Total positivity, Grassmannians, and networks

Details Total positivity, Grassmannians, and networks Total positivity, Grassmannians, and networks

2006-09-27

arxiv.org/abs/math/0609764v1

Mathematics

Combinatorics

79 pages

Scientific paper

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on the cells, and describe how they are glued to each other.

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