Bases of tropical Plucker functions, wirings, tilings and Leclerc-Zelevinsky conjectures.
Tuesday October 13, 2009
11:00AM - 12:00PM
Speakers:
Gleb Koshevoy
Abstract:
The first part of my talk deals with tropical Plucker functions, a class of functions that obey tropical analogs of classical Plucker relations on minors of a matrix. We construct a basis for the set of tropical Plucker functions defined on integer points of truncated box $\mathbf B_m^{m'}$, that is a subset $B\subset \mathbf B_m^{m'}$, such that the restriction map $TP(\mathbf B_m^{m'})\to \mathbb R^B$$, bijective. Also we characterize, in terms of the restriction to the basis, the classes of submodular, so- called skew-submodular, and discrete concave functions in T P, and present a bijection between MV-polytopes and submodular TP-functions on a Boolean cube.
The second part of my talk deals with study of bases of tropical Plucker functions on the Boolean cube. For, we introduce and study a class of wirings diagrams in a disc, which generalizes the class of wirings diagram when two diffrent wires have no more one common point. Based on such diagramms we construct a class of plane acyclic digraphs. We will show that the vertices of such a digraph form a weakly separated set-system, and any pair of strongly separated sets in such a system is connected by path. Using these results we affirmatively answer all conjectures by Leclerc and Zelevinsky (1998) on weakly-separated set-systems. These results obtained in a joint work with V.Danilov and A.Karzanov.
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