Vol. 7, No. 9, 2013

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Network parametrizations for the Grassmannian

Vol. 7 (2013), No. 9, 2275–2311
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 ${\mathsc{ℛ}}_{D}$ of the Grassmannian are in bijection with certain tableaux $D$ called Go-diagrams, and each component is isomorphic to ${\left({\mathbb{K}}^{\ast }\right)}^{a}×{\left(\mathbb{K}\right)}^{b}$ 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 ${N}_{\phantom{\rule{0.3em}{0ex}}D}$ and its weight matrix ${W}_{\phantom{\rule{0.3em}{0ex}}D}$, whose entries enumerate directed paths in ${N}_{\phantom{\rule{0.3em}{0ex}}D}$. By letting the weights in the network vary over $\mathbb{K}$ or ${\mathbb{K}}^{\ast }$ as appropriate, one gets a parametrization of the Deodhar component ${\mathsc{ℛ}}_{D}$. 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.

Keywords
Grassmannian, network, total positivity, Deodhar decomposition
Primary: 14M15
Secondary: 05E10