aa r X i v : . [ m a t h . C O ] M a r COMBINATORICS OF SYMMETRIC PLABIC GRAPHS
RACHEL KARPMAN AND YI SU
Abstract. A plabic graph is a planar bicolored graph embedded in a disk,which satisfies some combinatorial conditions. Postnikov’s boundary measure-ment map takes the space of positive edge weights of a plabic graph G toa positroid cell in a totally nonnegative Grassmannian. In this note, we in-vestigate plabic graphs which are symmetric about a line of reflection, up toreversing the colors of vertices. These symmetric plabic graphs arise naturallyin the study of total positivity for the Lagrangian Grassmannian. We charac-terize various combinatorial objects associated with symmetric plabic graphs,and describe the subset of a Grassmannian which can be realized by symmetricweightings of symmetric plabic graphs. Introduction
A plabic graph is a planar graph embedded in a disk, with vertices colored blackor white. Postnikov introduced plabic graphs as a tool for studying the positroidstratification of the totally nonnegative Grassmannian Gr ≥ ( k, n ) [7].The totally nonnegative Grassmannian is the region of the Grassmannian Gr( k, n )where all Pl¨ucker coordinates are nonnegative real numbers, up to multiplicationby a common scalar. Postnikov defined a stratification of Gr ≥ ( k, n ) by positroidcells , each defined as the locus in Gr ≥ ( k, n ) where some set of Pl¨ucker coordinatesvanish. The resulting positroid stratification of Gr ≥ ( k, n ) has a rich geometric andcombinatorial structure. There are numerous combinatorial objects which indexpositroid cells, including bounded affine permutations , Grassmann necklaces , and aclass of matroids called positroids [7, 3].Given a plabic graph G , Postnikov’s boundary measurement map takes the spaceof positive real edge weights of G surjectively to a positroid cell Π G in Gr ≥ ( k, n )for some values of k and n . Moreover, the bounded affine permutation, Grassmannnecklace, and matroid of Π G are encoded in structure of G . Note that plabic graphsare not in bijection with positroid cells; rather, for each positroid cell, we have afamily of plabic graphs.In this note, we study plabic graphs which satisfy a symmetry condition. SeeFigure 1 for an example. These symmetric plabic graphs arise in the theory of totalpositivity for the Lagrangian Grassmannian Λ(2 n ), the moduli space of maximalisotropic subspaces with respect to a symplectic bilinear form. The connectionwith Λ(2 n ), as well as additional results on the combinatorics of symmetric plabicgraphs, will be discussed in a forthcoming paper by the first author [2].Let G denote a plabic graph, and let N denote the underlying uncolored network.Suppose N is symmetric with respect to reflection through a distinguished diameter d of the disc. Let V be the vertex set of E , and let r : V → V map each vertex v ∈ V to its mirror image across the line d . Then G is a symmetric plabic graph ifthe following conditions hold: Figure 1.
A symmetric plabic graph.(1) G has no vertices on the line d , although edges may cross d (2) For each v ∈ V , the vertices v and r ( v ) have opposite colors.We briefly summarize our results. In Theorem 3.1, we characterize the positroids,Grassmann necklaces, and bounded affine permutations associated with symmetricplabic graphs. We then consider symmetric weightings of symmetric plabic graphs;that is, weightings where each edge ( u, v ) has the same weight as its reflection( r ( u ) , r ( v )). Theorem 4.2 and Corollary 4.3 give a complete description of theset of points in Gr ≥ ( k, n ) corresponding to such weightings of symmetric plabicgraphs. This is a subvariety, cut out set-theoretically by linear equations, whichwe call the symmetric part of Gr ( k, n ). With the right choice of conventions,the symmetric part of Gr ≥ ( k, n ) is precisely the totally nonnegative part of theLagrangian Grassmannian [2]. Finally, Theorem 4.4 gives an explicit constructionwhich yields a symmetric weighting of a symmetric plabic graph for each point inthe symmetric part of Gr ≥ ( k, n ). In particular, we construct a symmetric bridgegraph for each point in Gr ≥ ( k, n ). Bridge graphs are a special class of plabic graphswhich appear as a computational tool in particle physics [1].2. Background
Notation.
For natural numbers k ≤ n , let [ n ] denote the set { , , . . . , n } , andlet (cid:0) [ n ] k (cid:1) denote the set of all k -element subsets of [ n ]. For a ∈ [ n ], let ≤ a denotethe cyclic shift of the usual linear order on n given by(1) a < a + 1 < . . . < n < < . . . < a − . Note that ≤ is the usual order ≤ . We extend this to a partial order on (cid:0) [ n ] k (cid:1) , bysetting I ≤ a J if we have i ℓ ≤ a j ℓ for all ℓ ∈ [ k ], where(2) I = { i < a i < a . . . < a i k } and J = { j < a j < a . . . < a j k } . For a, b ∈ [ n ], we define the cyclic interval [ a, b ] cyc by(3) [ a, b ] cyc = ( { a, a + 1 , . . . , b } a ≤ b { a, a + 1 , . . . , n − , n, . . . , b } a > b . OMBINATORICS OF SYMMETRIC PLABIC GRAPHS 3 (Note that this differs slightly from Postnikov’s convention.) If we arrange theelements of [ n ] clockwise around a circle, then [ a, b ] cyc represents a sequence ofconsecutive numbers.Let S n be the symmetric group in n letters, and let s i denote the simple trans-position ( i, i + 1) which switches i and i + 1. Let R > denote the positive reals.2.2. Grassmannians and Pl¨ucker coordinates.
Let Gr( k, n ) denote the Grass-mannian of k -dimensional linear subspaces of the vector space C n . We may realizeGr( k, n ) as the space of full-rank k × n matrices modulo the left action of GL( k ),the group of invertible k × k matrices; a matrix M represents the space spanned byits rows.The Pl¨ucker embedding , which we denote p , maps Gr( k, n ) into the projectivespace P ( nk ) − with homogeneous coordinates x J indexed by the elements of (cid:0) [ n ] k (cid:1) . For J ∈ (cid:0) [ n ] k (cid:1) let ∆ J denote the minor with columns indexed by J . Let V be a k -dimensional subspace of C n with representative matrix M . Then p ( V ) is the pointdefined by x J = ∆ J ( M ). This map embeds Gr( k, n ) as a smooth projective varietyin P ( nk ) − . The homogeneous coordinates ∆ J are known as Pl¨ucker coordinates on Gr( k, n ). The totally nonnegative Grassmannian , denoted Gr ≥ ( k, n ) , is thesubset of Gr( k, n ) whose Pl¨ucker coordinates are all nonnegative real numbers, upto multiplication by a common scalar.Let V ∈ Gr ≥ ( k, n ). The indices of the non-vanishing Pl¨ucker coordinates of V give a set J ⊆ (cid:0) [ n ] k (cid:1) called the matroid of V . We define the matroid cell M J asthe locus of points V ∈ Gr ≥ ( k, n ) with matroid J . The matroids J for which M J is nonempty are called positroids , and the corresponding matroid cells arecalled positroid cells. Positroid cells form a stratification of Gr ≥ ( k, n ). That is,the closure of a positroid cell Π in Gr ≥ ( k, n ) is the union of Π and some lower-dimensional positroid cells [7].2.3. Plabic graphs. A plabic graph is a planar graph embedded in a disk, witheach vertex colored black or white. The boundary vertices are numbered 1 , , . . . , n in clockwise order, and all boundary vertices have degree one. We call the edgesadjacent to boundary vertices legs of the graph, and a leaf adjacent to a boundaryvertex a lollipop . A white lollipop is a white leaf adjacent to a black boundaryvertex, while a black lollipop is the opposite.Postnikov introduced plabic graphs in [7, Section 11.5]. We follow the conven-tions of [5], which are more restrictive than Postnikov’s. In particular, we require aplabic graph to be bipartite, with the black and white vertices forming the partitesets. An almost perfect matching on a plabic graph is a subset of its edges whichcovers each interior vertex exactly once; boundary vertices may or may not be cov-ered. We consider only plabic graphs which admit an almost perfect matching.Finally, we require that no edge in a plabic graph connects two boundary vertices.We define a collection of paths and cycles in G , called trips , as follows. Webegin by traversing an edge { u, v } of G , from u to v . We then proceed accordingto the rules of the road : turn (maximally) left at every white internal vertex, and(maximally) right at every black internal vertex. Continuing in this fashion, weeventually reach a boundary vertex. The resulting directed path is a trip in G . SeeFigure 2 for an example. We repeat this process for every boundary vertex. If theresulting collection of trips covers every edge of G twice, once in each direction, weare done. RACHEL KARPMAN AND YI SU
Figure 2.
A trip in a plabic graph G . We have ¯ f G (2) = 4.Otherwise, we find an internal edge e = { u, v } such that no trip covers e in thedirection u → v . We begin by tracing e in this direction, and proceed according tothe rules of the road, until we find ourselves one against about to trace the edge u → v . The resulting directed cycle is a trip. Repeat this process until each edgeof G is covered twice by trips, once in each direction.Given a plabic graph G with n boundary vertices, we define the trip permutation ¯ f G ∈ S n of G by setting ¯ f G ( a ) = b if the trip that starts at boundary vertex a endsat boundary vertex b .The boundary measurement map is defined only for reduced plabic graphs. Post-nikov defined reduced plabic graphs in terms of certain local transformations ofplabic graphs, and gave a criterion for a plabic graph G to be reduced [7, Section13]. We take this criterion as the definition of a reduced graph. Definition 2.1.
A plabic graph G is reduced if it satisfies the following criteria:(1) G has no trips which are cycles.(2) G has no leaves, except perhaps some which are adjacent to boundary ver-tices.(3) No trip uses the same edge twice, once in each direction, unless that tripstarts (and ends) at a boundary vertex connected to a leaf.(4) No trips T and T share two edges e , e such that e comes before e inboth trips. If G is a reduced graph, each fixed point of ¯ f G corresponds to a lollipop [7].2.4. The boundary measurement map.
Let G be a reduced plabic networkwith e edges, and assign weights t , . . . , t e to the edges of G . Postnikov defined asurjective map from the space of positive real edge weights of G to some positroidcell Π G in Gr ≥ ( k, n ), called the boundary measurement map [7, Section 11.5].Postnikov, Speyer and Williams re-cast this construction in terms of almost perfectmatchings [8, Section 4-5], an approach Lam developed further in [5]. We use thedefinition of the boundary measurement map found in [5].As mentioned above, an almost perfect matching of a plabic graph G is a col-lection of edges of G which covers each internal vertex exactly once. (Boundaryvertices may or may not be covered.) For P an almost perfect matching on a plabic OMBINATORICS OF SYMMETRIC PLABIC GRAPHS 5 graph G with e edges, let ∂ ( P ) = { black boundary vertices used in P }∪ { white boundary vertices not used in P } (4)We define the boundary measurement map (5) ∂ G : ( R > ) e → P ( nk ) − to be the map which sends ( t , . . . , t e ) to the point with homogeneous coordinates(6) ∆ J = X ∂ ( P )= J t P where the sum is over all matchings P of G , and t P is the product of the weightsof all edges used in P .The boundary measurement map ∂ G is surjective onto the totally nonnegativecell Π G . However, it is almost never injective, due to the existence of gauge trans-formations . Let v be a vertex of G , and let ω be a weighting of G with ∂ G ( ω ) = X .Multiplying the weights of all edges incident at v by λ ∈ R > simply multiplies allPl¨ucker coordinates by λ , and hence yields a new weighting ω ′ with ∂ G ( ω ′ ) = X .Conversely, if two weightings of G map to the same point in Gr ≥ ( k, n ), then theyare related by some sequence of gauge transformations [7].2.5. Bounded affine permutations.
Bounded affine permutations are one ofmany families of combinatorial objects which index positroid cells. There is anatural bijection between bounded affine permutations and Postnikov’s decoratedpermutations , so Postnikov’s results about the latter translate easily to statementsabout the former [7, 3].
Definition 2.2. An affine permutation of order n is a bijection f : Z → Z whichsatisfies the condition (7) f ( i + n ) = f ( i ) + n for all i ∈ Z . The affine permutations of order n form a group, which we denote e S n . We may embed S n in e S n by extending each permutation periodically, in accor-dance with (7). Conversely, for f an affine permutation, there is a unique permu-tation ¯ f ∈ S n such that for all i ∈ [ n ], we have ¯ f ( i ) ∼ = f ( i ) (mod n ). Definition 2.3.
An affine permutation f is a bounded affine permutation of type ( k, n ) if it satisfies the following conditions(1) n n X i =1 ( f ( i ) − i ) = k. (2) i ≤ ¯ f ( i ) ≤ i + n for all i ∈ Z . We write
Bound( k, n ) for the set of all bounded affine permutations of type ( k, n ) . Let G be a reduced plabic graph with n boundary vertices, and trip permutation¯ f G . Suppose we have(8) k = |{ i ∈ [ n ] : ¯ f G ( i ) < i or i is a white lollipop of G }| . RACHEL KARPMAN AND YI SU j ¯ f ( i ) i ¯ f ( j ) j = ¯ f ( i ) i ¯ f ( j ) j ¯ f ( i ) i = ¯ f ( j ) Figure 3.
Crossings in a chord diagram.Then G has an associated bounded affine permutation f G of type ( k, n ) defined bysetting(9) f G ( i ) = ( ¯ f G ( i ) ¯ f G ( i ) > i or G has a black lollipop at i ¯ f G ( i ) + n ¯ f G ( i ) < i or G has a white lollipop at i for i ∈ [ n ] , and extending periodically using (7). For f a bounded affine permuta-tion, we say f has a white fixed point at i if f ( i ) = i + n , and that f has a blackfixed point at i if f ( i ) = i .Bounded affine permutations of type ( k, n ) are in bijection with positroid cells inGr ≥ ( k, n ) [7]. Let G be as above, and weight the edges of G with indeterminates.The boundary measurement map carries the space of positive real edge weights of G to the positroid cell Π G corresponding to ¯ f G [7]. There is a family of reducedgraphs for each bounded affine permutation, and hence for each positroid cell.We represent bounded affine permutations visually using chord diagrams , intro-duced in [7, Section 16]. A chord diagram for a bounded affine permutation f isa circle with vertices labeled 1 , , . . . , n in clockwise order. We draw arrows fromvertex i to vertex ¯ f ( i ) for all i ∈ [ n ]. By convention, if i is a white fixed point of f ,we draw a clockwise loop at i ; if i is a black fixed point, we draw a counter-clockwiseloop.Let ( i, ¯ f ( i )) and ( j, ¯ f ( j )) be a pair of chords in the chord diagram of a boundedaffine permutation f . This pair represents a crossing if ¯ f ( j ) ∈ [ i, ¯ f ( i )] cyc and j ∈ [ ¯ f ( i ) , i ] cyc , so the two chords intersect. See Figure 3.Let Π be a positroid cell in Gr ≥ ( k, n ) with bounded affine permutation f , andlet V be a matrix representing a point in Π. Let v , . . . , v n be the columns of G . Weextend this periodically to a sequence of vectors in v i ∈ C k by setting v i = v j if i ≡ j (mod n ). The following lemma, which is implicit in [3], gives a characterization of f in terms of the v i . Lemma 2.4.
With the notation above, f ( i ) is the smallest r ≥ i with v i ∈ Span( v i +1 , v i +2 , . . . , v r ) . Similarly f − ( i ) is the largest r ≤ i such that v i ∈ Span( v r , v r +1 , . . . , v i − ) . Grassmann and reverse Grassmann necklaces.
We now introduce a finalpair of indexing sets for positroid cells: Grassmann necklaces and dual Grassmannnecklaces. The combinatorics of Grassmann necklaces are vital to the proof ofTheorem 4.4, which shows that we can construct a symmetric plabic graph withsymmetric weights for any point in the symmetric part of Gr ≥ ( k, n ). Definition 2.5. A Grassmann necklace I = ( I , . . . , I n ) of type ( k, n ) is a sequenceof k -element subsets of [ n ] such that the following hold, with indices taken modulo n : OMBINATORICS OF SYMMETRIC PLABIC GRAPHS 7 (1) If i ∈ I i then I i +1 = ( I i ∪ { j } ) − { i } for some j ∈ [ n ] . (2) If i I i , then I i +1 = I i . Postnikov defined a combinatorial bijection between Grassmann necklaces oftype ( k, n ) and bounded affine permutations. Let I be a Grassmann necklace oftype ( k, n ). We construct the bounded affine permutation f corresponding to I asfollows, with indices taken modulo n :(1) If I i +1 = ( I i ∪ { j } ) − { i } for some j = i , then ¯ f ( i ) = j .(2) If I i +1 = I i and i I i , then i is a black fixed point of f .(3) If I i +1 = I i and i ∈ I i , then i is a white fixed point of f .Next, we describe how to recover the Grassmann necklace I from the boundedaffine permutation f . We say i ∈ [ n ] is an anti-exceedance of f if either ¯ f − ( i ) > i or i is a white fixed point. We say i is an a -anti-exceedance if we have ¯ f − ( i ) > a i or i is a white fixed point. The Grassmann necklace I corresponding to f is givenby setting(10) I a = { i ∈ [ n ] | i is an a -anti-exceedance of f } . Let M be a positroid of type ( k, n ). Then M is a collection of k -element subsetsof [ n ]. For each 1 ≤ i ≤ n , let I i be the minimal element of M with respect tothe shifted linear order ≤ i . Then M is a Grassmann necklace of type ( k, n ). Thisprocedure gives a bijection between Grassmann necklaces and positroids [7]. Forthe inverse bijection, let I = ( I , . . . , I n ) be a Grassmann necklace of type ( k, n ),and let M be the set of all k -element subsets J ∈ (cid:0) [ n ] k (cid:1) such that I i ≤ i J for all i ∈ [ n ]. Then M is the positroid corresponding to I [6].Let M and I be as above. If V is a matrix representing some point in thepositroid cell Π corresponding to M , with columns v , . . . , v n , then the columnsindexed by I i represent the minimal basis for C k among the columns of V withrespect the the cyclic order ≤ i . We will use this fact frequently in the proof ofTheorem 4.4.We also recall the dual notion, defined in [4, Section 3.6]. Definition 2.6. A dual Grassmann necklace of type ( k, n ) is a sequence J =( J , . . . , J n ) of k -element subsets of n such that the following hold, with indicestaken modulo n :(1) If i ∈ J i +1 , then J i = ( J i +1 ∪ { j } ) − { i } for some j (2) If i J i +1 , then J i = J i +1 . We have bijections between reverse Grassmann necklaces, decorated permuta-tions, and positroids, which commute with the bijections given above for Grassmannnecklaces. We describe these briefly below.For J a dual Grassmann necklace, we define the corresponding bounded affinepermutation f by setting(1) If J i = ( J i +1 ∪ { j } ) − { i } for some j = i , then ¯ f − ( i ) = j .(2) If J i = J i +1 and i J i +1 , then i is a black fixed point of f .(3) If J i = J i +1 and i ∈ J i +1 , then i is a white fixed point of f where again indices are taken modulo n .If Π is a positroid cell with positroid M and dual Grassmann necklace J , then J i gives the maximal element of M with respect to the shifted order ≤ i . RACHEL KARPMAN AND YI SU
Bridge graphs.
We now describe a way to build up plabic graphs inductively.For more details on this construction, see [4].Let G be a reduced plabic graph with bounded affine permutation f of type( k, n ), corresponding to a positroid cell Π. If f ( i ) > f ( i + 1) for some i ∈ [ n ], then f s i is a bounded affine permutation of type ( k, n ), with corresponding positroidcell Π ∗ . Moreover, we have dim(Π ∗ ) = dim(Π) + 1. We may add an edge, calleda bridge , between the two edges of G incident i and i + 1, to produce a graph G ∗ corresponding to Π ∗ . The bridge has a white vertex adjacent to i and a blackvertex adjacent to i + 1. If G has a boundary leaf, or lollipop, at i or i + 1, thatleaf becomes one endpoint of the bridge. Note that after adding the bridge, we adddegree-two vertices as needed to make the graph bipartite, as in Figure 4.For i ∈ [ n ], let x i ( c ) be the elementary matrix with 1’s along the diagonal, anonzero entry c at position ( i, i + 1), and 0’s everywhere else. Let G be as above,and suppose f ( i ) < f ( i + 1). Assign positive real weights to the edges of G , and let M be a matrix representing the corresponding point in Gr ≥ ( k, n ) . Applying gaugetransformations, we can assume the edges adjacent to boundary vertices i and i + 1have weight 1. Adding a bridge at ( i, i +1) with weight c corresponds to multiplying M on the right by x i ( c ). Let X denote the point in Gr ≥ ( k, n ) corresponding to G with its original weighting, and X ∗ denote the point corresponding to the weightedgraph obtained by adding the bridge. The boundary measurements change asfollows:(11) ∆ I ( X ∗ ) = ( ∆ I ( X ) + c ∆ ( I ∪{ i } ) −{ i +1 } ( X ) if i + 1 ∈ I but i I ∆ I ( X ) otherwise . Note that we are abusing notation slightly, since the ∆ I represent homogeneouscoordinates rather than functions. Figure 4.
Adding a bridge to a plabic graph.The following proposition, from [5], gives a way to “undo” the operation ofadding a bridge, at least on the level of matrix representatives.
Proposition 2.7.
Let X ∈ Gr ≥ ( k, n ) , and suppose X is contained in the positroidcell Π with bounded affine permutation f . Suppose i < f ( i ) < f ( i + 1) < i + n + 1 .Then (12) c = ∆ I i +1 ( X )∆ ( I i +1 ∪{ i } ) −{ i +1 } ( X ) is positive and well defined, and X ∗ = X · x i ( − c ) is in Π ∗ , where Π ∗ ⊂ Gr ≥ ( k, n ) is the positroid with bounded affine permutation f s i and dim(Π ∗ ) = dim(Π) − . OMBINATORICS OF SYMMETRIC PLABIC GRAPHS 9
Figure 5.
A symmetric weighting of a symmetric plabic graph.The distinguished diameter d is shown in gray. Unlabeled edgeshave weight 1.Note that we have(13) i < f ( i ) < f ( i + 1) < i + n + 1if and only if the chords in the chord diagram for f which start at i and i + 1respectively form a crossing as in Figure 3. If this occurs, we say that f has abridge at ( i, i + 1).3. Characterizing symmetric plabic graphs
Let G be a plabic graph with 2 n boundary vertices, corresponding to a positroidcell Π. Label the boundary vertices of G with the numbers 1 , , . . . , n in clockwiseorder, and fix a distinguished diameter d of G with one end between vertices 2 n and1, and the other between vertices n and n + 1. For each i ∈ n , let i ′ = 2 n + 1 − i .For I = { i , i , . . . , i k } , we define R ( I ) = [2 n ] \{ i ′ | i ∈ I } . Let G be a symmetricplabic graph, and let I ⊆ [2 n ]. Reflection through d gives a bijection betweenthe set of almost perfect matchings P of G with ∂ ( P ) = I and the set of almostperfect matchings P ′ of G with ∂ ( P ′ ) = R ( I ). One immediate consequence is thatif I = ∂ ( P ) for some matching P of G then | I | = n , so Π lies in Gr ≥ ( n, n ). Theorem 3.1.
Let Π be a positroid cell in Gr ≥ ( n, n ) with positroid M andbounded affine permutation f . Let I = ( I , . . . , I n ) be the Grassmann necklace of Π , and let J = ( J , . . . , J n ) be the dual Grassmann necklace. The following areequivalent:(i) Π can be represented by a symmetric plabic graph.(ii) I ∈ M if and only if R ( I ) ∈ M .(iii) For a ∈ [2 n ] , if f ( a ) = b , then f (2 n + 1 − a ) = 4 n + 1 − b .(iv) R ( I i ) = I i ′ +1 for all i ∈ [2 n ] with indices taken modulo n .(v) R ( J i ) = J i ′ +1 for all i ∈ [2 n ] with indices taken modulo n . Proof.
The above discussion shows that (i) implies (ii).Next, we show that (ii) and (iii) are equivalent. Consider a reduced plabic graph H corresponding to Π, not necessarily symmetric, but with distinguished diameter d as above. Reflect H about d , and reverse the colors of all vertices. Let H ′ bethe resulting plabic graph. We note that H ′ is reduced. Indeed, suppose H hasa trip from boundary vertex a to boundary vertex t . Then this trip corresponds,under reflection though d and reversing colors of vertices, to a trip from a ′ to t ′ in H ′ . Hence, H ′ satisfies the criterion for being reduced, which is phrased entirely interms of forbidden intersections between trips.Let ¯ f be the trip permutation of H , and ¯ f ′ the decorated permutation of H ′ .Then ¯ f ( a ) = t implies ¯ f ′ ( a ′ ) = t ′ . Note also that a < t if and only if a ′ > t ′ .Moreover, a is a black fixed point of f if and only if a ′ is a white fixed point of f ′ ,and vice versa. It follows that f ( a ) = b implies(14) f ′ (2 n + 1 − a ) = 4 n + 1 − b. The reduced plabic graph H ′ corresponds to some positroid cell Π ′ , with positroid M ′ . It is clear that M ′ = { I | R ( I ) ∈ M} . Hence M ′ = M if and only if M satisfies condition (ii) above, while ¯ f = f ′ if and only if f satisfies condition (iii).But the statement M ′ = M and ¯ f = f ′ are both, in turn, equivalent to Π = Π ′ .Hence (iii) and (ii) are equivalent as desired.Next, we show (iii) implies (i). Suppose f ( a ) = b implies f (2 n +1 − a ) = 4 n +1 − b .We claim that we can construct a symmetric plabic graph with bounded affinepermutation f , using the bridge graph construction given above. We proceed byinduction on the dimension of the positroid Π corresponding to G . Note first thatif a is a black fixed point of f , then (iii) implies that a ′ is a white fixed point, andvice versa. We may therefore assume that f has no fixed points. Hence there issome pair ( i, i + 1) with 1 ≤ i ≤ n such that the bounded affine permutation f has a bridge at ( i, i + 1). If i = n , let g = f s i . Otherwise, let g = f s i ′ − s i . In eachcase, g is a bounded affine permutation which satisfies (iii). By induction, we canbuild a symmetric plabic graph corresponding to g . Adding a bridge at ( n, n + 1)or a pair of bridges at ( i, i + 1) and ( i ′ − , i ′ ), respectively, we obtain a symmetricplabic graph for f .Next, we show that (iii) implies (iv) and (v). Indeed, if (iii) holds, then the chorddiagram for f has an arrow from a to t if and only if it has an arrow from a ′ to t ′ .Hence for each i ∈ [2 n ], we have ¯ f − ( a ) > i a if and only if ¯ f − ( a ′ ) < ( i ′ +1) a ′ , andso I i = R ( I i ′ +1 ). The argument for the dual Grassmann necklace is analogous.Finally, we show that (iv) and (v) each imply (iii). Suppose (iv) holds. Notefirst that for a ∈ [2 n ], we have a ∈ I i if and only if a ′ / ∈ I i ′ +1 . It follows inparticular that if a ∈ I i for all i ∈ [2 n ] (that is, a corresponds to a “white” fixedpoint of f ) then a ′ I i for all i , and a ′ corresponds to a black fixed point. Hence f ( a ′ ) = 4 n + 1 − f ( a ) as desired. The analogous argument holds if a is a black fixedpoint.Suppose a is not a fixed point of f . Then we have I a = ( I a +1 ∪ { a } ) − { ¯ f ( a ) } (15) I a ′ +1 = ( I a ′ ∪ { ¯ f ( a ′ ) } ) − { a ′ } (16)Applying R to the first line above, we have(17) R ( I a ) = ( R ( I a +1 ) ∪ { ( ¯ f ( a )) ′ } ) − { a ′ } OMBINATORICS OF SYMMETRIC PLABIC GRAPHS 11
By (iv), this implies(18) I a ′ +1 = ( I a ′ ∪ { ( ¯ f ( a )) ′ } ) − { a ′ } Comparing this with our previous expression for I a ′ +1 above, we have(19) ¯ f ( a ′ ) = ( ¯ f ( a )) ′ It follows that f is a bounded affine permutation of type ( n, n ) which satisfies (iii).Again, note that we could make an analogous argument using the dual Grassmannnecklace. This completes the proof. (cid:3) Network Realizations of Symmetric Points
Let G be a symmetric plabic graph. Suppose we assign weights to the edgesof G such that each edge ( u, v ) has the same weight as its reflection ( r ( u ) , r ( v ))about the distinguished diameter d . For I ∈ (cid:0) [ n ] k (cid:1) , reflection across d then givesa weight-preserving bijection between almost perfect matchings P with ∂ ( P ) = I and almost perfect matchings P ′ with ∂ ( P ′ ) = R ( I ). Definition 4.1.
A point X ∈ Gr ≥ ( n, n ) is a symmetric point if ∆ I ( P ) =∆ R ( I ) ( P ) for all I . The subvariety of symmetric points in Gr ≥ ( n, n ) is the sym-metric part of Gr ≥ ( n, n ) . Similarly, for Π a positroid cell, the symmetric part of Π is the intersection of Π with the symmetric part of Gr ≥ ( n, n ) . Clearly, the image of every symmetric weighting of a symmetric plabic graph isa symmetric point. We show the converse.
Theorem 4.2.
Let X ∈ Gr ≥ ( n, n ) be a symmetric point. Then X may be repre-sented by a symmetric plabic graph with symmetric weights.Proof. Since X is a symmetric point, the positroid of X satisfies condition (ii) fromTheorem 3.1. Hence X may be represented by some weighting ω of a symmetricplabic graph G . Choose a collection F of edges of G which has all of the followingproperties:(1) F is symmetric about the diameter d . That is, an edge ( u, v ) of G is in F if and only if ( r ( u ) , r ( v )) is also in F .(2) Every vertex of G is covered by F at least once.(3) F consists of a disjoint collection of trees, each of which has exactly onevertex on the boundary of G .It is not hard to show that such a collection of edges exists, since F is symmetric.Since F is a disjoint union of trees, each with exactly one vertex on the boundary,we may gauge fix all edges in F to 1. Let ν be the resulting weighting of G , and let ν ′ be the weighting of G obtained by swapping the weights of ( u, v ) and ( r ( u ) , r ( v ))for each edge ( u, v ) of G .By symmetry, ν ′ is another weighting of G with ∂ G ( ν ′ ) = X . It follows that ν and ν ′ are the same up to gauge transformations. Moreover, ν ′ assigns the weight1 to each edge of F . Since F satisfies conditions 2 and 3 above, a sequence of gaugetransformations which fixes all edges in F must be trivial. Hence ν = ν ′ and ν is asymmetric weighting. This completes the proof. (cid:3) The following is immediate.
Corollary 4.3.
Let Π be a positroid cell. Then the symmetric part of Π is nonemptyif and only if Π can be realized as the image of a symmetric plabic graph G , wherethe symmetric part of Π is precisely the image of all symmetric weightings of G under the boundary measurement map. A lollipop graph is a plabic graph which has a lollipop at each boundary vertex.A bridge graph is a plabic graph obtained by starting with a lollipop graph, andrepeatedly adding bridges adjacent to the boundary. See Figure 4.4 for an example.Given X ∈ Gr ≥ ( k, n ), we can apply Proposition 2.7 repeatedly to construct aweighted plabic graph whose image under the boundary measurement map is thepoint X [5]. We now give a symmetric version of this result.4 3 2 1 t t t t Figure 6.
A symmetric bridge graph with symmetric weights.
Theorem 4.4.
Let X ∈ Gr ≥ ( k, n ) be a symmetric point. Then we can iterativelyconstruct a weighted bridge graph corresponding to X which is symmetric and hassymmetric weights.Proof. Let Π be the positroid cell containing X , and let f be the decorated per-mutation of Π. We induct on the dimension of Π. If the dimension is 0, then X may be represented by a lollipop graph G [7]. In this case, X has a single non-zeroPl¨ucker coordinate I , corresponding to the positions of the white lollipops, whichnecessarily satisfies I = R ( I ). Thus G has a white lollipop at i if and only if G hasa black lollipop at i ′ , and G is symmetric.Suppose dim(Π) >
0. We note first that Π satisfies condition (ii) from Theorem3.1. Hence i is a white fixed point of f if and only if i ′ is a black fixed point of f .We may therefore reduce to the case where f has no fixed points. It follows that f has a bridge at ( i, i + 1) for some 1 ≤ i ≤ n − f has a bridge at ( n, n +1). Let ( I , . . . , I n ) denote the Grassmannnecklace of X . By Proposition 2.7, the quantity(20) c = ∆ I n +1 ( X ) / ∆ ( I n +1 ∪{ n } ) −{ n +1 } ( X )is positive and well-defined, and the point X ∗ = X · x n ( − c ) lies in Gr ≥ ( n, n ) . OMBINATORICS OF SYMMETRIC PLABIC GRAPHS 13
The action of x n ( − c ) fixes the Pl¨ucker coordinates of X, except for those ∆ I with n + 1 ∈ I and n I . Note that I ∈ (cid:0) [2 n ] n (cid:1) has this property if and only if R ( I )does. For each I with n + 1 ∈ I and n I , we have∆ R ( I ) ( X ∗ ) = ∆ R ( I ) ( X ) − c ∆ ( R ( I ) ∪{ n } ) −{ n +1 } ( X )(21) = ∆ I ( X ) − c ∆ R (( I ∪{ n } ) −{ n +1 } ) ( X )(22) = ∆ I ( X ) − c ∆ ( I ∪{ n } ) −{ n +1 } ( X )(23) = ∆ I ( X ∗ )(24)So X ∗ is a symmetric point. By induction, we may build a symmetric bridge graphrepresenting the point X ∗ , which has symmetric weights. Adding a bridge ( n, n +1)with edge weight c gives the desired weighted graph, and we are done in this case.Next, suppose f does not have a bridge at ( n, n +1). Then f has a bridge ( i, i +1)for some 1 ≤ i ≤ n − i = n . Hence by symmetry, f has a pair of commutingbridges at ( i, i + 1) and ( i ′ − , i ′ ) , respectively. Without loss of generality, assume1 ≤ i ≤ n −
1. By Theorem 2.1, we have I i ′ = R ( I i +1 ), which implies(25) ( I i ′ ∪ { i ′ − } ) − { i ′ } = R (( I i +1 ∪ { i } ) − { i + 1 } ) . Since X satisfies ∆ R ( I ) ( X ) = ∆ I ( X ) for all I , we have(26) c := ∆ I i +1 ( X ) / ∆ ( I i +1 ∪{ i } ) −{ i +1 } ( X ) = ∆ I i ′ ( X ) / ∆ ( I i ′ ∪{ i ′ − } ) −{ i ′ } ( X )where both quantities are positive and well-defined, as above.Once again, let X ∗ = X · x i ( − c ). Let ( I ∗ , . . . , I ∗ n ) denote the Grassmann neck-lace of X ∗ , and let(27) c ∗ = ∆ I ∗ i ′ ( X ∗ ) / ∆ ( I ∗ i ′ ∪{ i ′ − } ) −{ i ′ } ( X ∗ ) . We claim that c ∗ = c . First, note that I ∗ j = I j unless j = i + 1, so I ∗ i ′ = I i ′ . Ifeither i ∈ I i ′ or i + 1 I i ′ , then we have(28) ∆ I i ′ ( X ) = ∆ I i ′ ( X ∗ )(29) ∆ ( I i ′ ∪{ i ′ − } ) −{ i ′ } ( X ) = ∆ ( I i ′ ∪{ i ′ − } ) −{ i ′ } ( X ∗ ) , and so c ∗ = c , as desired.Suppose i I i ′ and i + 1 ∈ I i ′ . Let M be a matrix representative for X with columns v , . . . , v n . Suppose f ( i ′ −
1) = i ′ . Then v i ′ is a scalar multiple of v i ′ − , and in particular we have v i ′ = cv i ′ − . Let v ∗ , . . . , v ∗ n be the columns of M ∗ = M · x i ( − c ). Then v ∗ i ′ = v i ′ and v ∗ i ′ − = v i ′ − . It follows that we have c ∗ = c as desired.Next, suppose f ( i ′ − = i ′ , and consider the submatrix S of M with columnsindexed by I i ′ . Since i I i ′ , it follows that v i is in the span of the columns of S indexed by elements of [ i ′ , i − cyc . Hence adding a multiple of v i to the column v i +1 does not change the determinant of S , and ∆ I i ′ ( X ∗ ) = ∆ I i ′ ( X ). Thus theratios c ∗ and c have the same numerator.We must now show that c ∗ and c have the same denominator. Since f ( i ′ − = i ′ , we have i ′ ∈ I i ′ − ∩ I i ′ . It follows that(30) ( I i ′ ∪ { i ′ − } ) − { i ′ } = ( I i ′ − ∪ { ¯ f ( i ′ − } ) − { i ′ } . Let S ∗ be the submatrix of M with columns indexed by this set. There are twocases to consider, depending on whether ¯ f ( i ′ ) lies in the cyclic interval [ i ′ , i ] cyc . Throughout this proof, we write v a , . . . , v b to denote the columns of S indexed byelements of [ a, b ] cyc .For the first case, suppose ¯ f ( i ′ ) ∈ [ i ′ , i ] cyc . Since G has a bridge at ( i ′ − , i ′ ) and¯ f ( i ′ − = i ′ , this means ¯ f ( i ′ −
1) must be in the cyclic interval [ i ′ + 1 , i − cyc .Let y = ¯ f ( i ′ − v y as a linear combination of the columnsindexed by I i ′ − which form a basis of h v i ′ − , . . . , v y − i . Note that we must havea nonzero coefficient of v i ′ in this linear combination, since the columns of S ∗ arelinearly independent.Hence v i ′ lies in the the span of the columns of S ∗ indexed by elements of[ i ′ − , y ] cyc . In particular, v i ′ lies in the span of the columns of S ∗ indexed byelements of [ i ′ − , i − cyc , which therefore span h v i ′ − , v i ′ , . . . , v i − i . Thus v i liesin the span of the columns of S ∗ indexed by elements of [ i ′ − , y ] cyc and we aredone with this case.For the second case, suppose ¯ f ( i ′ ) does not lie in the cyclic interval [ i ′ , i ] cyc .Then v i ′ is linearly independent of the columns v i ′ +1 , . . . , v i , so ( I i ′ ∪ { i ′ − } ) − { i ′ } contains a basis for h v i ′ +1 , . . . , v i − i . Since v i I i ′ , the corresponding columns of S ∗ contain v i in their span, and this case is complete.Hence c ∗ = c , and so X ∗∗ = X · x i ( − c ) x i ′ − ( − c ) lies in a positroid cell of dimen-sion two less than Π. It is straightforward to check that X ∗∗ is a symmetric point.By induction, we may build a symmetric plabic graph for X ∗∗ with symmetric edgeweights. Adding two bridges of weight c at ( i ′ − , i ′ ) and ( i, i + 1) respectivelygives a symmetric plabic network for X , and the proof is complete. (cid:3) Note that the sequence of bridges added above depends only on the boundedaffine permutation of Π, and the fact that X is symmetric. Hence, the proof ofTheorem 4.4 yields a method for constructing a symmetric bridge graph G corre-sponding to Π, and explicitly realizing each point in the symmetric part of Π witha symmetric weighting of G . We have thus demonstrated a symmetric analog ofthe bridge-graph construction for Gr ≥ ( k, n ) found in [5]. Acknowledgements
We are grateful to Francesca Gandini for helpful suggestions.
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