Positivity in Real Grassmannians: Combinatorial Formulas.

TL;DRAbstract

The main results of this dissertation are explicit formulas for a combinatorial approach to the study of totally nonnegative Grassmannians. A totally nonnegative Grassmannian consists of the points in a real Grassmannian where all Plücker coordinates can be taken to be simultaneously nonnegative. The combinatorial approach to the study of totally nonnegative Grassmannians was initiated by Postnikov, who introduced the concept of a boundary measurement matrix associated with a planar network. Using indirect recursive arguments, he showed that these matrices represent points in a totally nonnegative Grassmannian, and that every such point can be obtained in this way. This dissertation strengthens Postnikov's results by providing explicit combinatorial formulas which immediately imply these two key properties. First, we obtain subtraction-free formulas for the Plücker coordinates of a general boundary measurement matrix, thereby giving a constructive proof that any such matrix defines a t

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