Network parametrizations for the Grassmannian

Abstract

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, as shown by Kodama and the second author. Deodhar components S_D of the Grassmannian are in bijection with certain tableaux D called Go-diagrams, and each component is isomorphic to (K*)^a \times (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 D we construct a weighted network N_D and its weight matrix W_D, whose entries enumerate directed paths in N_D. By letting the weights in the network vary over K or K* as appropriate, one gets a parameterization of the Deodhar component S_D. One application of such a parameterization is that one may immediately determine which Plucker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plucker coordinates. A main tool for us is the work of Marsh and Rietsch on Deodhar components in the flag variety.