aa r X i v : . [ m a t h . C O ] J a n A POSITIVE GRASSMANNIAN ANALOGUE OF THE PERMUTOHEDRON
LAUREN K. WILLIAMS
Abstract.
The classical permutohedron Perm n is the convex hull of the points ( w (1) , . . . , w ( n )) ∈ R n where w ranges over all permutations in the symmetric group S n . This polytope has many beautiful properties –for example it provides a way to visualize the weak Bruhat order: if we orient the permutohedron so thatthe longest permutation w is at the “top” and the identity e is at the “bottom,” then the one-skeleton ofPerm n is the Hasse diagram of the weak Bruhat order. Equivalently, the paths from e to w along the edgesof Perm n are in bijection with the reduced decompositions of w . Moreover, the two-dimensional faces ofthe permutohedron correspond to braid and commuting moves, which by the Tits Lemma, connect any tworeduced expressions of w .In this note we introduce some polytopes Br k,n (which we call bridge polytopes ) which provide a positiveGrassmannian analogue of the permutohedron. In this setting, BCFW bridge decompositions of reducedplabic graphs play the role of reduced decompositions. We define Br k,n and explain how paths along itsedges encode BCFW bridge decompositions of the longest element π k,n in the circular Bruhat order . Wealso show that two-dimensional faces of Br k,n correspond to certain local moves for plabic graphs, which bya result of Postnikov [Pos06], connect any two reduced plabic graphs associated to π k,n . All of these resultscan be generalized to the positive parts of Schubert cells. A useful tool in our proofs is the fact that ourpolytopes are isomorphic to certain Bruhat interval polytopes . Conversely, our results on bridge polytopesallow us to deduce some corollaries about the structure of Bruhat interval polytopes.
Contents
1. Introduction 12. Background 23. Bridge polytopes and their one-skeleta 74. Bruhat interval polytopes and the proof of Theorem 3.2 95. The two-dimensional faces of bridge polytopes 11References 121.
Introduction
The totally nonnegative part ( Gr k,n ) ≥ of the real Grassmannian Gr k,n is the locus where all Pl¨uckercoordinates are non-negative [Lus94, Rie99, Pos06]. Postnikov initiated the combinatorial study of ( Gr k,n ) ≥ :he showed that if one stratifies the space based on which Pl¨ucker coordinates are positive and which arezero, one gets a cell decomposition. The cells are in bijection with several families of combinatorial objects,including decorated permutations, and equivalence classes of reduced plabic graphs . Reduced plabic graphsare a certain family of planar bicolored graphs embedded in a disk. These graphs are very interesting: theycan be used to parameterize cells in ( Gr k,n ) ≥ [Pos06] and to understand the cluster algebra structure onthe Grassmannian [Sco06]; they also arise as soliton graphs associated to the KP hierarchy [KW11]. Mostrecently reduced plabic graphs and the totally non-negative Grassmannian ( Gr k,n ) ≥ have been studiedin connection with scattering amplitudes in N = 4 super Yang-Mills [AHBC + Gr k,n ) ≥ , which they called the BCFW-bridge construction .In many ways, reduced plabic graphs behave like reduced decompositions of the symmetric group: forexample, just as any two reduced decompositions of the same permutation can be related by braid andcommuting moves, any two reduced plabic graphs associated to the same decorated permutation can be
Date : January 6, 2015.The author was partially supported by an NSF CAREER award DMS-1049513. related via certain local moves [Pos06]. The goal of this note is to highlight another way in which reducedplabic graphs behave like reduced decompositions. We will introduce a polytope called the bridge polytope ,which is a positive Grassmannian analogue of the permutohedron in the following sense: just as the one-skeleton of the permutohedron encodes reduced decompositions of permutations, the one-skeleton of thebridge polytope encodes BCFW-bridge decompositions of reduced plabic graphs. Moreover, just as thetwo-dimensional faces of permutohedra encode braid and commuting moves for reduced decompositions, thetwo-dimensional faces of bridge polytopes encode local moves for reduced plabic graphs.
Acknowledgments:
The author is grateful to Nima Arkani-Hamed, Jacob Bourjaily, and Yan Zhangfor conversations concerning the BCFW bridge construction.2.
Background
Reduced decompositions, Bruhat order, and the permutohedron.
Recall that the symmetricgroup S n is the group of all permutations on n letters. If we let s i denote the simple transposition exchanging i and i + 1, then S n is a Coxeter group generated by the s i (for 1 ≤ i < n ), subject to the relations s i = 1,as well as the commuting relations s i s j = s j s i for | i − j | ≥ braid relations s i s i +1 s i = s i +1 s i s i +1 .Given w ∈ S n , there are many ways to write w as a product of simple transpositions. If w = s i . . . s i ℓ isa minimal-length such expression, we call s i . . . s i ℓ a reduced expression of w , and ℓ the length ℓ ( w ) of w .The well-known Tits Lemma asserts that any two reduced expressions can be related by a sequence of braidand commuting moves if and only if they are reduced expressions for the same permutation.There are two important partial orders on S n , both of which are graded, with rank function given by thelength of the permutation. The strong Bruhat order is the transitive closure of the cover relation u ≺ v ,where u ≺ v means that ( ij ) u = v for i < j and ℓ ( v ) = ℓ ( u )+1. Here ( ij ) is the transposition (not necessarilysimple) exchanging i and j . We use the symbol (cid:22) to indicate the strong Bruhat order. The strong Bruhatorder has many nice combinatorial properties: it is thin and shellable, and hence is a regular CW poset[Ede81, Pro82, BW82]. Moreover, this partial order is closely connected to the geometry of the completeflag variety Fl n : it encodes when one Schubert cell is contained in the closure of another. Somewhat lesswell-known is its connection to total positivity. The totally non-negative part U + ≥ of the unipotent part of GL n has a decomposition into cells, indexed by permutations, and the strong Bruhat order describes whenone cell is contained in the closure of another [Lus94].The other partial order associated to S n is the (left) weak Bruhat order . This is the transitive closure ofthe cover relation u ⋖ v , where u ⋖ v means that s i u = v and ℓ ( v ) = ℓ ( u )+ 1. We use the symbol ≤ to indicatethe weak Bruhat order. The weak Bruhat order is related to reduced decompositions in the following way:given any w ∈ S n , the maximal chains from the identity e to w in the weak Bruhat order are in bijection withreduced decompositions of w . Moreover, the weak order has the advantage of being conveniently visualizedin terms of a polytope called the permutohedron, as we now describe. Definition 2.1.
The permutohedron Perm n in R n is the convex hull of the points { ( z (1) , . . . , z ( n )) | z ∈ S n } . See Figure 1 for the permutohedron Perm . We remark that the permutohedron is also connected to thegeometry of the complete flag variety Fl n : it is the image of Fl n under the moment map.Given a permutation z ∈ S n , we will often refer to it using the notation ( z (1) , . . . , z ( n )), or ( z , . . . , z n ),where z i denotes z ( i ).The following proposition is well-known, see e.g. [BLVS + Proposition 2.2.
Let u and v be permutations in S n with ℓ ( u ) ≤ ℓ ( v ) . There is an edge between vertices u and v in the permutohedron Perm n if and only if v = s i u for some i and ℓ ( v ) = ℓ ( u ) ± , in which casewe label the corresponding edge of the permutohedron by s i . Moreover, the two-dimensional faces of Perm n are either squares (with edges labeled in an alternating fashion by s i and s j , where | i − j | ≥ ) or hexagons(with edges labeled in an alternating fashion by s i and s i +1 for some i ). See Figure 1 for the example of Perm . Proposition 2.2 implies that edges of the permutohedron corre-spond to cover relations in the weak Bruhat order, and the two-dimensional faces correspond to braid andcommuting relations for reduced words.Let w be the longest permutation in S n . This permutation is unique, and can be defined by w ( i ) = n + 1 − i . Proposition 2.2 can be equivalently restated as follows. POSITIVE GRASSMANNIAN ANALOGUE OF THE PERMUTOHEDRON 3 (12)(23)(34)(12) (23) (12) (34)(34) (34) (23) (12)(23)(34)(23)(12)(12)(34) (23) (34)(23) (12) (12) (23)(34)
Figure 1.
The permutohedron Perm . Edges are labeled by the values swapped when wego between the permutations labeling the two vertices. Proposition 2.3.
The shortest paths from e to w along the one-skeleton of the permutohedron Perm n arein bijection with the reduced decompositions of w . For example, we can see from Figure 1 that there is a path e ⋖ (2 , , , ⋖ (3 , , , ⋖ (4 , , , ⋖ (4 , , , ⋖ (4 , , , ⋖ (4 , , ,
1) = w in the permutohedron Perm . Reading off the edge labels of this path gives rise to the reduced decomposition s s s s s s of w .2.2. The positive Grassmannian, permutations, circular Bruhat order, and plabic graphs.
Inthis section we introduced the positive Grassmannian, and some combinatorial objects associated to it,including permutations and plabic graphs. As we will see, reduced plabic graphs are in many ways analogousto reduced decompositions of permutations.The real Grassmannian Gr k,n is the space of all k -dimensional subspaces of R n . An element of Gr k,n canbe viewed as a full-rank k × n matrix modulo left multiplication by nonsingular k × k matrices. In otherwords, two k × n matrices represent the same point in Gr k,n if and only if they can be obtained from eachother by row operations. Let (cid:0) [ n ] k (cid:1) be the set of all k -element subsets of [ n ] := { , . . . , n } . For I ∈ (cid:0) [ n ] k (cid:1) , let∆ I ( A ) be the Pl¨ucker coordinate , that is, the maximal minor of the k × n matrix A located in the columnset I . The map A (∆ I ( A )), where I ranges over (cid:0) [ n ] k (cid:1) , induces the Pl¨ucker embedding Gr k,n ֒ → RP ( nk ) − .The totally non-negative part of the Grassmannian (sometimes called the positive Grassmannian ) ( Gr k,n ) ≥ is the subset of Gr k,n such that all Pl¨ucker coordinates are non-negative.If we partition ( Gr k,n ) ≥ into strata based on which Pl¨ucker coordinates are positive and which are zero,we obtain a decomposition into positroid cells [Pos06]. Postnikov showed that the cells are in bijection with,and naturally labeled by, several families of combinatorial objects, including Grassmann necklaces , decoratedpermutations and equivalence classes of reduced plabic graphs . He also introduced the circular Bruhat order on decorated permutations, which describes when one cell is contained in another. Like the strong Bruhatorder, the circular Bruhat order is graded, thin, shellable, and hence is a regular CW poset [Wil07]. Definition 2.4. A Grassmann necklace is a sequence I = ( I , . . . , I n ) of subsets I r ⊂ [ n ] such that, for i ∈ [ n ] , if i ∈ I i then I i +1 = ( I i \ { i } ) ∪ { j } , for some j ∈ [ n ] ; and if i / ∈ I r then I i +1 = I i . (Here indices i are taken modulo n .) In particular, we have | I | = · · · = | I n | , which is equal to some k ∈ [ n ] . We then saythat I is a Grassmann necklace of type ( k, n ) . To construct the Grassmann necklace associated to a positroid cell, one uses the following construction.
Lemma 2.5.
Define the shifted linear order < i by i < i i +1 < i · · · < i n < i < i · · · < i i − . Given A ∈ Gr k,n ,let I ( A ) = ( I , . . . , I n ) be the sequence of subsets in [ n ] such that, for i ∈ [ n ] , I i is the lexicographicallyminimal subset of (cid:0) [ n ] k (cid:1) with respect to the shifted linear order such that ∆ I i ( A ) = 0 . Then I ( A ) is aGrassmann necklace of type ( k, n ) . LAUREN K. WILLIAMS
Definition 2.6. A decorated permutation π on n letters is a permutation on n letters in which fixed pointshave one of two colors, “clockwise” and “counterclockwise”. A position i of π is called a weak excedance if π ( i ) > i or π ( i ) = i and i is a counterclockwise fixed point. Definition 2.7.
Given a Grassmann necklace I , define a decorated permutation π = π ( I ) by requiring that(1) if I i +1 = ( I i \ { i } ) ∪ { j } , for j = i , then π ( j ) = i .(2) if I i +1 = I i and i ∈ I i then π ( i ) = i is a counterclockwise fixed point.(3) if I i +1 = I i and i / ∈ I i then π ( i ) = i is a clockwise fixed point.As before, indices are taken modulo n . Using the above constructions, one may show the following.
Theorem 2.8. [Pos06]
The cells of ( Gr k,n ) ≥ are in bijection with Grassmann necklaces of type ( k, n ) andwith decorated permutations on n letters with exactly k weak excedances. The totally non-negative Grassmannian ( Gr k,n ) ≥ has a unique top-dimensional cell, consisting of elementswhere all Pl¨ucker coordinates are strictly positive. This subset is called the totally positive Grassmannian ( Gr k,n ) > , and it is labeled by the decorated permutation π k,n := ( n − k + 1 , n − k + 2 , . . . , n, , , . . . , n − k ).More generally, there is a nice class of positroid cells called the TP or totally positive Schubert cells . Definition 2.9. A TP Schubert cell is the unique positroid cell of greatest dimension which lies in theintersection of a usual Schubert cell with ( Gr k,n ) ≥ . The TP Schubert cells in ( Gr k,n ) ≥ are in bijectionwith k -element subsets J of [ n ] = { , , . . . , n } . To calculate the decorated permutation associated to the TPSchubert cell indexed by J , write J = { j < · · · < j k } , and the complement of J as J c = { h < · · · < h n − k } .Then the corresponding decorated permutation π = π ( J ) is defined by π ( h i ) = i for all ≤ i ≤ n − k , and π ( j i ) = n − k + i for all ≤ i ≤ k . (Any fixed points that arise are considered to be weak excedances if andonly if they are in positions labeled by J .) A Grassmannian permutation is a permutation π = ( π (1) , . . . , π ( n )) which has at most one descent, i.e.at most one position i such that π ( i ) > π ( i + 1). Note that the permutations of the form π ( J ) defined aboveare precisely the inverses of the Grassmannian permutations, since π ( J ) − = ( h , h , . . . , h n − k , j , j , . . . , j k ).In the case that J = { , , . . . , k } , the corresponding TP Schubert cell is ( Gr k,n ) > and π ( J ) = π k,n .To describe the partial order on positroid cells, it is convenient to represent each decorated permutationby a related affine permutation, as was done in [KLS13]. Then, the circular Bruhat order for ( Gr k,n ) ≥ canbe viewed as the restriction of the affine Bruhat order to the corresponding affine permutations, togetherwith a new bottom element ˆ0 added to the poset. Definition 2.10.
Given a decorated permutation π on n letters, we construct its affinization ˜ π as follows.If π ( i ) < i , set ˜ π ( i ) := π ( i ) + n . If π ( i ) = i and i is a clockwise fixed point, set ˜ π ( i ) := i . If π ( i ) = i and i isa counterclockwise fixed point, set ˜ π ( i ) := i + n . Finally, if π ( i ) > i , set ˜ π ( i ) := π ( i ) . Clearly the affine permutation constructed above is a map from { , , . . . , n } to { , , . . . , n } such that i ≤ ˜ π ( i ) ≤ i + n . And modulo n , it agreeds with the underlying permutation π .We now discuss plabic graphs, which e.g. are useful for constructing parameterizations of positroid cells[Pos06], and for understanding the cluster structure on the Grassmannian [Sco06]. Definition 2.11. A planar bicolored graph (or plabic graph ) is an undirected graph G drawn inside a disk(and considered modulo homotopy) plus n boundary vertices on the boundary of the disk, labeled , . . . , n inclockwise order. The remaining internal vertices are strictly inside the disk and are colored in black andwhite. We require that each boundary vertex is incident to a single edge. We will always assume that a plabic graph is leafless , i.e. that it has no non-boundary leaves, and thatit has no isolated components. The following map gives a connection between plabic graphs and decoratedpermutations.
Definition 2.12.
Given a plabic graph G , the trip T i is the directed path which starts at the boundary vertex i , and follows the “rules of the road”: it turns right at a black vertex and left at a white vertex. Note that T i will also end at a boundary vertex. The decorated trip permutation π G is the permutation such that π G ( i ) = j whenever T i ends at j . Moreover, if there is a white (respectively, black) boundary leaf at boundaryvertex i , then π G ( i ) = i is a counterclockwise (respectively, clockwise) fixed point. POSITIVE GRASSMANNIAN ANALOGUE OF THE PERMUTOHEDRON 5
Figure 2.
A plabic graphWe now define some local transformations of plabic graphs, which are analogous in some ways to braidand commuting moves for reduced expressions in the symmetric group.(M1) SQUARE MOVE. If a plabic graph has a square formed by four trivalent vertices whose colorsalternate, then we can switch the colors of these four vertices.
Figure 3.
Square move(M2) UNICOLORED EDGE CONTRACTION/UNCONTRACTION. If a plabic graph contains an edgewith two vertices of the same color, then we can contract this edge into a single vertex with the same color.We can also uncontract a vertex into an edge with vertices of the same color.
Figure 4.
Unicolored edge contraction(M3) MIDDLE VERTEX INSERTION/REMOVAL. If a plabic graph contains a vertex of degree 2, thenwe can remove this vertex and glue the incident edges together; on the other hand, we can always insert avertex (of any color) in the middle of any edge.
Figure 5.
Middle vertex insertion/ removal(R1) PARALLEL EDGE REDUCTION. If a network contains two trivalent vertices of different colorsconnected by a pair of parallel edges, then we can remove these vertices and edges, and glue the remainingpair of edges together.
Definition 2.13. [Pos06]
Two plabic graphs are called move-equivalent if they can be obtained from eachother by moves (M1)-(M3). The move-equivalence class of a given plabic graph G is the set of all plabicgraphs which are move-equivalent to G . A leafless plabic graph without isolated components is called reduced if there is no graph in its move-equivalence class to which we can apply (R1). The following result is analogous to the Tits Lemma for reduced decompositions of permutations.
Theorem 2.14. [Pos06, Theorem 13.4]
Two reduced plabic graphs are move-equivalent if and only if theyhave the same decorated trip permutation.
A priori, it is not so easy to detect whether a given plabic graph is reduced. One characterization wasgiven in [Pos06, Theorem 13.2]. Another very simple characterization was given in [KW11].
Definition 2.15.
Given a plabic graph G with n boundary vertices, start at each boundary vertex i and labelevery edge along trip T i with i . After doing this for each boundary vertex, each edge will be labeled by up totwo numbers (between and n ). If an edge is labeled by two numbers i < j , write [ i, j ] on that edge.We say that a plabic graph has the resonance property , if after labeling edges as above, the set E of edgesincident to a given vertex has the following property: LAUREN K. WILLIAMS
Figure 6.
Parallel edge reduction • there exist numbers i < i < · · · < i m such that when we read the labels of E , we see the labels [ i , i ] , [ i , i ] , . . . , [ i m − , i m ] , [ i , i m ] appear in clockwise order. This property and the following characterization of reduced plabic graphs were given in [KW11].
Theorem 2.16. [KW11, Theorem 10.5]
A plabic graph is reduced if and only if it has the resonance property.
BCFW bridge decompositions.
Given a reduced plabic graph, it is easy to construct its corre-sponding trip permutation, as explained in Definition 2.12. But how can we go backwards, i.e. given thepermutation, how can we construct a corresponding reduced plabic graph? One procedure to do this wasgiven in [Pos06, Section 20]. Another elegant solution – the
BCFW-bridge construction – was given in[AHBC +
12, Section 3.2].To explain the BCFW-bridge construction, we use the representation of each decorated permutation π as an affine permutation ˜ π , see Definition 2.10. We say that position i of ˜ π is a fixed point if ˜ π ( i ) = i or˜ π ( i ) = i + n . And we say that ˜ π is a decoration of the identity if it has a fixed point in each position. Definition 2.17 ( The BCFW-bridge construction ) . Given a decorated permutation π on n letters, whichis not simply a decoration of the identity, we start by choosing a pair of adjacent positions i < j such that ˜ π ( i ) < ˜ π ( j ) , and i and j are not fixed points. Here adjacent means that either j = i + 1 , or i < j and everyposition h such that i < h < j is a fixed point. We record the transposition τ = ( ij ) and swap the entriesin positions i and j in ˜ π . Any entries in the resulting permutation which are fixed points are designated as frozen , and henceforth ignored. We continue this process on the resulting affine permutation, until the endresult is a decoration of the identity. Finally we use the sequence of transpositions τ as a recipe for adding“bridges,” thereby constructing a plabic graph. See Table 1 and Figure 7 for an example of a bridge decomposition of the permutation π = (4 , , , , , π = (4 , , , , , bridge is the subgraph shown at the leftof Figure 7, and a bridge decomposition is a graph built by attaching successive bridges. The sequence oftranspositions τ gives a recipe for where to place successive bridges.1 2 3 4 5 6 τ ↓ ↓ ↓ ↓ ↓ ↓ (34) 4 6 5 7 8 9(23) 4 6 7 5 8 9(12) 4 7 6 5 8 9(56) 7 4 6 5 8 9(45) 7 4 6 5 9 8(34) 7 4 6 9 5 8(46) 7 4 9 6 5 8(24) 7 4 9 8 5 67 8 9 4 5 6 Table 1.
A BCFW-bridge decomposition of ˜ π = (4 , , , , , Proposition 2.18. [AHBC +
12, Section 3.2]
Given a decorated permutation π , the BCFW-bridge construc-tion constructs a reduced plabic graph whose trip permutation is π . POSITIVE GRASSMANNIAN ANALOGUE OF THE PERMUTOHEDRON 7
Figure 7.
A single bridge, and the plabic graph G associated to the bridge decom-position from Table 1. Note that the trip permutation of G equals the product(24)(46)(34)(45)(56)(12)(23)(34) of transpositions τ in the bridge decomposition, whichequals the corresponding decorated permutation π = (4 , , , , , Bridge polytopes and their one-skeleta
The goal of this section is to introduce some polytopes that we will call bridge polytopes , because their one-skeleta encode BCFW-bridge decompositions of reduced plabic graphs. This statement is analogous to thefact that the one-skeleton of the permutohedron encodes reduced decompositions of w . More specifically, foreach k -element subset J of [ n ], we will introduce a bridge polytope Br J which encodes bridge decompositionsof reduced plabic graphs for the permutation π ( J ) labeling the corresponding TP Schubert cell. In thecase that J = { , , . . . , k } , we will also denote Br J by Br k,n – this will be the polytope encoding bridgedecompositions of π k,n , the decorated permutation labeling the totally positive Grassmannian. (23) (12) (34)(12) (34)(23) (23)(34)(13)(23)(12)(34)(23)(12) (24) (13)(12)(34)(24) (23)(14) (23) (23) (12) (34)(12) (34)(23) (23)(34)(13)(23)(12)(34)(23)(12) (24) (13)(12)(34)(24) (23)(14) (23) Figure 8.
Two labelings of a bridge polytope. At the left, vertices are labeled by ordinarypermutations, and edges are labeled by the pair of values which are swapped. At the right,vertices are labeled by affine permutations, and edges are labeled by the pair of positions which are swapped. The map from vertex-labels at the left to vertex-labels at the right is z g z − . In both cases, the paths along the one-skeleton from top to bottom encode bridgedecompositions of the permutation π , = (3 , , , π = (3 , , , { ( z , z , z , z ) | z ≥ , z ≥ , z ≤ , z ≤ } . The edge between two permutations is labeled by the values of the swapped entries. The figure at the rightshows the same polytope, but now vertices are labeled by affine permutations. A vertex which was labeledby z at the left is labeled at the right by the affinization g z − of z − . Note that we took the inverse in order LAUREN K. WILLIAMS to get a vertex-labeling of the polytope such that edges correspond to the positions of the swapped entries,agreeing with the sequence of transpositions defined in Definition 2.17.The main result of this section will imply that the minimal chains in the one-skeleton of the bridge polytopefrom “top” to “bottom” (from (3 , , ,
2) to (1 , , ,
4) using the vertex-labeling at the left, and from (3 , , , , , ,
4) using the vertex-labeling at the right) are in bijection with the bridge decompositions of thereduced plabic graphs with trip permutation π , = (3 , , , Figure 9.
The plabic graph coming from the sequence of bridges (12), (23), (12), (24).Note that its trip permutation is (3 , , ,
2) = (24)(12)(23)(12).We now turn to the general construction.
Definition 3.1.
Let J ⊂ { , , . . . , n } and set (1) S J = { π ∈ S n | π ( j ) ≥ j for j ∈ J and π ( j ) ≤ j for j / ∈ J. } In other words, any permutation in S J is required to have a weak excedance in position j for each j ∈ J ,and a weak non-excedance in position j for each j / ∈ J .We define a polytope Br J by (2) Br J = Conv { ( π (1) , π (2) , . . . , π ( n )) | π ∈ S J } ⊂ R n . In the special case that J = { , , . . . , k } , we writeBr k,n = Br J = { π ∈ S n | π ( j ) ≥ j for 1 ≤ j ≤ k and π ( j ) ≤ j for k + 1 ≤ j ≤ n. } Recall the definition of π ( J ) from Definition 2.9, the decorated permutation labeling the correspondingTP Schubert cell. Let e ( J ) be the decoration of the identity which has counterclockwise fixed points preciselyin positions J .The following is our main result. We describe it using the vertex-labeling of the bridge polytope withordinary permutations, as in Definition 3.1, but it can be easily translated using the vertex-labeling by affinepermutations. Theorem 3.2.
Let J be a k -element subset of [ n ] . The shortest paths from π ( J ) to the identity permutation e along the one-skeleton of the bridge polytope Br J are in bijection with the BCFW-bridge decompositions of thepermutation π ( J ) − , where a sequence of edge-labels in a path is interpreted as a sequence of transpositions τ in the bridge decomposition. Equivalently, there is an edge between two vertices π and ˆ π of the bridge polytope Br J if and only if there exists some pair i < ℓ such that ( iℓ ) π = ˆ π , and π ( j ) = ˆ π ( j ) = j for i < j < ℓ . Inother words, π and ˆ π differ by swapping the values i and ℓ , and i + 1 , i + 2 , . . . , ℓ − are fixed points of both π and ˆ π . We will prove Theorem 3.2 in Section 4.The following corollary is immediate from Theorem 3.2.
Corollary 3.3.
The shortest paths in the bridge polytope Br k,n from π k,n down to the identity e are inbijection with the BCFW-bridge decompositions of the permutation π n − k,n . In other words, we can read offall BCFW-bridge decompositions of π n − k,n by recording the edge-labels on all shortest paths from π k,n to e in the one-skeleton of Br k,n . POSITIVE GRASSMANNIAN ANALOGUE OF THE PERMUTOHEDRON 9 Bruhat interval polytopes and the proof of Theorem 3.2
Bruhat interval polytopes are a class of polytopes which were recently studied by Kodama and the secondauthor in [KW13], in connection with the full Kostant-Toda lattice on the flag variety. The combinatorialproperties of Bruhat interval polytopes were further investigated by Tsukerman and the second author in[TW14]. We will show that bridge polytopes are isomorphic to certain Bruhat interval polytopes, and usea result from [KW13] as a tool for proving Theorem 3.2. We will also deduce a characterization of theone-skeleta of a large class of Bruhat interval polytopes.
Definition 4.1.
Let u, v ∈ S n such that u ≤ v in (strong) Bruhat order. The Bruhat interval polytope Q u,v is defined as the convex hull Q u,v = Conv { ( z (1) , . . . , z ( n )) | for z ∈ S n such that u ≤ z ≤ v } . Note that when u is the identity and v = w , the longest permutation in S n , the Bruhat interval polytope Q u,v equals the permutohedron Perm n . Lemma 4.2.
The bridge polytope Br J is isomorphic to the Bruhat interval polytope Q e,π ( J ) − . More specifi-cally, recall that for J a k -element subset of [ n ] , we write J = { j < · · · < j k } , and J c = { h < · · · < h n − k } .Consider the map ψ : R n → R n defined by ψ ( x , . . . , x n ) = ( x h , x h , . . . , x h n − k , x j , x j , . . . , x j k ) . Then Ψ is an isomorphism from Br J to Q e,π ( J ) − .Proof . Since ψ simply permutes the coordinates of R n , it obviously acts as an isomorphism on anypolytope. Moreover, ψ takes the vertex ( z (1) , . . . , z ( n )) labeled by the permutation z in Br k,n to the vertexlabeled by the permutation zπ ( J ) − of Q e,π ( J ) − . In particular, it takes the vertices π ( J ) and e of Br k,n to thevertices e and π ( J ) − of Q e,π ( J ) − , respectively. It is not hard to show that ψ maps the set of permutations S J to the set of permutations in the interval [ e, π ( J ) − ].Once we have established Theorem 3.2, Lemma 4.2 immediately implies the following description of theone-skeleton of any Bruhat interval polytopes Q e,w , when w is a Grassmannian permutation. Corollary 4.3.
Let w be a Grassmannian permutation in S n , i.e. a permutation with at most one descent.Then there is an edge between vertices u and v in Q e,w if and only if u and v satisfy the following properties: • v = ( iℓ ) u • in the permutations u and v , each of the values i +1 , i +2 , . . . , ℓ − are in precisely the same positionsthat they are in w . We now prove Theorem 3.2, by establishing Lemma 4.4, Proposition 4.6, and Theorem 4.7, below.
Lemma 4.4.
If there is an edge in Br J between π and ˆ π , then there exists some transposition ( iℓ ) such that ( iℓ ) π = ˆ π .Proof . By [KW13, Theorem A.10], every edge of a Bruhat interval polytope connects two vertices u and v , where v = ( iℓ ) for some transposition ( iℓ ). Lemma 4.4 now follows from Lemma 4.2. Remark 4.5.
It was shown more generally in [TW14] that every face of a Bruhat interval polytope is aBruhat interval polytope.
Proposition 4.6.
Suppose there’s an edge in Br J between π and ˆ π where ( iℓ ) π = ˆ π for some i < ℓ , i.e. π and ˆ π differ by swapping the values i and ℓ . Then i + 1 , i + 2 , . . . , ℓ − must be fixed points of π and ˆ π .Proof . We use a proof by contradiction. Suppose that not all of i + 1 , i + 2 , . . . , ℓ − j be the smallest element of { i + 1 , i + 2 , . . . , ℓ − } which is not a fixed point. Let a, b , and c be thepositions of i, ℓ , and j in π .Without loss of generality, π c = j > c . So c ∈ J and is a position of a weak excedance in any permutation in S J . Since we have an edge in the polytope between π and ˆ π , there exists a λ ∈ R n such that λ · x := P h λ h x h (applied to x ∈ S J ) is maximized precisely on π and ˆ π . In particular, we must have λ a = λ b .If λ a = λ b < λ c , then define ˜ π so that it agrees with π except in positions b and c , where we have ˜ π b = j and ˜ π c = ℓ . Since π b = ℓ and ˆ π b = i and i < j < ℓ , permutations in S J are allowed to have a b th coordinateof j . Since j > c and j < ℓ , we have ℓ > c . So permutations in S J are allowed to have a c th coordinate of ℓ .Therefore ˜ π ∈ S J . But λ · ˜ π > λ · π , which is a contradiction. If λ a = λ b > λ c , then define ˜ π so that it agrees with π except in positions a and c , where we have ˜ π a = j and ˜ π c = i . Since π a = i and ˆ π a = ℓ and i < j < ℓ , permutations in S J are allowed to have an a th coordinateof j . Since π c = j > c , and i + 1 , i + 2 , . . . , j − π , c / ∈ { i + 1 , i + 2 , . . . , j − } . So c ≤ i ,and permutations in S J are allowed to have a c th coordinate of i (recall that c ∈ J and hence c must be aweak excedance in any element of S J ). Therefore ˜ π ∈ S J . But λ · ˜ π > λ · π , which is a contradiction. Theorem 4.7.
Consider the bridge polytope Br J and two permutations π and ˆ π in S J ⊂ S n such that ( iℓ ) π = ˆ π and i + 1 , i + 2 , . . . , ℓ − are fixed points of π and ˆ π . Choose a vector λ = ( λ , . . . , λ n ) ∈ R n whosecoordinates are some permutation of the numbers , n , n , . . . , n n − , and such that λ π − (1) < λ π − (2) < · · · < λ π − ( i − < ✷ < λ π − ( i ) = λ π − ( ℓ ) < ✷ < λ π − ( ℓ +1) < · · · < λ π − ( n − < λ π − ( n ) . Here the ✷ at the left represents the coordinates λ π − ( j ) for j ∈ J and i < j < ℓ , sorted from left to rightin increasing order of j , and the ✷ at the right represents the coordinates λ π − ( j ) for j / ∈ J and i < j < ℓ ,sorted from left to right in increasing order of j .Then when we calculate the dot product λ · x for each x ∈ S J , λ · x is maximized precisely on π and ˆ π . Inparticular, there is an edge in Br J between π and ˆ π .Proof . To prove Theorem 4.7, we will first use the fact that the coordinates of λ are 1 , n , . . . , n n − toshow that to solve for the x ∈ S J on which λ · x is maximized, it suffices to use the greedy algorithm. Inother words, we will show that we can construct such x by maximizing individual coordinates x i one by one,starting with the x i where λ i is maximal, and continuing in decreasing order of the values of λ j . Next wewill show that when we run the greedy algorithm, we will get precisely the outcomes π and ˆ π .Let us show now that in order to maximize the dot product λ · x for x ∈ S J , the greedy algorithmworks. Recall that the coordinates of λ are the values 1 , n , . . . , n n − (in some order). Let h ∈ [ n ] besuch that λ h = n n − . We first claim that we need to maximize x h subject to the condition x ∈ S J . Let w h be that maximum possible value. Let y ∈ S J be some other permutation with y h ≤ w h −
1. Then λ h x h − λ h y h ≥ n n − . And since the maximum difference of x j and y j is n −
1, we have X ≤ j ≤ n,j = h λ j y j − X ≤ j ≤ n,j = h λ j x j ≤ ( n − n + · · · + n n − ) . Now note that since n − < n and 1+ n + · · · + n n − has n − n − n + · · · + n n − )
Looking at the ordering of the coordinates of λ as described in Theorem 4.7, we first need tomaximize x π − ( n ) , then x π − ( n − , . . . , then x π − ( ℓ +1) . Therefore we place n in position π − ( n ), n − π − ( n − . . . , ℓ + 1 in position π − ( ℓ + 1). (Note that there exist permutations x ∈ S J with thesecoordinates in these positions – for example, π and ˆ π are two examples of such permutations.) Step 2.
Next we need to maximize the values that we put in positions π − ( j ), for i < j < ℓ and π − ( j ) = j / ∈ J . But these are positions of weak non-excedances in S J , so the best we can do is to put fixedpoints there. Note that π and ˆ π also have fixed points in these positions, so the x we are building so faragrees with both π and ˆ π . Step 3.
Now we want to maximize the values that we can put in positions π − ( i ) and π − ( ℓ ). Thegreatest unused value so far is ℓ so we can put that in either position π − ( ℓ ) (agreeing with π ) or π − ( i )(agreeing with ˆ π ).Now let j be the maximum value that we have not yet placed in any position of the x we are building.If i < j < ℓ , then necessarily j ∈ J , i.e. j is a position where all permutations of S J must have a weakexcedance. So we must make j a fixed point in x (the only other option is to place a value smaller than j inposition j ). Similarly for any other i < j < ℓ , we must set x j = j .Having done this, the maximum value that we have not yet placed in x is i . So we now put i in position π − ( i ) (or position π − ( ℓ )). Note that the x we are building agrees with either π or ˆ π so far. POSITIVE GRASSMANNIAN ANALOGUE OF THE PERMUTOHEDRON 11
Step 4.
Finally we want to maximize the values that we place in positions π − ( i − , . . . , π − (1). Theremaining unused values are i − , . . . ,
1. So for i − ≥ j ≥
1, we place j in position π − ( j ).This greedy algorithm has constructed precisely two permutations, π and ˆ π , and clearly λ · π = λ · ˆ π .Therefore if we consider the values λ · x for x ∈ S J , λ · x is maximized precisely on π and ˆ π . It follows thatthere is an edge in Br J between π and ˆ π .5. The two-dimensional faces of bridge polytopes
In this section we describe the two-dimensional faces of bridge polytopes, and explain how they are relatedto the moves for reduced plabic graphs.
Theorem 5.1.
A two-dimensional face of a bridge polytope is either a square, a trapezoid, or a regularhexagon, with labels as in Figure 10. (jk) (jk) (ij)(jk)(ij) (ij)(kl)(kl) (ij) (jk)(ik) (ij) (jk)(ij)
Figure 10.
The possible two-dimensional faces of a bridge polytope. Here we have i 2. A simple calculation with dot productsshow that the possible angles among the edges are π , π , π : more precisely, two vectors e i − e j and e k − e l (where i, j, k, l are distinct) have angle π ; two vectors e i − e j and e i − e k have angle π ; and two vectors e i − e j and − e i + e k have angle π .The only possibilities for such a polygon are: a square, a trapezoid (with angles π , π , π , π ), a regularhexagon (all angles are π ), an equilateral triangle, or a parallelogram. In the first three cases, the labelson the edges of the polygons must be as in Figure 10. In the case of the square, it follows from the rules ofbridge decompositions that the intervals [ i, j ] and [ k, l ] must be disjoint, and hence without loss of generality i < j < k < l . (E.g. k cannot lie in between i and j , because in order to perform the swap i and j , allelements between i and j must be fixed points. And we are not allowed to swap fixed points.) A similarargument explains the ordering on i, j, k for the trapezoid and the hexagon.We now argue that it is impossible for a two-dimensional face to be a triangle or a parallelogram. Notethat if one traverses any cycle in the one-skeleton of a bridge polytope, the product of the correspondingedge labels must be 1. It follows that there cannot be a two-face with an odd number of sides, because theproduct of an odd number of transpositions is an odd permutation, and hence is never the identity. Thereforean equilateral triangle is impossible. Finally consider the case of a face which is a parallelogram. Its edgelabels must have alternating labels ( ij ), ( ik ), ( ij ), ( ik ). But the product ( ij )( ik )( ij )( ik ) is not equal to theidentity permutation. Remark 5.2. The same proof shows that the face of any Bruhat interval polytope is a square, a trapezoid,or a hexagon, as shown in Figure 10. Theorem 5.3. The three kinds of two-dimensional faces of bridge polytopes correspond to simple applicationsof the local moves for plabic graphs. More specifically, consider a shortest path p from π k,n (or more generally π ( J ) ) to e in the one-skeleton of the bridge polytope Br k,n (or more generally Br J ). Choose a two-dimensionalface F such that p traverses half the sides of F , and modify p along F , obtaining a new path p ′ which goesaround the other sides of F . Then the reduced plabic graphs corresponding to p and p ′ are related by homotopyand by the local moves for plabic graphs as in Figure 11. Proof . The proof of Theorem 5.3 is illustrated in Figure 11. The two reduced plabic graphs related by asquare face in a bridge polytope are related by homotopy. The two plabic graphs related by a trapezoidalface are related by moves of type (M3). And the two plabic graphs related by a hexagonal face are relatedby a combination of moves (M1) and (M3). Note that in the latter case, the dashed edge in Figure 11 mustbe present, or else the plabic graph associated to our bridge decomposition would not be reduced. i j k i j k i j k (M3) (M1) i j k (M3) i j k i j k i j k (M3) (M3) i j k l i j k l Figure 11. A graphical proof of Theorem 5.3. References [AHBC + 12] Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, andJaroslav Trnka. Scattering amplitudes and the positive Grassmannian, 2012. Preprint, arXiv:1212.5605 .[BLVS + 99] Anders Bj¨orner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and G¨unter M. Ziegler. Oriented matroids ,volume 46 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, secondedition, 1999.[BW82] Anders Bj¨orner and Michelle Wachs. Bruhat order of Coxeter groups and shellability. Adv. in Math. , 43(1):87–100,1982.[Ede81] Paul H. Edelman. The Bruhat order of the symmetric group is lexicographically shellable. Proc. Amer. 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