Network parametrizations for the Grassmannian

Abstract

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 R D of the Grassmannian are in bijection with certain tableaux D called Go-diagrams, and each component is isomorphic to (K * ) a × (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 R D . One application of such a parameterization is that one may immediately determine which Plücker 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 Plücker coordinates. A main tool for us is the work of Marsh and Rietsch [8] on Deodhar components in the flag variety.

The Deodhar decomposition of the Grassmannian and the regularity of KP solitons

“…Their relationship is as follows: the matroid stratification refines the positroid stratification which refines the Schubert decomposition. In Section 3.4 we will describe the Deodhar decomposition, which is a refinement of the positroid stratification, and (as verified in [35]) is refined by the matroid stratification.…”

The geometry of on-shell diagrams

“…Postnikov showed that the positroid cells are indexed by Γ diagrams and planar bipartite graphs [22]. Each Deodhar component was shown to be indexed by so-called Go-diagrams [27] and subsequently by (generally non-planar) networks [24], which have a direct mapping to elements of the Grassmannian. The graph that represents a Deodhar component actually is in a specific matroid stratum, but each Deodhar component will have only one representative.…”

“…A graph is reduced or irreducible if it has the minimum number of independent closed paths within a given equivalence class. 24 Being defined up to equivalence transformations, reduced graphs are clearly not unique. More practically, a graph is reducible if it is possible to remove edges without changing its matroid polytope, modulo multiplicities.…”

“…The matching of moduli spaces is however a well-defined mathematical question regarding natural geometric objects in the field theory. 24 The notion of independent closed paths generalizes the one of internal faces, which is typically used for planar graphs. 25 We will assume this definition is equivalent to the one of irreducible graphs.…”

“…To put a label in each entry, we select the lexicographically minimal non-zero element with respect to the permutation in question. For example, the permutation 2413 will select the matroid element (24), if present, otherwise select (21), if present, etc.…”