Harvard-MIT Combinatorics: Bijections for the regions of hyperplane arrangements of Coxeter type


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November 9, 2022 4:15 pm - 5:15 pm
Science Center 232

Olivier Bernardi - Brandeis

A hyperplane arrangement of braid type is a collection of hyperplanes in R^n of the form {x_i-x_j=s}, where i,j are indices in [n] and s is an integer. Classical families include the Catalan, Shi, Semi-order and Linial arrangements. In this talk we will discuss some bijections between the regions of braid type arrangements and some labeled plane trees. This bijective framework applies to the braid type arrangements which satisfy a particular property that we call "transitivity" (the above classical families are all transitive). Time permitting we will then discuss some recent progress in extending this bijective framework in two directions: (a) extension of the bijections to lower dimensional faces, and (b) extension to arrangements of other Coxeter types (which include hyperplanes of the form {x_i+x_j=s}). Part of this work is joint with Te Cao.