Network parametrizations for the Grassmannian
Kelli Talaska, Lauren Williams · Algebra & Number Theory · 2013
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 D of the Grassmannian are in bijection with certain tableaux D called Go-diagrams, and each component is isomorphic to ދ( * ) a × )ދ( b for some nonnegative integers a and b.Our main result is an explicit parametrization 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 ދ or ދ * as appropriate, one gets a parametrization of the Deodhar component D .One application of such a parametrization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindström-Gessel-Viennot lemma.We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates.A main tool for us is the work of Marsh and Rietsch [Represent. Theory 8 (2004), 212-242] on Deodhar components in the flag variety.