Higher Bruhat Orders and Higher Stasheff-Tamari Posets

Hugh Thomas

Abstract: The higher Bruhat orders B(n,d) were introduced by Manin and
Schechtman as generalizations of weak Bruhat order:  B(n,1) is weak Bruhat
order on the symmetric group S_n, and B(n,0) is the Boolean lattice Q_n.  
The higher Stasheff-Tamari posets are defined as partial orders on the set
of triangulations of a cyclic polytope.  There are reformulations of these
families of posets which make the connections between them more obvious: a
convex-geometric reformulation of the higher Bruhat orders due to Kapranov
and Voevodsky, and a combinatorial reformulation of the Stasheff-Tamari
posets due to myself.  I will discuss various maps between higher
Stasheff-Tamari posets and higher Bruhat orders which specialize to
familar maps between S_n and Q_{n-1}, and from S_n to planar binary trees.