Nick Early, University of Minnesota.

Canonical bases for permutohedral plates

Abstract:

There is a natural construction according to which the set of all faces of an arrangement of hyperplanes can be made into a vector space, by taking linear combinations of their characteristic functions.  Our space is equipped with a natural basis of characteristic functions of certain polyhedral cones called permutohedral cones passing through the origin, studied as plates by A. Ocneanu, which are labeled by ordered set partitions; these are in duality with faces of the arrangement of reflection hyperplanes xi=xj.  It is interesting to study the affine case as well: recently we conjectured stating that plates situated compatibly with the lattice of integer points in a generalized hypersimplex provide a new combinatorial interpretation of the h*-vector for that generalized hypersimplex.

In this talk, we construct directly a certain canonical basis which is compatible with one or both of two quotients: neglecting characteristic functions of (1) nonpointed cones, and (2) cones of codimension at least 1.  The essential feature here is that subsets of the canonical basis map to bases of the quotients.  As a consequence, we obtain the straightening relations which were originally computed by Ocneanu through the introduction of an auxiliary space of formal linear combinations of layered trees.  Time permitting, we will describe the circumstances which led us to formulate the conjecture for the h*-vector.

For more details, see Discrete CATS website