The Poset of Shortest Paths in an Interval of the Bruhat Order

A Coxeter group W is a group generated by reflections; examples are the symmetric group and the hyperoctahedral group. These groups have many interesting combinatorial properties. For instance, one can define a partial order, called the Bruhat order, on the elements of W . Let [u,v] be an interval in the Bruhat order. The Bruhat graph of [u,v] , B(u,v) , includes the Hasse diagram of the poset [u,v] with edges directed upwards, as well as other edges that I will describe in the talk. A u - v path is a chain in [u,v] , but while not every u - v chain is a u - v path, every maximal u - v chain is such a path (of greatest length).