About this Event
Microsoft Research, MSR First Floor Conference Center , One Memorial Drive, Cambridge, MA
Free Event
Free Event
SPEAKER: Nick Early (MIT)
TITLE: 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 cones are in duality with faces of the arrangement of reflection hyperplanes xi=xj.
In this talk, we construct directly a 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. The proof involves a matrix which is upper unitriangular with respect to a certain lexicographic order on ordered set partitions. We finally compute the enumeration of plates in the canonical basis, graded by dimension.
