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
of the Grassmannian are in bijection with certain tableaux
called
Go-diagrams, and each component is isomorphic to
for some
nonnegative integers
and
.
Our main result is an explicit parametrization of each Deodhar component in the
Grassmannian in terms of networks. More specifically, from a Go-diagram
we construct a weighted
network
and its
weightmatrix , whose entries
enumerate directed paths in
.
By letting the weights in the network vary over
or
as appropriate, one gets a parametrization of the Deodhar
component .
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
