are valid for n large enough.) The methods we use are geometrical. For the upper bound, we consider some area in the n-strand braid diagram naturally associated with an n-expression. For the lower bound, we consider van Kampen diagrams and introduce certain curves 1991 Mathematics Subject Classification. 20B30, 05E15, 20F55, 20F36.