Abstract
A parametrization of a positroid variety Π of dimension d is a regular map (C×)d→Π which is birational onto a dense subset of Π. There are several remarkable combinatorial constructions which yield parametrizations of positroid varieties. We investigate the relationship between two families of such parametrizations, and prove they are essentially the same. Our first family is defined in terms of Postnikov's boundary measurement map, and the domain of each parametrization is the space of edge weights of a planar network. We focus on a special class of planar networks called bridge graphs, which have applications to particle physics. Our second family arises from Marsh and Rietsch's parametrizations of Deodhar components of the flag variety, which are indexed by certain subexpressions of reduced words. Projecting to the Grassmannian gives a family of parametrizations for each positroid variety. We show that each Deodhar parametrization for a positroid variety corresponds to a bridge graph, while each parametrization arising from a bridge graph agrees with some projected Deodhar parametrization.
Highlights
Lusztig defined the totally nonnegative part of an abstract flag manifold and conjectured that it was made up of cells, a conjecture later proved by Rietsch (Lusztig, 1994, 1998; Rietsch, 1999)
Open positroid varieties are the images of open Richardson varieties in the flag variety F (n) under the natural projection πk : F (n) → Gr(k, n)
We investigate a particular class of network parametrizations, which arise from bridge graphs
Summary
Lusztig defined the totally nonnegative part of an abstract flag manifold and conjectured that it was made up of cells, a conjecture later proved by Rietsch (Lusztig, 1994, 1998; Rietsch, 1999). We have a family of Deodhar components D ⊂ F (n) such that the natural projection from F (n) to Gr(k, n) maps each D isomorphically to a dense subset ofΠ These are precisely the top-dimensional Deodhar components of the Richardson varieties Xuw which project birationally toΠ. We will show that these two ways of parametrizing positroid varieties–via bridge graphs, and via projected Deodhar parametrizations–are essentially the same This result was first conjectured by Thomas Lam (Lam, 2013a). Talaska and Williams explored the link between distinguished subexpressions and network parametrizations further in (Talaska and Williams, 2013) They considered Deodhar components of F (n) indexed by all distinguished subexpressions of Grassmannian permutations, not just PDS’s. This is an extended abstract for a longer paper, which may be found at arXiv:1411.2997 [math.CO]
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