Network parameterizations for the GrassmannianAlgebra & Discrete Mathematics
|Lauren Williams, Dept. of Mathematics, UC Berkeley
|Thu, Feb 28 2013, 3:10PM
Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation. Deodhar components of the Grassmannian are in bijection with certain tableaux called Go-diagrams, and each component is isomorphic to (K^*)^a × K^b for some non-negative integers a and b. Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram we construct a weighted network and its weight matrix, whose entries enumerate directed paths in the network. By letting the weights in the network vary over K or K* as appropriate, one gets a parameterization of the corresponding Deodhar component. We also give a (minimal) characterization of each Deodhar component in terms of Plucker coordinates. Note that in his study of the totally non-negative part of the Grassmannian, Postnikov constructed parameterizations of positroid cells that used planar networks associated to Le-diagrams. Our construction generalizes his. This is joint work with Kelli Talaska.