Perpendicular dissections, composed partitions, and deformations of the braid arrangement

Thomas Zaslavsky

Binghamton University

September 15,
refreshments at 3:45pm


Given $n$ reference points in real $d$-space, we specify a finite set of hyperplanes that are perpendicular to lines that join pairs of the $n$ points. These hyperplanes dissect the space into a number of regions which is determined by the intersection semilattice of the hyperplanes. The semilattice in turn is, for generic reference points, determined by $d$ and the lift matroid of a gain graph that corresponds to the specifications of the hyperplanes.
Dissections of this kind arise from generalizing a problem in geometric voting theory. Particular examples of possible interest for voting lead to dissections whose matroids are truncations of those of simple, symmetrical deformations of the braid hyperplane arrangement of rank $n-1$. (Our dissections themselves are not deformations of the braid arrangement.) The intersection semilattices of these examples consist of composed partitions, which are partitions in which each block has an ordered partition, and generalizations to $k$-composed partitions (sometimes called ``generalized partitions'').
The talk will to a great extent depend on pictures and will not assume any knowledge of weird technical machinery.

Speaker's Contact Info: zaslav(at-sign)

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