From generalized permutahedra to Grothendieck polynomials via flow polytopes
FFROM GENERALIZED PERMUTAHEDRA TO GROTHENDIECKPOLYNOMIALS VIA FLOW POLYTOPES
KAROLA M´ESZ ´AROS AND AVERY ST. DIZIER
Abstract.
We prove that for permutations 1 π (cid:48) where π (cid:48) is dominant, the Grothendieckpolynomial G π (cid:48) ( x ) is a weighted integer-point transform of its Newton polytope with allweights nonzero. We also show that the Newton polytopes of the homogeneous componentsof G π (cid:48) ( x ) are generalized permutahedra. Moreover, the Schubert polynomial S π (cid:48) ( x ) fordominant π (cid:48) equals the integer-point transform of a generalized permutahedron. Theseresults imply recent conjectures of Monical, Tokcan and Yong regarding the supports ofSchubert and Grothendieck polynomials for the special case of permutations 1 π (cid:48) , where π (cid:48) is dominant. We connect Grothendieck polynomials and generalized permutahedra via afamily of dissections of flow polytopes obtained from the subdivision algebra. We naturallylabel each simplex in a dissection by a sequence, called a left-degree sequence, and show thatthe left-degree sequences arising from simplices of a fixed dimension in our dissections of flowpolytopes are exactly the integer points of generalized permutahedra. This connection of left-degree sequences and generalized permutahedra together with the connection of left-degreesequences and Grothendieck polynomials established in earlier work of Escobar and the firstauthor reveal the beautiful relation between generalized permutahedra and Grothendieckpolynomials. Contents
1. Introduction 12. Background information 33. Triangular arrays and left-degree sequences 64. Newton polytopes of left-degree polynomials 155. Newton polytopes of Schubert and Grothendieck polynomials 256. Left-degree sequences are invariants of the graph 28Acknowledgements 32References 331.
Introduction
The flow polytope F G associated to a directed acyclic graph G is the set of all flows f : E ( G ) → R ≥ of size one. Flow polytopes are fundamental objects in combinatorialoptimization [16], and in the past decade they were also uncovered in representation theory[1, 11], the study of the space of diagonal harmonics [7, 12], and the study of Schubert andGrothendieck polynomials [4, 5]. In this paper we establish the deep connection between flowpolytopes and generalized permutahedra and use this connection to prove that for certain M´esz´aros is partially supported by a National Science Foundation Grant (DMS 1501059). a r X i v : . [ m a t h . C O ] J un KAROLA M´ESZ ´AROS AND AVERY ST. DIZIER permutations, the supports of Schubert polynomials as well as the homogeneous componentsof Grothendieck polynomials are integer points of generalized permutahedra.A natural way to analyze a convex polytope is to dissect it into simplices. The relationsof the subdivision algebra, developed in a series of papers [8, 9, 10], encode dissections of afamily of flow (and root) polytopes (see Section 2 for details). The key to connecting flowpolytopes and generalized permutahedra lies in the study of the dissections of flow polytopesobtained via the subdivision algebra: (1) How are the dissections of a flow polytope obtained via the subdivision algebra related toeach other?
In Theorem A we give a full characterization of the left-degree sequences (Definition 2.5) ofany dissection of a flow polytope obtained via the subdivision algebra, and we show thatwhile the dissections themselves are different their left-degree sequences are the same. Thatthe left-degree sequences do not depend on the dissection was previously proved in specialcases by Escobar and the first author [4], and independently from the authors, Grinberg[6] recently showed it for arbitrary graphs in his study of the subdivision algebra. Ourcharacterization of the left-degree sequences of any reduction tree of any graph serves as thecornerstone of the rest of the work in this paper.Since by Theorem A the left-degree sequences are an invariant of the underlying flowpolytope and do not depend on the choice of dissection, it is natural to ask: (2) What is the significance of the left-degree sequences associated to a flow polytope F G ? The answer to this question is both inspiring and revealing. In Theorem B, we prove thatleft-degree sequences of F G with fixed sums are exactly lattice points of generalized permu-tahedra, which were introduced by Postnikov in his beautiful paper [14]. Moreover, we showthat the left-degree polynomial L G ( t ) (Definition 4.2) has polytopal support (Definition 4.1).In earlier work of Escobar and the first author [4], it was shown that some left-degreepolynomials are Grothendieck polynomials. This brings us to: (3) What does the answer to (2) imply about Schubert and Grothendieck polynomials? In Theorem C, we conclude that for all permutations 1 π (cid:48) where π (cid:48) is dominant, the Grothendieckpolynomial G π (cid:48) ( x ) is a weighted integer-point transform of its Newton polytope, with allweights nonzero. Moreover, the homogeneous components of G π (cid:48) ( x ) are weighted integer-point transforms of their Newton polytopes, which are all generalized permutahedra. Forthe homogeneous component corresponding to the Schubert polynomial S π (cid:48) ( x ), somethingmore is true: it equals the integer-point transform of its Newton polytope, which is a gen-eralized permutahedron. Theorem C implies in particular that the recent conjectures ofMonical, Tokcan, and Yong [13, Conjecture 5.1 & 5.5] are true for permutations 1 π (cid:48) , where π (cid:48) is a dominant permutation.The outline of this paper is as follows. Sections 2 covers the necessary background. Sec-tions 3, 4 and 5 answer questions (1), (2) and (3) from above respectively. For ease of readingSections 3, 4 and 5 are phrased for simple graphs. In Section 6 we show that our techniquesextend to generalize all results to all graphs. ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 3 Background information
In this section, we summarize definitions, notations, and results that we use later. Through-out this paper, by graph we mean a loopless directed graph where multiple edges are allowed,as described below. Although we sometimes refer to edges by their endpoints, we keep inmind that E ( G ) is a multiset. We also adopt the convention of viewing each element of amultiset as being distinct, so that we may speak of subsets, though we will use the wordsubmultiset interchangeably to highlight the multiplicity. Due to this convention, all unionsin this paper are assumed to be disjoint multiset unions.For any integers m and n , we will frequently use the notation [ m, n ] to refer to the set { m, m + 1 , . . . , n } and [ n ] to refer to the set [1 , n ]. Flow polytopes.
Let G be a loopless graph on vertex set [0 , n ] with edges directed fromsmaller to larger vertices. For each edge e , let in( e ) denote the smaller (initial) vertexof e and fin( e ) the larger (final) vertex of e . Imagine fluid moving along the edges of G .At vertex i let there be an external inflow of fluid a i (outflow of − a i if a i < a = ( a , . . . , a n ) the netflow vector . Formally, a flow on G with netflow vector a is anassignment f : E ( G ) → R ≥ of nonnegative values to each edge such that fluid is conservedat each vertex. That is, for each vertex i (cid:88) in( e )= i f ( e ) − (cid:88) fin( e )= i f ( e ) = a i . The flow polytope F G ( a ) is the collection of all flows on G with netflow vector a .Alternatively, let M G denote the incidence matrix of G , that is let the columns of M G bethe vectors e i − e j for ( i, j ) ∈ E ( G ), i < j , where e i is the ( i + 1)-th standard basis vector in R n +1 . Then, F G ( a ) = { f ∈ R n ≥ : M G f = a } . From this perspective, note that the number of integer points in F G ( a ) is exactly the numberof ways to write a as a nonnegative integral combination of the vectors e i − e j for edges( i, j ) in G , i < j , that is the Kostant partition function K G ( a ). For brevity, we write F G := F G (1 , , . . . , , − F G as the flow polytope of G , since in this paperour primary focus is on studying these particular flow polytopes.The following milestone result giving the volume of flow polytopes was shown by Postnikovand Stanley in unpublished work: Theorem 2.1 (Postnikov-Stanley) . Given a loopless connected graph G on vertex set { , , . . . , n } , let d i = indeg G ( i ) − for each vertex i , where indeg G ( i ) is the number of edgesincoming to vertex i in G . Then, the normalized volume of the flow polytope of G is Vol F G = K G (cid:32) , d , . . . , d n , − n (cid:88) i =1 d i (cid:33) . Baldoni and Vergne [1] generalized this result for flow polytopes with arbitrary netflowvectors. Theorem 2.1 beautifully connects the volume of the flow polytope of any graph toan evaluation of the Kostant partition function. We note that since the number of integerpoints of a flow polytope is already given by a Kostant partition function evaluation, thevolume of any flow polyope is given by the number of integer points of another.
KAROLA M´ESZ ´AROS AND AVERY ST. DIZIER
Recall that two polytopes P ⊆ R k and P ⊆ R k are integrally equivalent if thereis an affine transformation T : R k → R k that is a bijection P → P and a bijectionaff( P ) ∩ Z k → aff( P ) ∩ Z k . Integrally equivalent polytopes have the same face lattice,volume, and Ehrhart polynomial. We write P ≡ P to denote integral equivalence.While simple to prove, the following lemma is important. We leave its proof to the reader.For the rest of the paper, given a graph G and a set S of its edges, we use the notation G/S to denote the graph obtained from G by contracting the edges in S (and deleting loops) andwe use the notation G \ S to denote the graph obtained from G by deleting the edges in S .For a set V of vertices of G , we also use the notation G \ V to denote the graph obtainedfrom G by deleting the vertices in V and all edges incident to them. When S or V consistsof just one element, we simply write G/e or G \ v . Lemma 2.2.
Let G be a graph on [0 , n ] . Assume vertex j has only one outgoing edge e andnetflow a j ≥ . If e is directed from j to k ∈ [ n ] , then F G ( a , . . . , a n ) and F G/e ( a , . . . , a j − , a j +1 , a j +2 , . . . , a k − , a k + a j , a k +1 , . . . , a n ) are integrally equivalent. An analogous result holds if j has only one incoming edge and a j ≤ . Dissections of flow polytopes.
For graphs with a special source and sink, there is asystematic way to dissect the flow polytope F (cid:101) G studied in [10]. Let G be a graph on [0 , n ],and define (cid:101) G on [0 , n ] ∪ { s, t } with s being the smallest vertex and t the biggest vertexby setting E ( (cid:101) G ) = E ( G ) ∪ { ( s, i ) , ( i, t ) : i ∈ [0 , n ] } . The systematic dissections can beexpressed in the language of the subdivision algebra or equivalently in terms of reductiontrees [8, 9, 10]. We use the language of reduction trees in this paper.Let G be a graph on [0 , n ] with edges ( i, j ) and ( j, k ) for some i < j < k . By a reduction on G , we mean the construction of three new graphs G , G and G on [0 , n ] given by: E ( G ) = E ( G ) \{ ( j, k ) } ∪ { ( i, k ) } E ( G ) = E ( G ) \{ ( i, j ) } ∪ { ( i, k ) } (2.1) E ( G ) = E ( G ) \{ ( i, j ) , ( j, k ) } ∪ { ( i, k ) } We say G reduces to G , G and G . We also say that the above reduction is at vertex j ,on the edges ( i, j ) and ( j, k ). Proposition 2.3.
Let G be a graph on [0 , n ] which reduces to G , G and G as above.Then for each m ∈ [3] , there is a polytope Q m integrally equivalent to F (cid:101) G m such that Q and Q subdivide F (cid:101) G and intersect in Q . That is, the polytopes Q , Q , and Q satisfy F (cid:101) G = Q (cid:91) Q with Q o (cid:92) Q o = ∅ and Q (cid:92) Q = Q . Moreover, Q and Q have the same dimension as F (cid:101) G and Q has dimension one less.Proof. Let r and r denote the edges of G from i to j and from j to k respectively thatwere used in the reduction. Viewing R E ( (cid:101) G ) as functions f : E ( (cid:101) G ) → R , cut F (cid:101) G with thehyperplane H defined by the equation f ( r ) = f ( r ). Let Q be the intersection of F (cid:101) G withthe positive half-space f ( r ) ≥ f ( r ), let Q be the intersection of F (cid:101) G with the negativehalf-space f ( r ) ≤ f ( r ), and let Q be the intersection of F (cid:101) G with the hyperplane H . SeeFigure 1 for an illustration of the integral equivalence between Q m and F (cid:101) G m . Notice that ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 5 i j k i j ki j ki j kp q p − q q q − pppG G G G p > qp < qp = q Figure 1.
An illustration of the integral equivalence between Q m and F (cid:101) G m for m ∈ [3] used Proposition 2.3.since we are doing the reductions on the edges of G (as opposed to on the edges incident tothe source or sink in (cid:101) G ), it follows that the hyperplane H meets F (cid:101) G in its interior, givingthe claims on the dimensions of Q m , m ∈ [3]. (cid:3) Iterating this subdivision process produces a dissection of F (cid:101) G into simplices. This processcan be encoded using a reduction tree. A reduction tree of G is constructed as follows.Let the root node of the tree be labeled by G . If a node has any children, then it has threechildren obtained by performing a reduction on that node and labeling the children with thegraphs defined in (2.1). Continue this process until the graphs labeling the leaves of the treecannot be reduced. See Figure 2 for an example.Fix a reduction tree R ( G ) of G . Let L be a graph labeling one of the leaves in R ( G ).Lemma 2.2 implies that F (cid:101) L is a simplex, so the flow polytopes of the graphs labeling theleaves of R ( G ) dissect F (cid:101) G into simplices. All dissections we consider in this paper will bedissections into simplices. By full-dimensional leaves of R ( G ), we mean the leaves L with E ( L ) = E ( G ). By lower-dimensional leaves we mean all other leaves L of R ( G ). Notethat the full-dimensional leaves correspond to top-dimensional simplices in the dissection of F (cid:101) G , and the lower-dimensional leaves index intersections of the top-dimensional simplices.Since all simplices above are unimodular, it follows that: Corollary 2.4.
The normalized volume of F (cid:101) G equals the number of full-dimensional leavesin any reduction tree of G . Moreover, the number of leaves with a fixed number of edges isindependent of the reduction tree. Left-degree sequences.
Let G be a graph on [0 , n ], and let R ( G ) be a reduction treeof G . Denote by indeg G ( i ) the number of edges directed into vertex i . For each leaf L of KAROLA M´ESZ ´AROS AND AVERY ST. DIZIER
Figure 2.
A reduction tree for a graph on three vertices. The edges involvedin each reduction are shown in bold. The left-degree sequences of the leavesare shown in blue. R ( G ), consider the left-degree sequence (indeg L (1) , indeg L (2) , . . . , indeg L ( n )). By full-dimensional sequences we will mean left-degree sequences of full-dimensional leaves of R ( G ). The following definition is central to this paper. Definition 2.5.
Denote by LD( G ) the multiset of left-degree sequences of leaves in a reduc-tion tree of G .Although the actual leaves of a reduction tree are dependent on the individual reductionsperformed, we prove in Theorem A that LD( G ) is independent of the particular reductiontree considered. 3. Triangular arrays and left-degree sequences
In this section, we expand the technique described in [10] that characterized left-degreesequences of full-dimensional leaves in a specific reduction tree of a graph. We give a char-acterization of the left-degree sequences of all leaves of this reduction tree, not just the fulldimensional ones. This enables us to relate the left-degree sequences to generalized permu-tahedra in Section 4 and to use left-degree sequences and generalized permutahedra to showin Section 5 that the Schubert and Grothendieck polynomials have polytopal support. Themain theorem of this section is the following. The independence of LD( G ) on the reductiontree was first proved independently by Grinberg [6] in his study of the subdivision algebra. Theorem A.
For any graph G on [0 , n ] the multiset of left-degree sequences LD( G ) in anyreduction tree of G equals the first columns of Sol G ( F ) over all F ⊆ E ( G \ , also denotedby InSeq( T ( G )) . In particular, LD( G ) is independent of the choice of reduction tree. ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 7
For simplicity, throughout this section we restrict to the case where G is a simple graphon the vertex set [0 , n ]. The set Sol G ( F ) is defined in Definition 3.6 for simple graphs. Weaddress the general case in Section 6 where we also prove Theorem A.We start by generalizing [10, Lemma 3] to include the descriptions of the lower dimen-sional leaves of reductions performed at a special vertex v . The proof is a straightforwardgeneralization of that of [10, Lemma 3] illustrated in Figure 3. The key to the proof is the special reduction order , whereby we always perform a reduction on the longest edgespossible that are incident to the vertex at which we are reducing (the length of an edgebeing the absolute value of the difference of its vertex labels). We leave the details of theproof to the interested reader. Lemma 3.1.
Assume G has a distinguished vertex v with p incoming edges and one outgoingedge ( v, u ) . If we perform all reductions possible which involve only edges incident to v inthe special reduction order, then we obtain graphs H i , i ∈ [ p + 1] , and K j , j ∈ [ p ] , with (indeg H i ( v ) , indeg H i ( u )) = ( p + 1 − i, i ) and (indeg K j ( v ) , indeg K j ( u )) = ( p − j, j ) . G with v = 2 and u = 3 H K K H H Figure 3.
The graphs H i and K j of Lemma 3.1.We now construct a specific reduction tree T ( G ) and characterize the left-degree sequencesof its leaves. Denote by I i the set of incoming edges to vertex i in G . Let V i be the set ofvertices k with ( k, i ) ∈ I i , and let G [0 , i ] be the restriction of G to the vertices [0 , i ]. Forany reduction tree R ( G ), by InSeq( R ( G )) we mean the multiset of left-degree sequences ofthe leaves of R ( G ). Since we will build T ( G ) inductively from T ( H ) for smaller graphs H ,it is convenient to let InSeq n ( R ( H )) denote the multiset InSeq( R ( H )) with each sequencepadded on the right with zeros to have length n .We proceed using the following algorithm, analogous to the one described in [10]: • For the base case, define the reduction tree T ( G [0 , G [0 , T ( G [0 , { (indeg G (1)) } . • Having built T ( G [0 , i ]), construct the reduction tree T ( G [0 , i + 1]) from T ( G [0 , i ])by appending the vertex i + 1 and the edges I i +1 to all graphs in T ( G [0 , i ]) and then KAROLA M´ESZ ´AROS AND AVERY ST. DIZIER performing reductions at each vertex in V i +1 on all graphs corresponding to the leavesof T ( G [0 , i ]) in the special reduction order as described below. • Let V i +1 = { i < i < · · · < i k } and let ( s , . . . , s n ) be one of the sequences inInSeq n ( T ( G [0 , i ])). Applying Lemma 3.1 to each of the vertices i , . . . , i k , we seethat the leaves of T ( G [0 , i + 1]) which are descendants of the graph with n -left-degree sequence ( s , . . . , s n ) in T ( G [0 , i ]) will have n -left-degree sequences exactlygiven by ( s , . . . , s n ) + v i +1 [ i ] + · · · + v i +1 [ i k ]where v i +1 [ i l ] ∈ S ( i l ) ∪ S ( i l ) and S , S are given by: S ( i l ) = { ( c , . . . , c n ) : c i = 0 for i / ∈ { i l , i + 1 } , c i l = s i l − s, and c i +1 = s for s ∈ [ s i l + 1] } ,S ( i l ) = { ( c , . . . , c n ) : c i = 0 for i / ∈ { i l , i + 1 } , c i l = s i l − − s, and c i +1 = s for s ∈ [ s i l ] } . Definition 3.2.
For a simple graph G on [0 , n ], denote by T ( G ) the specific reduction treeconstructed using the algorithm described above. Definition 3.3.
To each leaf L of T ( G ), associate the triangular array of numbers Arr( L )given by a n, a n − , · · · a , a , a , a n, a n − , · · · a , a , ... ... . . . a n, n − a n − , n − a n, n where ( a i, , a i, , . . . , a i, i ) is the left-degree sequence of the leaf of T ( G [0 , i ]) preceding (orequaling if i = n ) L in the construction of T ( G ). Theorem 3.4 ([10], Theorem 4) . The arrays
Arr( L ) for full-dimensional leaves L of T ( G ) are exactly the nonnegative integer solutions in the variables { a i, j : 1 ≤ j ≤ i ≤ n } to theconstraints: • a , = E ( G [0 , • a i, j ≤ a i − , j if ( j, i ) ∈ E ( G ) • a i, j = a i − , j if ( j, i ) / ∈ E ( G ) • a i, i = E ( G [0 , i ]) − (cid:80) i − k =1 a i, k Example 3.5. If G is the graph on vertex set [0 ,
4] with E ( G ) = { (0 , , (0 , , (1 , , (2 , , (2 , , (3 , } , then from Theorem 3.4 we obtain theconstraints: 0 ≤ a , = a , = a , ≤ a , = 10 ≤ a , ≤ a , ≤ a , = 3 − a , ≤ a , ≤ a , = 4 − a , − a , ≤ a , = 6 − a , − a , − a , The solutions to these constraints yield the full-dimensional left-degree sequences( a , , a , , a , , a , ) of G . ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 9
Given a graph G , we write the constrains specified in Theorem 3.4 in the form shownin Example 3.5 and call them the triangular constraint array of G . We proceed bygeneralizing triangular constraint arrays to encode the lower-dimensional leaves of T ( G ) aswell. Definition 3.6.
Denote by Tri G ( ∅ ), or when the context is clear, by Tri( ∅ ), the triangularconstraint array of G . For each subset F ⊆ E ( G \
0) (recall that G is simple in this section),define a constraint array Tri( F ) by modifying Tri( ∅ ) as follows: for each ( j, i ) ∈ F and eachordered pair ( m, j ) with n ≥ m ≥ i , replace each occurrence of a m, j by a m, j + 1 and add 1to the constant at the leftmost edge of row j . Denote by Sol G ( F ), or when the context isclear, by Sol( F ), the collection of all integer solution arrays to the constraints Tri( F ). Example 3.7.
With G as in Example 3.5 and F = { (2 , , (2 , , (3 , } , we haveTri( F ) : 0 ≤ a , = a , = a , ≤ a , = 12 ≤ a , + 2 ≤ a , + 1 ≤ a , = 3 − a , ≤ a , + 1 ≤ a , = 3 − a , − a , ≤ a , = 3 − a , − a , − a , The characterization of InSeq( T ( G )) given in the construction of T ( G ) implies the followingtheorem. Theorem 3.8.
The leaves of T ( G ) are in bijection with the multiset union of solutions tothe arrays Tri( F ) , that is { Arr( L ) : L is a leaf of T ( G ) } = (cid:91) F ⊆ E ( G \ Sol G ( F ) . In particular,
InSeq( T ( G )) is the (multiset) image of the right-hand side under the map thattakes a triangular array to its first column ( a n, , . . . , a n, n ) . Since Theorem A shows that InSeq( R ( G )) = LD( G ) for any reduction tree R ( G ) of G ,we can now state the following important definition. Definition 3.9.
For any F ⊆ E ( G \ G, F ) the submultiset of LD( G )consisting of sequences occurring as the first column of an array in Sol( F ).As a consequence of Theorem 3.8,InSeq( T ( G )) = (cid:91) F ⊆ E ( G \ LD(
G, F ) . Combinatorially, we can think of LD(
G, F ) in the following way. Construct the reductiontree T ( G ) of G . Take any graph H appearing as a node of T ( G ). Let H have descendants H , H and H in T ( G ) obtained by the reduction on edges ( i, j ) and ( j, k ) in H with i < j < k , so that H has edge set E ( H ) \{ ( i, j ) , ( j, k ) } ∪ { ( i, k ) } ). Label the edge in T ( G )between H and H by ( j, k ). To each leaf L of T ( G ), associate the set of all labels on theedges of the unique path from L to the root G of T ( G ). The left-degree sequences of leavesassigned a set F in this manner are exactly the elements of the multiset LD( G, F ).To understand the multisets Sol( F ) and LD( G, F ), we study the constraint arrays Tri( F ).We begin by investigating the case where G = K n +1 is the complete graph on [0 , n ]. Given F ⊆ E ( K n +1 \ f i, j = { ( j, k ) ∈ F : k ≤ i } . (3.1)Observe that for each F ⊆ E ( K n +1 \ F ) is obtained from Tri( ∅ ) by replacing a i,j in Tri( ∅ ) by a i, j + f i, j and replacing the 0 in the leftmost spot of row j by f n, j . Also notethat f j, j = 0 for each j . Modify Tri( F ) to obtain a new constraint array denoted A K n +1 ( F )with the same solutions by subtracting f n, j from each term in row j for each j , so thatthe leftmost column becomes all zeros. For notational compactness, let b i, j = a i, j + f i, j . A K n +1 ( F ) is given by0 ≤ b n, − f n, ≤ · · · ≤ b , − f n, ≤ b , − f n, = E ( K n +1 [0 , − f n, ≤ b n, − f n, ≤ · · · ≤ b , − f n, = E ( K n +1 [0 , − f n, − b , ... ... . . .0 ≤ b n, n − f n, n = E ( K n +1 ) − f n, n − n − (cid:88) k =1 b n, k Note that the real solution set in variables { a i, j } to A K n +1 ( F ) is a polytope in R ( n +12 ).We first show that it is a flow polytope. For any constraint array A , denote by Poly( A ) the polytope defined by the inequalities in A . Lemma 3.10.
Let K n +1 be the complete graph on [0 , n ] . Fix F ⊆ E ( K n +1 \ and let Q be the polytope Q = Poly( A K n +1 ( F )) . Then, there exists a graph denoted Gr( K n +1 ) and anetflow vector a FK n +1 such that Q is integrally equivalent to F Gr( K n +1 ) (cid:0) a FK n +1 (cid:1) . G Leaves L of T ( G ) F ⊆ E ( G \ ∅{ (1 , } Tri G ( F )0 ≤ a , ≤ a , = 10 ≤ a , = 2 − a , ≤ a , + 1 ≤ a , = 10 ≤ a , = 2 − a , Sol G ( F )1 110 120 11 Figure 4.
A small example demonstrating Theorem 3.8. In general, Sol G ( F )will be empty for many F . ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 11
Proof.
For { ( i, j ) : 1 ≤ j < i ≤ n } , we introduce slack variables z i, j to convert the inequali-ties in A K n +1 ( F ) into equations Y i, j via Y i, j : a i, j + f i, j + z i, j = a i − , j + f i − , j if i > j i (cid:88) k =1 a i, k = E ( K n +1 [0 , i ]) if i = j. Define an equivalent system of equations { Z (cid:48) i, j } by setting Z (cid:48) i, j : Y i, j if i > j or i = j = 1 Y i, j − Y i − , j − − j − (cid:88) k =1 Y j, k if i = j > . We then modify each equation Z (cid:48) i, j by rearranging negated terms to get equations Z i, j givenby Z i, j : a i, j + z i, j = a i − , j + f i − , j − f i, j if i > ja i, j = indeg K n +1 (1) if i = j = 1 a i, j = indeg K n +1 ( j ) + j − (cid:88) k =1 z j, k if i = j > E ( K n +1 [0 , j ]) − E ( K n +1 [0 , j − K n +1 ( j ).We now construct the graph Gr( K n +1 ). Let the vertices of Gr( K n +1 ) be { v i, j : 1 ≤ j ≤ i ≤ n } ∪ { v n +1 , n +1 } with the ordering v , < v , < · · · < v n, < v , < · · · < v n, n < v n +1 , n +1 .Let the edges of Gr( K n +1 ) be labeled suggestively by the flow variables a i,j and z i,j . Set E (Gr( K n +1 )) = E a ∪ E z where E a consists of edges a i, j : v i, j → v i +1 , j for 1 ≤ j ≤ i ≤ n and E z consists of edges z i, j : v i, j → v i, i for 1 ≤ j < i ≤ n and we take indices ( n + 1 , j ) to refer to ( n + 1 , n + 1).To define the netflow vector a FK n +1 , we assign netflow indeg K n +1 ( j ) to vertices v j, j with j < n + 1, we assign netflow − E ( K n +1 ) + n − (cid:88) k =1 f n, k to v n +1 , n +1 , and we assign netflow f i − , j − f i, j to each remaining vertex v i, j .The netflow vector a FK n +1 is given by reading each row of the triangular array f n − , − f n, f n − , − f n − , · · · f , − f , indeg K n +1 (1) f n − , − f n, · · · f , − f , indeg K n +1 (2)... . . .indeg K n +1 ( n ) v , v , v , v , v , v , v , z , a , a , a , a , a , a , z , z , a , a , a , a , a , a , a , z , z , z , v , v , v , v , v , v , v , Figure 5.
Two drawings of the graph Gr( K n +1 ) of Lemma 3.10. The drawingon the right has the netflow vector a ∅ K n +1 .right to left starting with the first row, moving top to bottom, and then appending − E ( K n +1 )+ (cid:80) n − k =1 f n, k at the end.By construction, the flow equation at vertex v i,j in Gr( K n +1 ) is exactly the equation Z i,j for ( i, j ) (cid:54) = ( n + 1 , n + 1). At v n +1 , n +1 , the flow equation is Y n, n , which follows from theequations Z i, j and adds no additional restrictions. (cid:3) We now generalize Lemma 3.10 to any simple graph G on [0 , n ]. Note that for F ⊆ E ( G \ G ( F ) can be obtained from Tri K n +1 ( F ) by turning certain inequalities into equalities andchanging all occurrences of E ( K n +1 [0 , j ]) to E ( G [0 , j ]) for each j . In the language of theproof of Lemma 3.10, this amounts to setting z i, j = 0 whenever ( j, i ) / ∈ E ( G ). Relative tothe graph Gr( K n +1 ), this is equivalent to deleting the edges labeled z i, j for ( j, i ) / ∈ E ( G ).Thus, we have the following extension of the construction given in the proof of Lemma 3.10. Definition 3.11.
For a simple graph G on [0 , n ] define a graph Gr( G ) on vertices { v i, j : 1 ≤ j ≤ i ≤ n } ∪ { v n +1 , n +1 } ordered v , < v , < · · · < v n, < v , < · · · < v n, n < v n +1 , n +1 and with edges E a ∪ E z where E a consists of edges a i, j : v i, j → v i +1 , j for 1 ≤ j ≤ i ≤ n and E z consists of edges z i, j : v i, j → v i, i for ( j, i ) ∈ E ( G \ . For any F ⊆ E ( G \ a FG for Gr( G ) by reading each row of thetriangular array f n − , − f n, f n − , − f n − , · · · f , − f , indeg G (1) f n − , − f n, · · · f , − f , indeg G (2)... . . .indeg G ( n )right to left starting with the first row, moving top to bottom, and then appending − E ( G ) + (cid:80) n − k =1 f n, k at the end, where f i, j = { ( j, k ) ∈ F : k ≤ i } . Proposition 3.12.
Let G be a simple graph on [0 , n ] and F ⊆ E ( G \ . Then, Poly(Tri G ( F )) is integrally equivalent to F Gr( G ) ( a FG ) . In particular, the multiset of solutions Sol( F ) to Tri( F ) ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 13 consists precisely of the projections of integral flows on
Gr( G ) with netflow a FG onto the edgeslabeled { a i, j } . Example 3.13.
Let G be the graph on vertex set [0 ,
4] with edge set E ( G ) = { (0 , , (0 , , (1 , , (2 , , (2 , , (3 , } and F = { (2 , } . The graph Gr( G ) andnetflow a FG are: v , a , a , a , a , a , a , a , z , z , z , a , z , a , a , G ) v , v , v , v , v , − v , v , v , v , v , − Observe that contracting the edges { a , , a , , a , , a , , a , , a , } yields the graph below,which is exactly (cid:101) G \{ s, } . The next result shows that this occurs in general. z , z , z , a , a , a , a , z , − For a graph G and a subset F ⊆ E ( G \ F as a subgraph of G on the same vertexset. Note that for each j , f n, j = { ( j, k ) ∈ F : k ≤ n } = outdeg F ( j )and the number − E ( G ) + n − (cid:88) k =1 f n, k appearing as the last entry of a FG equals − E ( G \ F ). Theorem 3.14.
Let G be a simple graph on [0 , n ] and F ⊆ E ( G \ . Then, the flow polytopes F Gr( G ) (cid:0) a FG (cid:1) and F (cid:101) G \{ s, } (cid:0) d F , d F , · · · , d Fn , − E ( G \ F ) (cid:1) are integrally equivalent, where d Fj = indeg G ( j ) − outdeg F ( j ) for j ∈ [ n ] .Proof. First, note that in Gr( G ), the edges { a i,j : i < n } are each the only edges incomingto their target vertex. Contracting these edges via Lemma 2.2 identifies vertices v i, j and v i (cid:48) , j . Label the representative vertices v j, j by j for j ∈ [ n ] and v n +1 , n +1 by t . The remainingedges are z i,j : j → i for ( j, i ) ∈ E ( G ) and a n, j : j → t for j ∈ [ n ] , which, are exactly the edges of (cid:101) G − { s, } .Viewing the netflow vector a FG as the array f n − , − f n, f n − , − f n − , · · · f , − f , indeg G (1) f n − , − f n, · · · f , − f , indeg G (2)... . . .indeg G ( n ) − E ( G \ F ) , Lemma 2.2 implies the entries of the netflow vector after contracting are given by readingthe sums of each row from top to bottom. (cid:3)
Recall from Definition 3.9 that LD(
G, F ) is the multiset of left-degree sequences occurringas the first column ( a n, , . . . , a n, n ) of an array in Sol( F ). Corollary 3.15.
Let G be a simple graph on [0 , n ] and F ⊆ E ( G \ . If b FG is the vector b FG = (indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ) , − E ( G \ F )) and ψ is the map that takes a flow on (cid:101) G \{ s, } to the tuple of its values on the edges { ( j, t ) : j ∈ [ n ] } , then LD(
G, F ) equals the (multiset) image under ψ of all integral flows on (cid:101) G \{ s, } with netflow vector b FG .In particular, LD(
G, F ) is in bijection with integral flows on (cid:101) G \{ s, } with netflow b FG . We note that the preceding result implies a formula for the Ehrhart polynomial of flowpolytopes of graphs with special source and sink vertices. In particular, a special case ofTheorem 2.1 follows readily.
Theorem 3.16.
Let G be a simple graph on [0 , n ] and let d i = indeg G ( i ) . Then, the nor-malized volume of the flow polytope on (cid:101) G is (3.2) Vol F (cid:101) G = K (cid:101) G \{ s, } ( d , . . . , d n , − E ( G )) . Moreover, the Ehrhart polynomial of F (cid:101) G is (3.3) Ehr( F (cid:101) G , t ) = ( − d d (cid:88) i =0 ( − i (cid:88) F ⊆ E ( G \ F = d − i K (cid:101) G \{ s, } (cid:0) b FG (cid:1) (cid:18) t + ii (cid:19) , where b FG = (indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ) , − E ( G \ F )) .Proof. From the dissection of F (cid:101) G obtained via the reduction tree T ( G ), it follows thatVol F (cid:101) G is the number of full-dimensional left-degree sequences. By Corollary 3.15, theseare in bijection with the integer points in the flow polytope F (cid:101) G \{ s, } ( d , . . . , d n , − E ( G )),proving (3.2).To prove (3.3) note that F ◦ (cid:101) G = (cid:70) σ ◦ ∈ D T ( G ) σ ◦ , where D T ( G ) is the set of open simplices corre-sponding to the leaves of the reduction tree T ( G ). Then, Ehr( F ◦ (cid:101) G , t ) = (cid:80) σ ◦ ∈ D T ( G ) Ehr( σ ◦ , t ).Since all simplices in D T ( G ) are unimodular, it follows that for a k -dimensional simplex ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 15 σ ◦ ∈ D T ( G ) , Ehr( σ ◦ , t ) = Ehr(∆ ◦ , t ), where ∆ is the standard k -simplex. By [3, Theo-rem 2.2] Ehr(∆ ◦ , t ) = (cid:0) t − k (cid:1) . Thus, Ehr( F ◦ (cid:101) G , t ) = (cid:80) ∞ i =0 f i (cid:0) t − i (cid:1) , where f i is the number of i -simplices in D T ( G ) . If we let d = E ( (cid:101) G ) − V ( (cid:101) G ) + 1, which is the dimension of F (cid:101) G , thenfor i ∈ [0 , d ], f i = (cid:88) F ⊆ E ( G \ F = d − i LD ( G, F ) . Corollary 3.15 then implies f i = (cid:88) F ⊆ E ( G \ F = d − i K (cid:101) G \{ s, } (cid:0) b FG (cid:1) for i ∈ [0 , d ] . Therefore, Ehr( F ◦ (cid:101) G , t ) = d (cid:88) i =0 (cid:88) F ⊆ E ( G \ F = d − i K (cid:101) G \{ s, } (cid:0) b FG (cid:1) (cid:18) t − i (cid:19) . From the Ehrhart-Macdonald reciprocity [3, Theorem 4.1]Ehr( F (cid:101) G , t ) = ( − d Ehr( F ◦ (cid:101) G , − t ) , it follows that Ehr( F (cid:101) G , t ) = ( − d d (cid:88) i =0 (cid:88) F ⊆ E ( G \ F = d − i K (cid:101) G \{ s, } (cid:0) b FG (cid:1) (cid:18) − t − i (cid:19) = ( − d d (cid:88) i =0 ( − i (cid:88) F ⊆ E ( G \ F = d − i K (cid:101) G \{ s, } (cid:0) b FG (cid:1) (cid:18) t + ii (cid:19) . (cid:3) Newton polytopes of left-degree polynomials
In this section, we study the Newton polytopes of polynomials L G ( t ) built from left-degree sequences (see Definition 4.2). We first show that each of these polynomials havepolytopal support (Definition 4.1). Then, we investigate the Newton polytopes of theirhomogeneous components and certain homogeneous subcomponents and prove that theseNewton polytopes are generalized permutahedra. We can summarize some of our results as: Theorem B.
Let G be a graph on [0 , n ] . Then the left-degree polynomial L G ( t ) has polytopalsupport, and the Newton polytope of each homogeneous component L kG ( t ) of L G ( t ) of degree E ( G ) − k is a generalized permutahedron. Theorems 4.5, 4.9 and 4.21 imply Theorem B, and also contain a lot more detail regardingthe players in Theorem B.
Definition 4.1.
Recall that for a polynomial f = (cid:88) α ∈ Z n ≥ c α t α , the Newton polytope isNewton( f ) = Conv ( { α : c α (cid:54) = 0 } ) . We say the polynomial f has polytopal support if c α (cid:54) = 0 whenever α ∈ Newton( f ), that iswhenever the integer points of Newton( f ) are exactly the exponents of monomials appearingin f with nonzero coefficients.The question of when a polynomial has polytopal support is a very natural one, andhas recently been investigated for various polynomials from algebra and combinatorics byMonical, Tokcan and Yong in [13], who refer to this notion as the SNP property (saturatedNewton polytope property).Recall from Definition 3.9 that for a simple graph G and a subset F ⊆ E ( G \ G, F )denotes the submultiset of LD( G ) consisting of sequences occurring as the first column of anarray in Sol( F ). Just as in Section 3, for the remainder of this section we add the simplifyingassumption that G has no multiple edges. All of the results of this section are also valid forgraphs with multiple edges, with similar proof and notation modifications to those describedin Section 6. Definition 4.2.
Let G be a graph on [0 , n ]. For α ∈ LD( G ), let codim( α ) = E ( G ) − (cid:80) ni =1 α i . Define the left-degree polynomial L G ( t ) in variables t = ( t , t , . . . , t n ) by L G ( t ) = (cid:88) α ∈ LD( G ) ( − codim( α ) t α . Similarly, for F ⊆ E ( G \ L G,F ( t ) by L G,F ( t ) = (cid:88) α ∈ LD(
G,F ) ( − codim( α ) t α = (cid:88) α ∈ LD(
G,F ) ( − F t α . Note that the ( − codim( α ) in Definition 4.2 has no effect on the Newton polytope. It ispresent so the definition of the left-degree polynomial agrees with the definition of right-degree polynomials utilized in [4] that we address in Section 5.Restating Theorem 3.8 in terms of left-degree sequences gives the multiset union decompo-sition LD( G ) = (cid:91) F ⊆ E ( G \ LD(
G, F ) . Relative to Newton polytopes, this impliesNewton( L G ( t )) = Conv (cid:91) F ⊆ E ( G \ Newton ( L G,F ( t )) . (4.1)We first study the polytope Newton( L G ( t )) and then the component pieces Newton ( L G,F ( t )).To start, we define a new constraint array. Definition 4.3.
Let G be a simple graph on [0 , n ]. Proceed as follows: • Start with the trianglular constraint array Tri G ( ∅ ) of G as in Theorem 3.4. • Replace the zero on the left of row j by y n, j + y n − , j + · · · + y j +1 , j for j ∈ [ n − n is left unchanged. ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 17 • For each ( i, j ) with n ≥ i > j ≥
1, replace all occurrences of a i, j in the array by a i, j + (cid:80) ik = j +1 y k, j . • For every ( j, i ) / ∈ E ( G \ y i,j = 0 throughout.We refer to this array as the augmented constraint array of G and view it as havingvariables a i, j and y i, j subject to the additional constraints that for all 1 ≤ j < i ≤ n ,0 ≤ y i, j ≤ . Example 4.4. If G is the graph on vertex set [0 ,
4] with E ( G ) = { (0 , , (0 , , (1 , , (2 , , (2 , , (3 , } , then we start with the constraints:0 ≤ a , = a , = a , ≤ a , = 10 ≤ a , ≤ a , ≤ a , = 3 − a , ≤ a , ≤ a , = 4 − a , − a , ≤ a , = 6 − a , − a , − a , After performing the modifications, we arrive at: y , ≤ a , + y , = a , + y , = a , + y , ≤ a , = 1 y , + y , ≤ a , + y , + y , ≤ a , + y , ≤ a , = 3 − a , − y , y , ≤ a , + y , ≤ a , = 4 − a , − y , − a , − y , ≤ a , = 6 − a , − y , − a , − y , − y , − a , − y , Theorem 4.5.
Let A denote the augmented constraint array of G and Poly( A ) the polytopedefined by the real valued solutions to A with the additional constraints ≤ y i, j ≤ for all i and j with ≤ j < i ≤ n . If ρ is the projection that maps a solution of A to its values ( a n, , . . . , a n, n ) , then Newton( L G ( t )) = ρ (Poly( A )) . Furthermore, each integer point in the right-hand side is in
LD( G ) , so L G has polytopalsupport. For the proof of Theorem 4.5 and later Theorem 4.20, it is convenient to replace Poly( A )by an integrally equivalent flow polytope using the proof techniques from Lemma 3.10 andTheorem 3.14. Begin with the case where G is a complete graph. By introducing slackvariables z i, j for the inequalites in the augmented constraint array (not 0 ≤ y i, j ≤ Y i, j given by Y i, j : a i, j + y i, j + z i, j = a i − , j if i > ja i, j = E ( G [0 , i = j = 1 i (cid:88) k =1 a i, k + i (cid:88) m =2 m − (cid:88) k =1 y m, k = E ( G [0 , i ]) if i = j > Applying the exact same transformation used in the proof of Lemma 3.10, we get equivalentequations Z i, j given by Z i, j : a i, j + y i, j + z i, j = a i − , j if i > ja i, j = indeg G (1) if i = j = 1 a i, j = indeg G ( i ) + i − (cid:88) k =1 z i, k if i = j > y i, j = 0 and z i, j = 0whenever ( j, i ) / ∈ E ( G ). We can realize the solutions to the Z i, j as points in a flow polytopeof some graph. However, to account for the additional restrictions 0 ≤ y i, j ≤
1, we viewit as a capacitated flow polytope . This is for convenience and is not mathematicallysignificant since any capacitated flow polytope is integrally equivalent to an uncapacitatedflow polytope [2, Lemma 1].
Definition 4.6.
Define the augmented constraint graph Gr aug ( G ) to have vertex set { v i, j : 1 ≤ j ≤ i ≤ n } ∪ { v n +1 , n +1 } with the ordering v , < v , < · · · < v n, < v , < · · · < v n, n < v n +1 , n +1 and edge set E a ∪ E z ∪ E y labeled by the variables a i, j , z i, j , and y i, j respectively, where E a consists of edges a i, j : v i, j → v i +1 , j for 1 ≤ j ≤ i ≤ n,E z consists of edges z i, j : v i, j → v i, i for ( j, i ) ∈ E ( G \ ,E y consists of edges y i, j : v i, j → v n +1 , n +1 for ( j, i ) ∈ E ( G \ , and we take indices ( n + 1 , j ) to refer to ( n + 1 , n + 1). Define a netflow vector a aug G byreading each row of the array0 0 · · · G (1)0 0 · · · G (2)... . . .indeg G ( n ) − E ( G )from right to left and reading the rows from top to bottom.Denote by F c Gr aug ( G ) ( a aug G ) the capacitated flow polytope of the graph Gr aug ( G ) with net-flow a aug G and with the capacity constraints 0 ≤ y i, j ≤ ≤ j < i ≤ n . By con-struction, the points in F c Gr aug ( G ) ( a aug G ) are exactly the solutions to the augmented constraintarray of G . Definition 4.7.
Similar to Theorem 3.14, contracting the edges { a i, j : 1 ≤ j ≤ i < n } ofGr aug ( G ) and relabeling the representative vertices v n, j by j and v n +1 , n +1 by t , we obtain agraph called the augmented graph of G . This graph is denoted G aug and is defined onvertices [ n ] ∪ { t } with labeled edges E a ∪ E z ∪ E y where E a consists of edges a n, j : j → t for j ∈ [ n ]; E z consists of edges z i, j : j → i for ( j, i ) ∈ E ( G \ E y consists of edges y i, j : j → t for ( j, i ) ∈ E ( G \ . ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 19
Example 4.8.
For G the complete graph on [0 , aug ( G ) and G aug are shownbelow. v , v , v , v , v , v , v , z , a , a , a , a , a , a , z , z , a , y , y , y , z , z , z , a , a , y , a , y , Gr aug ( G ) G aug y , Before proceeding to the proof of Theorem 4.5, recall that b FG = (indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ) , − E ( G \ F ))for any F ⊆ E ( G \ . Denote by F cG aug (cid:0) b ∅ G (cid:1) the capacitated flow polytope of the graph G aug with netflow b ∅ G and the capacity constraints 0 ≤ y i, j ≤ ≤ j < i ≤ n . Proof of Theorem 4.5.
By the constructions of Definitions 4.6 and 4.7, we have integralequivalences of capacitated flow polytopesPoly( A ) ≡ F c Gr aug ( G ) ( a aug G ) ≡ F cG aug (cid:0) b ∅ G (cid:1) . Thus, it suffices to prove Newton( L G ( t )) = ψ (cid:0) F cG aug (cid:0) b ∅ G (cid:1)(cid:1) . where ψ is the projection that takes a flow on F cG aug (cid:0) b ∅ G (cid:1) to its values on the edges labeled { a n, j : j ∈ [ n ] } . This is accomplished in Theorem 4.9. (cid:3) Theorem 4.9.
For G a graph on [0 , n ] , the Newton polytope of the left-degree polynomial L G ( t ) and the capacitated flow polytope F cG aug (cid:0) b ∅ G (cid:1) satisfy Newton( L G ( t )) = ψ (cid:0) F cG aug (cid:0) b ∅ G (cid:1)(cid:1) , where where ψ is the projection that takes a flow on F cG aug (cid:0) b ∅ G (cid:1) to its values on the edgeslabeled { a n, j : j ∈ [ n ] } .Proof. Let α ∈ LD(
G, F ) for F ⊆ E ( G \ G aug suchthat each edge y i, j has flow 1 if ( j, i ) ∈ F and zero otherwise. By the construction of G aug ,these are in bijection with the integer flows on (cid:101) G \{ s, } with netflow vector b FG , which inturn are in bijection to LD( G, F ) (Corollary 3.15). Thus α is the projection of a capacitatedflow on G aug with netflow b ∅ G .Conversely, let α = ( α , . . . , α n ) ∈ ψ (cid:0) F cG aug (cid:0) b FG (cid:1)(cid:1) be an integer point. Then, there existssome flow f (not necessarily integral) on G aug with netflow b ∅ G having the integer values α j on the edges ( j, t ). If we remove these edges and modify the netflow vector accordingly, thenew flow polytope we get is the (integrally capacitated) flow polytope of a graph with anintegral netflow vector. Any such polytope has integral vertices [16, Theorem 13.1]. Thus,we can choose f to be an integral flow.Since the edges labeled y i, j are constrained between 0 and 1, f takes value 0 or 1 on theseedges. If we let F = { ( j, i ) ∈ E ( G \
0) : f takes value 1 on the edge labeled by y i, j } , then f induces a flow on (cid:101) G \{ s, } with netflow vector b FG , so α ∈ LD(
G, F ). (cid:3) We now analyze the component polytopes Newton( L G,F ( t )) and show that they are gen-eralized permutahedra. We first briefly recall the relevant definitions from [14].A generalized permutahedron is a deformation of the usual permutahedron obtainedby parallel translation of the facets. Generalized permutahedra are parameterized by realnumbers { z I } I ⊆ [ n ] with z ∅ = 0 and satisfying the supermodularity condition z I ∪ J + z I ∩ J ≥ z I + z J for any I, J ⊆ [ n ] . For a choice of parameters { z I } I ⊆ [ n ] , the associated generalized permutahedron P zn ( { z I } ) isdefined by P zn ( { z I } ) = (cid:40) t ∈ R n : (cid:88) i ∈ I t i ≥ z I for I (cid:54) = [ n ] , and n (cid:88) i =1 t i = z [ n ] (cid:41) . There is a subclass of generalized permutahedra given by Minkowski sums of dilationsof the faces of the standard ( n − I ⊆ [ n ], let ∆ I = Conv( { e i : i ∈ I } ),where e i is the ith standard basis vector in R n and ∆ ∅ is the origin. Given a set { y I } I ⊆ [ n ] ofnonnegative real numbers with y ∅ = 0, denote by P yn ( { y I } ) the polytope P yn ( { y I } ) = (cid:88) I ⊆ [ n ] y I ∆ I . Proposition 4.10 ([14], Proposition 6.3) . For nonnegative real numbers { y I } I ⊆ [ n ] , the poly-tope P yn ( { y I } ) is a generalized permutahedron P zn ( { z I } ) with z I = (cid:80) J ⊆ I y J . We now return to left-degree polynomials. For F ⊆ E ( G \ f i, j givenby f i, j = { ( j, k ) ∈ F : k ≤ i } . By Corollary 3.15 (Theorem 6 . G, F ) is in bijection with integralflows on the graph (cid:101) G \{ s, } with the netflow vector b FG defined by b FG = (indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ) , − E ( G \ F ))via projection onto the edges ( i, t ). To each I ⊆ [ n ], associate the integer z FI given by z FI = min (cid:40)(cid:88) i ∈ I f ( i, t ) : f is a flow on (cid:101) G \{ s, } with netflow vector b FG (cid:41) . (4.2) Definition 4.11.
For a collection of vertices I of a graph G , define the outdegree outdeg G ( I )to be the number of edges from vertices in I to vertices not in I .Note that the parameters z FI of (4.2) satisfy the supermodularity condition since z FI = z FI (cid:48) where I (cid:48) is the largest subset of I satisfying outdeg G ( I (cid:48) ) = 0.Our goal is to show that Newton( L G,F ( t )) = P zn { z FI } I ⊆ [ n ] . The proof relies on the following fact about flow polytopes, which readily follows from themax-flow min-cut theorem.
ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 21
Lemma 4.12.
Let G be a graph on [0 , n ] and α = ( α , . . . , α n ) ∈ R n +1 . Then F G ( α ) isnonempty if and only if n (cid:88) i =0 α i = 0 and (cid:88) i ∈ S α i ≤ for all S ⊆ [0 , n ] with outdeg G ( S ) = 0 . (4.3) Proof.
Observe that the conditions (4.3) are necessary in order for F G ( α ) to be nonempty.We now show they are also sufficient. For this, we rephrase the problem as a max-flowproblem on another graph. Let G (cid:48) = ( V ( G ) ∪ { s, t } , E ( G ) ∪ { ( s, i ) | i ∈ [0 , n ] , α i > } ∪{ ( i, t ) | i ∈ [0 , n ] , α i < } ) with edges directed from smaller to larger vertices, where s isthe smallest and t is the largest vertex. Let the upper capacity of the edges ( s, i ), with i ∈ [0 , n ] , α i >
0, be α i and the upper capacity of the edges ( i, t ), with i ∈ [0 , n ] , α i <
0, be − α i . All edges have a lower capacity of 0 and the edges also belonging to G all have theupper capacity (cid:80) i ∈ [0 ,n ] ,α i > α i . If the maximum flow on G (cid:48) saturates the edges incident to s (equivalently, to t ) then F G ( α ) is nonempty. We thus proceed to show that if α satisfies(4.3) with the given G , then the maximum flow on G (cid:48) saturates the edges incident to s . Inother words, if α satisfies (4.3) with the given G , then the value of the maximum flow on G (cid:48) is (cid:80) i ∈ [0 ,n ] ,α i > α i .Recall that by the max-flow min-cut theorem [16, Theorem 10.3] the maximum value ofan s − t flow on G (cid:48) subject to the above capacity constraints equals the minimum capacityof an s − t cut in G (cid:48) . For the cut ( { s } , V ( G ) \{ s } ) the capacity is (cid:80) i ∈ [0 ,n ] ,α i > α i , and weshow that this is the minimum capacity of an s − t cut in G (cid:48) . If the cut contains any edgenot incident to s or t , then the capacity of that edge is already (cid:80) i ∈ [0 ,n ] ,α i > α i . On theother hand, if the cut does not contain any edge not incident to s or t , the partition ofvertices is of the form ( { s } ∪ S, S c ∪ { t } ) , where S ⊆ [0 , n ] with outdeg G ( S ) = 0 and S c =[0 , n ] \ S . Thus, by (4.3) we have (cid:80) i ∈ S α i ≤
0. The capacity of the cut ( { s } ∪ S, S c ∪ { t } ) is (cid:80) i ∈ S c , ( s,i ) ∈ G (cid:48) α i − (cid:80) i ∈ S, ( i,t ) ∈ G (cid:48) α i . Note that (cid:80) i ∈ S c , ( s,i ) ∈ G (cid:48) α i − (cid:80) i ∈ S, ( i,t ) ∈ G (cid:48) α i ≥ (cid:80) i ∈ [0 ,n ] ,α i > α i since it is equivalent to 0 ≥ (cid:80) i ∈ S,α i > α i + (cid:80) i ∈ S, ( i,t ) ∈ G (cid:48) α i = (cid:80) i ∈ S α i . In other words, thecapacity of any cut is at least (cid:80) i ∈ [0 ,n ] ,α i > α i , and we saw that this is achieved. Thus, thevalue of the maximum flow on G (cid:48) is (cid:80) i ∈ [0 ,n ] ,α i > α i , as desired. (cid:3) Theorem 4.13.
For a simple graph G , F ⊆ E ( G \ , and { z FI } the parameters defined by(4.2), Newton( L G,F ( t )) is the generalized permutahedron Newton( L G,F ( t )) = Conv(LD( G, F )) = P zn { z FI } I ⊆ [ n ] . Furthermore, each integer point of P zn { z FI } is in LD(
G, F ) , so Newton( L G,F ( t )) has polytopalsupport.Proof. Since LD(
G, F ) equals the projection of integral flows on (cid:101) G \{ s, } with netflow b FG onto the edges { ( j, t ) } j ∈ [ n ] , Conv(LD( G, F )) ⊆ P zn { z FI } . For the reverse direction, let d denote the truncation of b FG by its last entry, that is let d = ( d , . . . , d n ) where d i = indeg G ( i ) − outdeg F ( i ) . We must show that each point x = ( x , . . . , x n ) ∈ P zn { z FI } , the assignment a n, j = x j in (cid:101) G \{ s, } can be extended to a flow on (cid:101) G \{ s, } . This is equivalent to showing F G \ ( d − x ) (cid:54) = ∅ . By Lemma 4.12, it suffices to note that (cid:88) i ∈ S d i − x i ≤ S ⊆ [ n ] with outdeg G ( S ) = 0 . However, since outdeg G ( S ) = 0, we have (cid:88) i ∈ S x i ≥ z S = (cid:88) i ∈ S d i . (cid:3) We further show that Newton( L G,F ( t )) can be written as P yn { y I } for some parameters y I .Let L = { J ⊆ [ n ] : outdeg H ( J ) = 0 } . Then L is a lattice, so consider the set Q of joinirreducible elements of L . For J ⊆ [ n ], define y FJ = (cid:40) indeg G ( k ) if J ∈ Q , J covers J (cid:48) in L, J \ J (cid:48) = { k } J / ∈ Q (4.4) Proposition 4.14. P yn { y FJ } = P zn { z FI } Proof.
Note that z FI = z FI where I is the largest element of L contained in I . Thus, z FI = z FI = (cid:88) k ∈ I b k = (cid:88) J ∈ QJ ⊆ I y FJ = (cid:88) J ⊆ I y FJ . Apply Proposition 4.10. (cid:3)
From (4.4), we can read off the { y FI } decomposition of Newton( L G,F ( t )). Let δ ( i ) denoteall the vertices of G that can be reached from i by an increasing path (including i itself).Then, Newton( L G,F ( t )) = n (cid:88) i =1 indeg G ( i )∆ δ ( i ) . (4.5) Example 4.15.
For a simple graph G , recall that the transitive closure of G is the simplegraph formed by adding edges ( i, j ) to E ( G ) whenever the vertices i and j are connected byan increasing path in G . If G is a simple graph on [0 , n ] such that the transitive closure of G \{ } is complete, then for each F ⊆ E ( G \ L G,F ( t )) = Π n (indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ))where Π n ( x ) is the Pitman-Stanley polytope as defined in [17], but shifted up one dimensionin affine space, that isΠ n ( x ) = (cid:40) t ∈ R n ≥ : k (cid:88) p =1 t p ≤ k (cid:88) p =1 x p for k ∈ [ n − , and n (cid:88) p =1 t p = n (cid:88) p =1 x p (cid:41) = x n ∆ { n } + x n − ∆ { n − , n } + · · · + x ∆ [ n ] . Proposition 4.16. If T is a tree on [0 , n ] , then Newton( L T,F ( t )) is a simple polytope.Proof. By the Cone-Preposet Dictionary for generalized permutahedra, ([15], Proposition3.5) it is enough to show that each vertex poset Q v is a tree-poset, that is, its Hasse diagramhas no cycles. To show this, let I ⊆ [ n ] and consider the normal fan N (∆ I ) of the simplex∆ I . By (4.5), the normal fan of Newton( L G,F ( t )) is the refinement of normal fans N (∆ I ). ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 23
Thus, a maximal cone of the normal fan of Newton( L G,F ( t )) is given by an intersectionof of maximal cones in each N (∆ I ) for I = δ ( j ), j ∈ [ n ], indeg T ( j ) >
0. A maximal conein N (∆ I ) gives the vertex poset relations x i > x j for all j ∈ I and any chosen i ∈ I . Thus,relations in the Hasse diagram of a vertex poset lift to undirected paths in T .If some Q v has a cycle C , then we can lift the relations to get two different paths in T between two vertices. This subgraph will contain a cycle, contradicting that T is a tree. (cid:3) The Newton polytopes of the homogeneous components of L G ( t ) are also generalizedpermutahedra. Definition 4.17.
For each k ≥ L kG ( t ) denote the degree E ( G ) − k homogeneouscomponent of L G ( t ), that is L kG ( t ) = (cid:88) F ⊆ E ( G \ F = k L G,F ( t )For a simple graph G on [0 , n ], the proof of Theorem 4.5 showed that the augmented graph G aug of Definition 4.7 has the property that the projection of integral flows on G aug withnetflow b ∅ G = (indeg G (1) , . . . , indeg G ( n ) , − E ( G ))and capacitance 0 ≤ y i, j ≤ ≤ j < i ≤ n onto the edges labeled a n, j for j ∈ [ n ]is exactly LD( G ). The following construction is a variation on this theme designed so itsintegral flows will only project to left-degree sequences whose entries have a particular sum. Definition 4.18.
Given a simple graph G on [0 , n ] and k ≥
0, let G ( k ) be the graph on[1 , n + 1] ∪ { t } with labeled edges E a ∪ E z ∪ E y where E a consists of edges a n, j : j → t for j ∈ [ n ]; E z consists of edges z i, j : j → i for ( j, i ) ∈ E ( G \ E y consists of edges y i, j : j → n + 1 for ( j, i ) ∈ E ( G \ . The flow polytope F cG ( k ) ( b ( k ) G ) is the flow polytope of G ( k ) with netflow vector b ( k ) G =(indeg G (1) , . . . , indeg G ( n ) , − k, k − E ( G )) and capacities 1 on the edges y i, j . Example 4.19.
For G the complete graph on [0 , G ( k ) is shown below alongside G aug forcomparison. z , z , z , a , a , y , a , y , G aug y , z , z , z , a , y , a , y , G ( k ) y , a , – k k –6 Note that capacitated integral flows on G ( k ) with netflow b ( k ) G are in bijection with capac-itated integral flows on G aug with netflow b ∅ G where exactly k edges y i, j have flow 1, and thebijection preserves the values on the edges { a n, j : j ∈ [ n ] } . Theorem 4.20.
For k ≥ , if ψ is the projection that takes a flow on F cG ( k ) (cid:16) b ( k ) G (cid:17) to thetuple of its values on the edges labeled a n, j for j in [ n ] , then Newton (cid:0) L kG ( t ) (cid:1) = ψ (cid:16) F cG ( k ) (cid:16) b ( k ) G (cid:17)(cid:17) . Furthermore, each integer point in the right-hand side is a left-degree sequence with compo-nents that sum to E ( G ) − k , so L kG has polytopal support.Proof. Let α be an integer point in Newton (cid:0) L kG ( t ) (cid:1) , so α ∈ LD(
G, F ) for F ⊆ E ( G \
0) with F = k . Then, α corresponds to a capacitated integral flow on G aug with netflow b ∅ G , whichin turn corresponds to a capacitated integral flow on G ( k ) with netflow b ( k ) G that ψ takes to α . Conversely, let α be an integer point in ψ (cid:16) F cG ( k ) (cid:16) b ( k ) G (cid:17)(cid:17) . Lift α to an integral flow f on G ( k ) . The flow f corresponds to an integral flow on G aug , so if F = { ( j, i ) : y i, j = 1 in f } ,then F = k and α ∈ LD(
G, F ). (cid:3) Similar to the proof of Theorem 4.13, for k ≥ I ⊆ [ n ], define parameters z ( k ) I by z ( k ) I = min (cid:40)(cid:88) i ∈ I f ( i, t ) : f is a flow on G ( k ) with netflow vector b ( k ) G (cid:41) . (4.6) Theorem 4.21.
For k ≥ and { z ( k ) I } the parameters defined by (4.6), Newton( L kG ( t )) isthe generalized permutahedron Newton( L kG ( t )) = P zn { z ( k ) I } I ⊆ [ n ] . Furthermore, each integer point of P zn { z ( k ) I } is a left-degree sequence, so Newton( L G,F ( t )) has polytopal support. Additionally, if G is an acyclic graph, then L G ( t ) is the integer-pointtransform of its Newton polytope.Proof. The proof of the first two statements is analogous to that of Theorem 4.13.To prove the third statement we must show that if G is an acyclic graph, all nonzerocoefficients of L G are 1. It follows from Corollary 3.15 (Theorem 6.3) that LD( G, ∅ ) equalsthe multiset of projections of integral flows on (cid:101) G \{ s, } with the netflow vector b ∅ G . Then,the multiplicity of any particular α ∈ LD( T, ∅ ) is the number of flows on G \ b ∅ G − α . However, acyclic graphs admit at most one flow for any given netflow vector, soevery element of LD( G, ∅ ) has multiplicity 1. This implies all coefficients in L G are 0 or1. (cid:3) Theorems 4.5 and 4.21 imply:
Corollary 4.22.
Given a graph G on the vertex set [0 , n ] with m edges, we have that Newton( L G ( t )) ∩ { ( x , . . . , x n ) ∈ R n | n (cid:88) i =1 = m − k } = P zn { z ( k ) I } I ⊆ [ n ] , for the parameters { z ( k ) I } given in (4.6).Proof. We have that Newton( L G ( t )) ∩{ ( x , . . . , x n ) ∈ R n | (cid:80) ni =1 = m − k } = Newton( L kG ( t )) , which by Theorem 4.21 equals P zn { z ( k ) I } I ⊆ [ n ] . (cid:3) ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 25
Theorems 3.16 and 4.21 imply:
Corollary 4.23. If G is an acyclic graph on [0 , n ] , then the normalized volume of the flowpolytope of (cid:101) G is Vol F (cid:101) G = Ehr( P G , , where P G := Newton( L G ( t )) is the generalized permutahedron specified in Theorem 4.21. Corollary 4.23 is of the same flavor as Postnikov’s following beautiful result; for the detailsof the terminology used in this theorem refer to [14].
Theorem 4.24. [14, Theorem 12.9]
For a bipartite graph G , the normalized volume of theroot polytope Q G is Vol Q G = Ehr( P − G , , where P − G is the trimmed generalized permutahedron. Root polytopes and flow polytopes are closely related, as can be seen by contrasting thetechniques and results in the papers [8, 9, 10, 11, 14]. It is thus reasonable to expect thatCorollary 4.23 and Theorem 4.24 are related mathematically. We invite the interested readerto investigate their relationship.5.
Newton polytopes of Schubert and Grothendieck polynomials
In this section, we discuss the connection between left-degree sequences, Schubert poly-nomials, and Grothendieck polynomials discovered in [4] and relate it to their Newton poly-topes. Our main theorem is as follows:
Theorem C.
Let π ∈ S n +1 be of the form π = 1 π (cid:48) where π (cid:48) is a dominant permutation of { , , . . . n + 1 } . Then the Grothendieck polynomial G π has polytopal support and the Newtonpolytope of each homogeneous component of G π is a generalized permutahedron. In partic-ular, the Schubert polynomial S π has polytopal support and Newton( S π ) is a generalizedpermutahedron. Moreover, S π is the integer-point transform of its Newton polytope. Theorem C implies that the recent conjectures of Monical, Tokcan, and Yong [13, Con-jecture 5.1 & 5.5] are true for permutations 1 π (cid:48) , where π (cid:48) is a dominant permutation. Thefollowing conjecture, discovered jointly with Alex Fink, is a strengthening of [13, Conjecture5.5]. We have tested it for all π ∈ S n , for n ≤ Conjecture 5.1.
The Grothendieck polynomial G π has polytopal support and the Newtonpolytope of each homogeneous component of G π is a generalized permutahedron. Since [4] uses right-degree sequences and right-degree polynomials instead of their left-degree counterparts, we will adopt this convention throughout this section. To simplifynotation, all graphs in this section will be on the vertex set [ n + 1]. Note the following easyrelation between right-degree and left-degree.Given a graph G on vertex set [ n + 1], let G ∗ be the mirror image of the graph G withvertex set shifted to [0 , n ]. More formally, let G ∗ be the graph on vertices [0 , n ] with edges E ( G ∗ ) = { ( n + 1 − j, n + 1 − i ) : ( i, j ) ∈ E ( G ) } . The right-degree sequences of G are exactly the left-degree sequences of G ∗ read backwards.We can then define the right-degree multiset RD( G ) as the multiset of right-degree se-quences of leaves in any reduction tree of G , and RD( G, ∅ ) the submultiset of sequenceswhose components sum to E ( G ) (notation consistent with LD( G, F ) in Definition 3.9).
Definition 5.2.
For any graph G on [ n + 1], define the right-degree polynomial R G by R G ( t , t , . . . t n ) = L G ∗ ( t n , t n − , . . . , t ) = (cid:88) α ∈ RD( G ) ( − codim( α ) t α t α . . . t α n n where codim( α ) = E ( G ) − (cid:80) ni =1 α i .For k ≥
0, let R kG ( t ) denote the degree E ( G ) − k homogeneous component of R G ( t ).Define the reduced right-degree polynomial (cid:101) R G as follows: If { v i , . . . v i k } are the vertices of G with positive outdegree, then R G is a polynomial in t i , . . . , t i k . Obtain (cid:101) R G by relabelingthe variables t i m by t m for each m . Note that R G (resp. (cid:102) R G ) is the top homogeneouscomponent of R G (resp. (cid:101) R G ), and is given by R G ( t , . . . , t n ) = (cid:88) α ∈ RD( G, ∅ ) t α t α . . . t α n n The following statement collects the right-degree analogues of Theorem 4.5 and Theorem4.21 from the previous section.
Theorem 5.3.
Let G be a graph on [ n + 1] . Then, R G ( t ) has polytopal support, and theNewton polytope of each homogeneous component R kG is a generalized permutahedron. Ad-ditionally, if G is an acyclic graph, then R G ( t ) is the integer-point transform of its Newtonpolytope. Recall that for a polytope P ⊆ R m , the integer-point transform of P is L P ( x , . . . , x m ) = (cid:88) p ∈ P ∩ Z m x p . We now recall the definition of pipe dreams of a permutation and the characterization ofSchubert and Grothendieck polynomials in terms of pipe dreams.
Definition 5.4. A pipe dream for π ∈ S n +1 is a tiling of an ( n + 1) × ( n + 1) matrix withtwo tiles, crosses and elbows (cid:5)(cid:7) , such that(1) all tiles in the weak south-east triangle are elbows, and(2) if we write 1 , , . . . , n + 1 on the top and follow the strands (ignoring second crossingsamong the same strands), they come out on the left and read π from top to bottom.A pipe dream is reduced if no two strands cross twice. (cid:5)(cid:7) (cid:5) (cid:5)(cid:7) (cid:5)(cid:7) (cid:5) (cid:5)(cid:7) (cid:5) (cid:5) Figure 6.
A reduced pipe dream for π = 2143. All tiles not shown are elbows.For π ∈ S n +1 let PD( π ) denote the collection of all pipe dreams of π and RPD( π ) thecollection of all reduced pipe dreams of π . For P ∈ PD( π ), define the weight of P by wt ( P ) = (cid:89) ( i,j ) ∈ cross( P ) t i . ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 27
Recall that for any π ∈ S n +1 , the Grothendieck polynomial G π can be represented in termsof pipe dreams of π by: G π ( t , . . . , t n ) = (cid:88) P ∈ PD( π ) wt ( P )and the Schubert polynomial S π is the lowest degree homogeneous component of the Grothendieckpolynomial: S π ( t , . . . , t n ) = (cid:88) P ∈ RPD( π ) wt ( P ) . In [4], it is proved that RD( T ) is independent of the reduction tree for T a tree, and thefollowing connection to Grothendieck polynomials is shown. Theorem 5.5 ([4], Theorem 5.3) . Let π ∈ S n +1 be of the form π = 1 π (cid:48) where π (cid:48) is a dominantpermutation of { , , . . . n +1 } . Then, there is a tree T ( π ) and nonnegative integers g i = g i ( π ) such that (cid:101) R T ( π ) ( t ) = (cid:32) n (cid:89) i =1 t g i i (cid:33) G π ( t − , . . . , t − n ) . Explicitly, if C ( π ) denotes the set core( π ) ∪ { (1 , } , then g i ( π ) is the number of boxes incolumn i of C ( π ) . In terms of Newton polytopes, Theorem 5.5 impliesNewton ( G π ) = ϕ (cid:16) Newton (cid:16) (cid:101) R T ( π ) ( t ) (cid:17)(cid:17) and Newton ( S π ) = ϕ (cid:16) Newton (cid:16) (cid:101) R T ( π ) ( t ) (cid:17)(cid:17) where ϕ is the integral equivalence( x , . . . , x n ) (cid:55)→ ( g − x , . . . , g n − x n ) . Proof of Theorem C.
By Theorem 5.3, right-degree polynomials R G ( t ) have polytopal sup-port. Since Newton (cid:16) (cid:101) R T ( π ) (cid:17) is the image of Newton (cid:0) R T ( π ) (cid:1) by a projection forgetting coor-dinates that are always zero, it follows from Theorem 5.5 that G π has polytopal support.Theorem 5.3 and Theorem 5.5 also yield that each homogeneous component of G π haspolytopal support and that their Newton polytopes are generalized permutahedra. In par-ticular, this holds for the Schubert polynomial. Since by [4] the Schubert polynomial of π = 1 π (cid:48) , where π (cid:48) is a dominant permutation, has 0 , (cid:3) From the proof of Theorem 5.5 in [4], one can infer the following new transition rule forSchubert polynomials of permutations of the form 1 π (cid:48) with π (cid:48) dominant. Lemma 5.6. ( Transition rule for Schubert polynomials. ) Let π ∈ S n +1 be of theform π = 1 π (cid:48) with π (cid:48) a dominant permutation of { , . . . , n + 1 } . Let π (cid:48) have diagram givenby the partition λ ( π (cid:48) ) = ( λ , · · · , λ z ) with λ z = k . For ≤ l ≤ k , let w l be the permutationon { , . . . , n + 1 } whose diagram is the partition ( λ − ( k − l ) , . . . , λ z − − ( k − l )) . Then S π ( x ) = k (cid:88) l =0 (cid:32) l (cid:89) m =1 x m (cid:33) (cid:32) k +1 (cid:89) p = l +2 x zp (cid:33) S w l ( x φ l ) where x = ( x , x , . . . ) , x φ l = ( x φ l (1) , x φ l (2) , . . . ) , and φ l ( i ) = (cid:40) i if i ≤ l + 1 i + k − l if i ≥ l + 2We illustrate the above transition rule in the following example. Example 5.7.
Let π = 14523. Then, π (cid:48) = 4523, so λ ( π (cid:48) ) = (2 , ≤ l ≤ w l will have diagram given by the partition ( l ). These permutations are w =2345, w = 3245, and w = 3425. Hence, the terms in the transition rule are(1)( x x ) S w ( x , x , x , x ) = x x ( x )( x ) S w ( x , x , x , x ) = x x + x x x ( x x )(1) S w ( x , x , x , x ) = x x + x x x + x x x . Adding these terms together gives the expected polynomial S π ( x , x , x , x ) = x x + x x x + x x x + x x + x x x + x x . Left-degree sequences are invariants of the graph
In this section we generalize the results of the Section 3 to any graph G , not necessarilysimple. Similar accommodations can be made to generalize Sections 4 and 5. We also proveTheorem A, which characterizes the left-degree sequences of the leaves of a reduction tree of G , and concludes that they are independent of the choice of reduction tree, and are thereforean invariant of G itself. To deal with multiple edges in E ( G ), we view each element of E ( G )as being distinct. Formally, we may think of assigning a distinguishing number to each copyof a multiple edge. In this way, we may speak of subsets F ⊆ E ( G \
0) in the usual sense.For G any graph on the vertex set [0 , n ], we can still construct the reduction tree T ( G )using the same algorithm as before in Definition 3.2. As in the case of simple graphs, theleaves of this specific reduction tree can be encoded as solutions to some constraint arrays.The key is using a generalized version of Lemma 3.1 with multiple incoming and outgoingedges at vertex v . This generalization is derived the same way and is not harder, but farmore technical. The arrays we obtain are no longer necessarily triangular, but rather theymay be staggered. This is explained below and demonstrated in Examples 6.1 and 6.2. Weleave the proofs to the interested reader; they are straightforward generalizations of those inthe previous section. Triangular arrays
Tri G ( ∅ ) for arbitrary G . For the case of full-dimensional degreesequences, replace each a i, j by a (1) i, j in Definition 3.3 and Theorem 3.4, and add variables a ( k ) i, j with k > j, i ) appearing in G . When there are k > j, i ) ∈ E ( G ), also replace a (1) i, j ≤ a (1) i − , j in the constraint array by a (1) i, j ≤ a (2) i, j ≤ · · · ≤ a ( k ) i, j ≤ a (1) i − , j . The following example demonstrates these changes. Example 6.1.
Following Example 3.5, if G is the graph on vertex set [0 ,
4] with E ( G ) = { (0 , , (0 , , (0 , , (1 , , (1 , , (2 , , (2 , , (3 , , (3 , } , ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 29 we obtain the constraints:0 ≤ a (1)4 , = a (1)3 , = a (1)2 , ≤ a (2)2 , ≤ a (1)1 , = 20 ≤ a (1)4 , ≤ a (1)3 , ≤ a (1)2 , = 5 − a (1)2 , ≤ a (1)4 , ≤ a (2)4 , ≤ a (1)3 , = 6 − a (1)3 , − a (1)3 , ≤ a (1)4 , = 9 − a (1)4 , − a (1)4 , − a (1)4 , Triangular arrays
Tri G ( F ) for arbitrary G . Similarly, we can encode all left-degreesequences by introducing the arrays Tri( F ) used in Theorem 3.8. To do this we view E ( G )as a multiset, so we formally view each copy of a multiple edge ( j, i ) as a distinct element.Let F vary over subsets of E ( G \ G ( F ) from (the general version of) Tri G ( ∅ )as before using the numbers f i, j of (3.1) and treating each a ( m ) i, j identically for different m . Example 6.2.
With G as in Example 6.1 and F = { (1 , , (1 , , (2 , } , the array Tri( F )is given by 2 ≤ a (1)4 , + 2 = a (1)3 , + 2 = a (1)2 , + 2 ≤ a (2)2 , + 2 ≤ a (1)1 , = 21 ≤ a (1)4 , + 1 ≤ a (1)3 , + 1 ≤ a (1)2 , = 3 − a (1)2 , ≤ a (1)4 , ≤ a (2)4 , ≤ a (1)3 , = 3 − a (1)3 , − a (1)3 , ≤ a (1)4 , = 6 − a (1)4 , − a (1)4 , − a (1)4 , Using the definition of Tri G ( F ) for arbitrary graphs G , we can extend the definitions ofSol G ( F ) and LD( G, F ) from simple graphs to arbitrary graphs G . As in Proposition 3.12, foreach F ⊆ E ( G \
0) the polytope Poly(Tri G ( F )) is integrally equivalent to the flow polytopeof a graph Gr( G ), a straightforward generalization of Definiton 3.11. The proofs of Theorem3.14 and its Corollaries then go through with minor changes. In particular, we have thefollowing crucial result. Theorem 6.3.
Let G be a graph on [0 , n ] , ρ be the map that takes a triangular array in any Sol G ( F ) to its first column (cid:16) a (1) n, , . . . , a (1) n, n (cid:17) , and ψ be the map that takes a flow on (cid:101) G \{ s, } to the tuple of its values on the edges { ( j, t ) : j ∈ [ n ] } . For F ⊆ E ( G \ , recall the netflowvector b FG = (indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ) , − E ( G \ F )) . Then for each F ⊆ E ( G \ , LD(
G, F ) = ρ (Sol G ( F )) = ψ (cid:16) F (cid:101) G \{ s, } (cid:0) b FG (cid:1) ∩ Z E ( (cid:101) G \{ s, } ) (cid:17) , so InSeq ( T ( G )) = (cid:91) F ⊆ E ( G \ LD(
G, F )= (cid:91) F ⊆ E ( G \ ρ (Sol G ( F ))= (cid:91) F ⊆ E ( G \ ψ (cid:16) F (cid:101) G \{ s, } (cid:0) b FG (cid:1) ∩ Z E ( (cid:101) G \{ s, } ) (cid:17) In the proof of Theorem A below, it will be more convenient to use an equivalent for-mulation of Theorem 6.3: Instead of considering flows on (cid:101) G \{ s, } with netflow vector b FG ,consider flows on (cid:101) G \{ s } with netflow vector (0 , b FG ), where(0 , b FG ) = (0 , indeg G (1) − outdeg F (1) , . . . , indeg G ( n ) − outdeg F ( n ) , − E ( G \ F )) . Next, we use Theorem 6.3 to prove that for all graphs G on [0 , n ], LD( G ) depends only on G and not on the choice of reduction tree of G as stated in Theorem A. Before proceedingwith the proof, we first recall the relevant notation introduced previously. For a graph G on[0 , n ], let R ( G ) be any reduction tree of G and T ( G ) the specific reduction tree whose leavesare encoded by the arrays Sol G ( F ) (constructed in Definition 3.2). Recall that InSeq( R ( G ))denotes the multiset of left-degree sequences of the leaves of R ( G ). Since LD( G ) was definedas the left-degree sequences of leaves in any reduction tree of G , to show this definition isvalid it suffices to prove that InSeq( R ( G )) = InSeq( T ( G )). Proof of Theorem A.
We proceed by induction on the maximal depth of a reduction tree of G . For the base case, the only reduction tree possible is the single leaf G . For the induction,perform a single reduction on G using fixed edges r = ( i, j ) and r = ( j, k ) with i < j < k toget graphs G , G , and G , with notation as in (2.1). Note that we are selecting particularedges r and r even if there are multiple edges ( i, j ) or ( j, k ). Let r denote the new edge( i, k ) in G m for each m ∈ [3]. Let R ( G m ) be the reduction tree of G m , m ∈ [3], induced from R ( G ) by restriction to the node labeled by G m and all of its descendants.By the induction assumption, InSeq( R ( G m )) is exactly InSeq( T ( G m )), soInSeq( R ( G )) = (cid:91) m ∈ [3] InSeq( R ( G m )) = (cid:91) m ∈ [3] InSeq( T ( G m )) . Thus, we need to show that (cid:91) m ∈ [3] InSeq( T ( G m )) = InSeq( T ( G ))(6.1)regardless of the choice of r and r . However, if ρ is the map that takes an array to its firstcolumn, then Theorem 6.3 implies the disjoint union decompositionsInSeq ( T ( G )) = (cid:91) F ⊆ E ( G \ ρ (Sol G ( F )) , and for each m ∈ [3], InSeq ( T ( G m )) = (cid:91) F ⊆ E ( G m \ ρ (Sol G m ( F ))Thus, to prove (6.1), it suffices to show (cid:91) F ⊆ E ( G \ ρ (Sol G ( F )) = (cid:91) m ∈ [3] (cid:91) F ⊆ E ( G m \ ρ (Sol G m ( F )) . (6.2)To show (6.2), to each F ⊆ E ( G \ F m ) m ∈ I ( F,r ,r ) with I ( F, r , r ) ⊆ [3] and F m ⊆ E ( G m \ m ∈ [3], such that each subset of any E ( G m \
0) is in exactly one
ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 31 tuple and for each F ⊆ E ( G \ ρ (Sol G ( F )) = (cid:91) m ∈ I ( F,r ,r ) ρ (Sol G m ( F m )) . By Theorem 6 .
3, we verify the equivalent condition ψ (cid:16) F (cid:101) G \{ s } (cid:0) , b FG (cid:1) ∩ Z E ( (cid:101) G \{ s } ) (cid:17) = (cid:91) m ∈ I ( F,r ,r ) ψ (cid:16) F (cid:101) G m \{ s } (cid:0) , b FG m (cid:1) ∩ Z E ( (cid:101) G m \{ s } ) (cid:17) . To make the notation more compact, let H = (cid:101) G \{ s } and H m = (cid:102) G m \{ s } for m ∈ [3]. Weproceed in several cases depending on F, r , r . In each case, the argument is very similar tothe proof of Proposition 2.3. I. Suppose that r is not incident to vertex
0. The following four cases deal with this case.
Case 1: r , r / ∈ F : Associate to F the tuple ( F , F ) with F = F and F = F. Let h be an integral flow on H with netflow vector (0 , b FG ). For m ∈ [3], we define integralflows on H m with netflow (0 , b FG m ) having the same image under ψ . • If h ( r ) ≥ h ( r ), define h on H with netflow b F G by h ( e ) = h ( r ) if e = r h ( r ) − h ( r ) if e = r h ( e ) otherwise • If h ( r ) < h ( r ), define h on H with netflow b F G by h ( e ) = h ( r ) if e = r h ( r ) − h ( r ) − e = r h ( e ) otherwiseFor the inverse map, given integral flows h m on H m with netflow b F m G m for m ∈ [2], defineflows h ( m ) on H by h (1) ( e ) = h ( r ) + h ( r ) if e = r h ( r ) if e = r h ( e ) otherwise and h (2) ( e ) = h ( r ) if e = r h ( r ) + h ( r ) + 1 if e = r h ( e ) otherwise Case 2: r ∈ F, r / ∈ F : Associate to F the tuple ( F , F ) with F = F \{ r } ∪ { r } and F = F \{ r } ∪ { r } . Use the same maps on flows given in Case 1.
Case 3: r / ∈ F, r ∈ F : Associate to F the tuple ( F , F , F ) with F = F \{ r } ∪ { r } , F = F, and F = F \{ r } . Let h be an integral flow on H with netflow vector (0 , b FG ). For m ∈ [3], we define integralflows on H m with netflow (0 , b F m G m ) having the same image under ψ . • If h ( r ) > h ( r ), define h on H with netflow b F G by h ( e ) = h ( r ) if e = r h ( r ) − h ( r ) − e = r h ( e ) otherwise • If h ( r ) < h ( r ), define h on H with netflow b F G by h ( e ) = h ( r ) if e = r h ( r ) − h ( r ) − e = r h ( e ) otherwise • If h ( r ) = h ( r ), define h on H with netflow b F G by h ( e ) = (cid:40) h ( r ) if e = r h ( e ) otherwiseGiven integral flows h m on H m with netflows b F m G m for m ∈ [3], construct the inverse mapby defining flows h ( m ) on H for m ∈ [3]. Let h (2) be the same as in Case 1, and define h (1) ( e ) = h ( r ) + h ( r ) + 1 if e = r h ( r ) if e = r h ( e ) otherwise and h (3) ( e ) = h ( r ) if e = r h ( r ) if e = r h ( e ) otherwise Case 4: r , r ∈ F : Associate to F the tuple ( F , F , F ) with F = F \{ r } ∪ { r } , F = F \{ r } ∪ { r } , and F = F \{ r , r } ∪ { r } . Use the maps on flows given in Case 3.A straightforward check shows that every F ⊆ E ( G m \
0) for m ∈ [3] is reached exactlyonce by cases 1-4 . II. Suppose that r is incident to vertex
0. The following two cases deal with this case.
Case 1’: r / ∈ F : Associate to F the tuple ( F , F ) with F = F and F = F. Use the maps on flows given in Case 1.
Case 2’: r ∈ F : Associate to F the tuple ( F , F ) with F = F and F = F \{ r } Use the maps on flows for H and H given in Case 3.A straightforward check shows that every F ⊆ E ( G m \
0) for m ∈ [3] is reached exactlyonce by cases 1’-2’ . (cid:3) Acknowledgements
We thank Bal´azs Elek, Alex Fink and Allen Knutson for inspiring conversations.
ENERALIZED PERMUTAHEDRA TO GROTHENDIECK POLYNOMIALS VIA FLOW POLYTOPES 33
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