Network Parameterizations for the Grassmannian

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


Introduction
There is a remarkable subset of the real Grassmannian Gr k,n (R) called its totally non-negative part (Gr k,n ) ≥0 [7,9], which may be defined as the subset of the real Grassmannian where all Plücker coordinates have the same sign. Postnikov showed that (Gr k,n ) ≥0 has a decomposition into positroid cells, which are indexed by certain tableaux called Γ -diagrams. He also gave explicit parameterizations of each cell. In particular, he showed that from each Γ -diagram one can produce a planar network, and that one can write down a parameterization of the corresponding cell using the weight matrix of that network. This parameterization shows that the cell is isomorphic to R d >0 for some d. Such a parameterization is convenient, because for example, one may read off formulas for Plücker coordinates from non-intersecting paths in the network, using the Lindstrom-Gessel-Viennot Lemma.
A natural question is whether these network parameterizations for positroid cells can be extended from (Gr k,n ) ≥0 to the entire real Grassmannian Gr k,n (R). In this paper we give an affirmative answer to this question, by replacing the positroid cell decomposition with the Deodhar decomposition of the Grassmannian Gr k,n (K) (here K is an arbitrary field).
The components of the Deodhar decomposition are not in general cells, but nevertheless have a simple topology: by [2,3], each one is isomorphic to (K * ) a × (K) b . The relation of the Deodhar decomposition of Gr k,n (R) to Postnikov's cell decomposition of (Gr k,n ) ≥0 is as follows: the intersection of a Deodhar component S D ∼ = (R * ) a ×(R) b with (Gr k,n ) ≥0 is precisely one positroid cell isomorphic to (R >0 ) a if b = 0, and is empty otherwise. In particular, when one intersects the Deodhar decomposition with (Gr k,n ) ≥0 , one obtains the positroid cell decomposition of (Gr k,n ) ≥0 . There is a related positroid stratification of the real Grassmannian, and each positroid stratum is a union of Deodhar components.
As for the combinatorics, components of the Deodhar decomposition are indexed by distinguished subexpressions [2,3], or equivalently, by certain tableaux called Go-diagrams [6], which generalize Γ diagrams. In this paper we associate a network to each Go-diagram, and write down a parameterization of the corresponding Deodhar component using the weight matrix of that network. Our construction generalizes Postnikov's, but our networks are no longer planar in general.
Our main results can be summed up as follows. See Theorems 3.15 and 4.3 and the constructions preceding them for complete details.
Theorem. Let K be an arbitrary field.
• Every point in Gr k,n (K) can be realized as the weight matrix of a unique network associated to a Go-diagram, and we can explicitly construct the corresponding network. The networks corresponding to points in the same Deodhar component have the same underlying graph, but different weights.
• Every Deodhar component may be characterized by the vanishing and nonvanishing of certain Plücker coordinates. Using this characterization, we can also explicitly construct the network associated to a point given either by a matrix repsresentative or by a list of Plücker coordinates.
To illustrate the main results, we provide a small example here. More complicated examples may be seen throughout the rest of the paper.  of certain Plücker coordinates: Figure 1 shows two Go-diagrams D 1 and D 2 and their associated networks. Note that the network on the right is not planar. The weight matrices associated to these diagrams are The positroid stratum S I is the disjoint union of the two corresponding Deodhar components S D1 and S D2 , which can be characterized in terms of vanishing and nonvanishing of minors as: Note that if one lets the a i 's range over K * and lets c 4 range over K, then we see that S D1 ∼ = (K * ) 4 and There are several applications of our construction. First, as a special case of our theorem, one may parameterize all k × n matrices using networks. Second, by applying the Lindstrom-Gessel-Viennot Lemma to a given network, one may write down explicit formulas for Plücker coordinates in terms of collections of non-intersecting paths in the network. Third, building upon work of [6], we obtain (minimal) descriptions of Deodhar components in the Grassmannian, in terms of vanishing and nonvanishing of Plücker coordinates. It follows that each Deodhar component is a union of matroid strata.
Although less well known than the Schubert decomposition and matroid stratification, the Deodhar decomposition is very interesting in its own right. Deodhar's original motivation for introducing his decomposition was the desire to understand Kazhdan-Lusztig polynomials. In the flag variety, one may intersect two opposite Schubert cells, obtaining a Richardson variety, which Deodhar showed is a union of Deodhar components. Each Richardson variety R v,w (q) may be defined over a finite field K = F q , and in this case, the number of points determines the R-polynomials R v,w (q) = #(R v,w (F q )), introduced by Kazhdan and Lusztig [4] to give a recursive formula for the Kazhdan-Lusztig polynomials. Since each Deodhar component is isomorphic to (F * q ) a × (F q ) b for some a and b, if one understands the decomposition of a Richardson variety into Deodhar components, then in principle one may compute the R-polynonomials and hence Kazhdan-Lusztig polynomials.
Another reason for our interest in the Deodhar decomposition is its relation to soliton solutions of the KP equation. It is well-known that from each point A in the real Grassmannian, one may construct a soliton solution u A (x, y, t) of the KP equation. It was shown in recent work of Kodama and the second author [6] that when the time variable t tends to −∞, the combinatorics of the solution u A (x, y, t) depends precisely on which Deodhar component A lies in.
The outline of this paper is as follows. In Section 2, we give some background on the Grassmannian and its decompositions, including the Schubert decomposition, the positroid stratification, and the matroid stratification. In Section 3, we present our main construction: we explain how to construct a network from each diagram, then use that network to write down a parameterization of a subset of the Grassmannian that we call a network component. Our main result is that this network component coincides with the corresponding Deodhar component in the Grassmannian. Finally in Section 4 we give a characterization of Deodhar components in terms of the vanishing and nonvanishing of certain Plücker coordinates.
ACKNOWLEDGEMENTS: L.W. is grateful to Yuji Kodama for their joint work on soliton solutions of the KP equation, which provided motivation for this project.

Background on the Grassmannian
The Grassmannian Gr k,n is the space of all k-dimensional subspaces of an n-dimensional vector space K n . In this paper we will usually let K be an arbitrary field, though we will often think of it as R or C. An element of Gr k,n can be viewed as a full-rank k × n matrix modulo left multiplication by nonsingular k × k matrices. In other words, two k × n matrices represent the same point in Gr k,n if and only if they can be obtained from each other by row operations. Let [n] k be the set of all k-element subsets of k , let ∆ I (A) be the Plücker coordinate, that is, the maximal minor of the k × n matrix A located in the column set I. The map A → (∆ I (A)), where I ranges over [n] k , induces the Plücker embedding Gr k,n ֒→ KP ( n k )−1 into projective space. We now describe several useful decompositions of the Grassmannian: the Schubert decomposition, the positroid stratification, and the matroid stratification. Note that the matroid stratification refines the positroid stratification, which refines the Schubert decomposition. The main subject of this paper is the Deodhar decomposition of the Grassmannian, which refines the positroid stratification, and is refined by the matroid stratification (as we prove in Corollary 4.4).

The Schubert decomposition of Gr k,n
Throughout this paper, we identify partitions with their Young diagrams. Recall that the partitions λ contained in a k × (n − k) rectangle are in bijection with k-element subset I ⊂ [n]. The boundary of the Young diagram of such a partition λ forms a lattice path from the upper-right corner to the lowerleft corner of the rectangle. Let us label the n steps in this path by the numbers 1, . . . , n, and define I = I(λ) as the set of labels on the k vertical steps in the path. Conversely, we let λ(I) denote the partition corresponding to the subset I. Definition 2.1. For each partition λ contained in a k × (n − k) rectangle, we define the Schubert cell As λ ranges over the partitions contained in a k ×(n−k) rectangle, this gives the Schubert decomposition of the Grassmannian Gr k,n , i.e.
We now define the shifted linear order < i (for i ∈ [n]) to be the total order on [n] defined by One can then define cyclically shifted Schubert cells as follows.

Definition 2.2.
For each partition λ contained in a k × (n − k) rectangle, and each i ∈ [n], we define the cyclically shifted Schubert cell is the lexicographically minimal subset with respect to < i such that ∆ I(λ) = 0}.

The positroid stratification of Gr k,n
The positroid stratification of the Grassmannian Gr k,n is obtained by taking the simultaneous refinement of the n Schubert decompositions with respect to the n shifted linear orders < i . This stratification was first considered by Postnikov [9], who showed that the strata are conveniently described in terms of Grassmann necklaces, as well as decorated permutations and Γ -diagrams. Postnikov coined the terminology positroid because the intersection of the positroid stratification of the real Grassmannian with the totally non-negative part of the Grassmannian (Gr k,n ) ≥0 gives a cell decomposition of (Gr k,n ) ≥0 (whose cells are called positroid cells). In particular, we have |I 1 | = · · · = |I n |, which is equal to some k ∈ [n]. We then say that I is a Grassmann necklace of type (k, n). Equivalently, each positroid stratum is an intersection of n cyclically shifted Schubert cells, that is, Grassmann necklaces are in bijection with tableaux called Γ -diagrams. n) is a partition λ contained in a k × (n − k) rectangle together with a filling D : λ → {0, +} of its boxes which has the Γ -property: there is no 0 which has a + above it and a + to its left. (i) (Here, "above" means above and in the same column, and "to its left" means to the left and in the same row.) In Figure 2 we give an example of a Γ -diagram.
(i) This forbidden pattern is in the shape of a backwards L, and hence is denoted Γ and pronounced "Le." Given an element A ∈ Gr k,n , there is an associated matroid M A whose bases are the k-subsets I ⊂ [n] such that ∆ I (A) = 0. This gives a stratification of Gr k,n called the matroid stratification, or Gelfand-Serganova stratification.

The main result: network parameterizations from Go-diagrams
In this section we define certain tableaux called Go-diagrams, then explain how to parameterize the Grassmannian using networks associated to Go-diagrams. First we will define more general tableaux called diagrams. To each diagram D we associate a network N D as follows.

Definition 3.2.
Let λ be a partition with ℓ boxes contained in a k × (n − k) rectangle, and let D be a diagram in λ. Label the boxes of λ from 1 to ℓ, starting from the rightmost box in the bottom row, then reading right to left across the bottom row, then right to left across the row above that, etc. The (weighted) network N D associated to D is a directed graph obtained as follows: • Associate an internal vertex to each + and each ✈ ; • After labeling the southeast border of the Young diagram with the numbers 1, 2, . . . , n (from northeast to southwest), associate a boundary vertex to each number; • From each internal vertex, draw an edge right to the nearest +-vertex or boundary vertex; • From each internal vertex, draw an edge down to the nearest +-vertex or boundary vertex; • Direct all edges left and down. After doing so, k of the boundary vertices become sources and the remaining n − k boundary vertices become sinks.
• If e is a horizontal edge whose left vertex is a +-vertex (respectively a ✈ -vertex) in box b, assign e the weight a b (respectively c b ). We think of a b and c b as indeterminates, but later they will be elements of K * and K respectively.
• If e is a vertical edge, assign e the weight 1.
Note that in general such a directed graph is not planar, as two edges may cross over each other without meeting at a vertex. See Figure 3 for an example of a diagram and its associated network. We now explain how to associate a weight matrix to such a network.

Definition 3.3. Let N D be a network as in Definition 3.2. Let
denote the sources. If P is a directed path in the network, let w(P ) denote the product of all weights along P . If P is the empty path which starts and ends at the same boundary vertex, we let w(P ) = 1. If r is a source and s is any boundary vertex, define where the sum is over all paths P from r to s. The sign is chosen (uniquely) so that is the k × (n − k) weight matrix. We make the convention that the rows of W D are indexed by the sources i 1 , . . . , i k from top to bottom, and its columns are indexed by 1, 2, . . . , n from left to right. Figure 3 is

Distinguished expressions
We now review the notion of distinguished subexpressions, as in [2] and [8]. This definition will be essential for defining Go-diagrams. We assume the reader is familiar with the (strong) Bruhat order < on W = S n , and the basics of reduced expressions, as in [1]. Let w := s i1 . . . s im be a reduced expression for w ∈ W . A subexpression v of w is a word obtained from the reduced expression w by replacing some of the factors with 1. For example, consider a reduced expression in S 4 , say s 3 s 2 s 1 s 3 s 2 s 3 . Then s 3 s 2 1 s 3 s 2 1 is a subexpression of s 3 s 2 s 1 s 3 s 2 s 3 . Given a subexpression v, we set v (k) to be the product of the leftmost k factors of v, if k ≥ 1, and v (0) = 1.

Definition 3.6 (Distinguished subexpressions). [2, Definition 2.3] A subexpression
In other words, if right multiplication by s ij decreases the length of v (j−1) , then in a distinguished subex- In other words, it is distinguished and non-decreasing.
Lemma 3.8. [8] Given v ≤ w and a reduced expression w for w, there is a unique PDS v + for v in w.

Go-diagrams
In this section we explain how to index distinguished subexpressions by certain tableaux called Godiagrams, which were introduced in [6]. Go-diagrams are fillings of Young diagrams by pluses +, black stones ✈ , and white stones ❢ . (ii) (ii) In KW2, we used a slightly different convention and used blank boxes in place of +'s. Fix k and n. Let W k = s 1 , s 2 , . . . ,ŝ n−k , . . . , s n−1 be a parabolic subgroup of W = S n . Let W k denote the set of minimal-length coset representatives of W/W k . Recall that a descent of a permutation π is a position j such that π(j) > π(j + 1). Then W k is the subset of permutations of S n which have at most one descent; and that descent must be in position n − k.
It follows from [11] and [10] that elements w of W k can be identified with partitions λ w contained in a k × (n − k) rectangle. More specifically, let Q k be the poset whose elements are the boxes of a k × (n − k) rectangle; if b 1 and b 2 are two adjacent boxes such that b 2 is immediately to the left or immediately above b 1 , we have a cover relation b 1 ⋖ b 2 in Q k . The partial order on Q k is the transitive closure of ⋖. Now label the boxes of the rectangle with simple generators s i as in the figure below. If b is a box of the rectangle, then let s b denote its label by a simple generator. Let w k 0 ∈ W k denote the longest element in W k . Then the set of reduced expressions of w k 0 can be obtained by choosing a linear extension of Q k and writing down the corresponding word in the s i 's. We call such a linear extension a reading order; two linear extensions are shown in the figure below. Additionally, given a partition λ contained in the k × (n − k) rectangle (chosen so that the upper-left corner of its Young diagram is aligned with the upperleft corner of the rectangle), and a linear extension of the sub-poset of Q k comprised of the boxes of λ, the corresponding word in s i 's is a reduced expression of a minimal length coset representatives w ∈ W k . The element w ∈ W k depends only on the partition, not the linear extension, and all reduced expressions of w can be obtained by varying the linear extension. Finally, this correspondence is a bijection between partitions λ w contained in the k × (n − k) rectangle and elements w ∈ W k . Definition 3.9. [6, Section 4] Fix k and n. Let w ∈ W k , let w be a reduced expression for w, and let v be a distinguished subexpression of w. Then w and w determine a partition λ w contained in a k × (n − k) rectangle together with a reading order of its boxes. The Go-diagram associated to v and w is a filling of λ w with pluses and black and white stones, such that: for each k ∈ J • v we place a white stone in the corresponding box; for each k ∈ J • v we place a black stone in the corresponding box of λ w ; and for each k ∈ J + v we place a plus in the corresponding box of λ w . Remark 3.10. By [6,Section 4], the Go-diagram associated to v and w does not depend on w, only on w. Moreover, whether or not such a filling of a partition λ w is a Go-diagram does not depend on the choice of reading order of the boxes of λ w . Definition 3.11. We define the standard reading order of the boxes of a partition to be the reading order which starts at the rightmost box in the bottom row, then reads right to left across the bottom row, then right to left across the row above that, then right to left across the row above that, etc. This reading order is illustrated at the right of the figure below.
By default, we will use the standard reading order in this paper.

Remark 3.13. The Go-diagrams associated to PDS's are in bijection with
Γ -diagrams, see [6,Section 4]. Note that the Go-diagram associated to a PDS contains only pluses and white stones. This is precisely a Γ -diagram.

The main result
To state the main result, we now consider Go-diagrams (not arbitrary diagrams), the corresponding networks (Go-networks), and the corresponding weight matrices.
Definition 3.14. Let D be a Go-diagram contained in a k × (n − k) rectangle. We define a subset R D of the Grassmannian Gr k,n by letting each variable a i of the weight matrix (Definition 3.3) range over all nonzero elements K * , and letting each variable c i of the weight matrix range over all elements K. We call R D the network component associated to D.
We will not define the Deodhar decomposition of the Grassmannian, but refer to [2,3,8] for details.
Theorem 3.15. Let D be a Go-diagram contained in a k × (n− k) rectangle. Suppose that D has t pluses and u black stones. Then R D is isomorphic to the corresponding Deodhar component, and in particular is isomorphic to (K * ) t × K u . Furthermore, Gr k,n is the disjoint union of the network components R D , as D ranges over all Go-diagrams contained in a k × (n − k) rectangle. In other words, each point in the Grassmannian Gr k,n can be represented uniquely by a weighted network associated to a Go-diagram.
Corollary 3.16. Every matrix can be represented by a unique weighted network associated to a Godiagram.

A characterization of Deodhar components by minors
In this section we characterize Deodhar components in the Grassmannian by a list of vanishing and nonvanishing Plücker coordinates.