Bessenrodt-Stanley polynomials and the octahedron recurrence
aa r X i v : . [ m a t h . C O ] J un BESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRONRECURRENCE
PHILIPPE DI FRANCESCO
Abstract.
We show that a family of multivariate polynomials recently introduced byBessenrodt and Stanley can be expressed as solution of the octahedron recurrence withsuitable initial data. This leads to generalizations and explicit expressions as path or dimerpartition functions.
Contents
1. Introduction 22. A family of Laurent polynomials associated to Young diagrams 33. A ∞ / T -system and its solutions 53.1. T -system and initial data 53.2. Solution 53.3. Network interpretation 84. Application: computation of the Laurent polynomials p a,b ( θ · ) 104.1. Steepest stepped surface 104.2. Connection with the Laurent polynomials p a,b ( θ · ) 104.3. Network interpretation 124.4. Dimer interpretation 164.5. Back to the polynomials of Bessenrodt and Stanley 205. 3D Generalization 215.1. The general solution of the T -system 215.2. Pyramid of a partition 225.3. The family of Laurent polynomials for a pyramid 225.4. Generalized Bessenrodt-Stanley polynomials for a pyramid 246. Conclusion/Discussion 24References 28 Date : July 20, 2018. Introduction
In a recent publication, Bessenrodt and Stanley [2] introduced a family of multivariatepolynomials attached to any partition λ , generalizing a construction by Berlekamp [1].These were defined as weighted sums over sub-diagrams of the Young diagram of λ .In this paper, we show that these polynomials may be viewed as the restriction of par-ticular solutions of the octahedron recurrence relation in a three-dimensional half-space.The octahedron recurrence is a special case of so-called T -system, introduced in the con-text of generalized Heisenberg integrable quantum spin chains with Lie group symmetries[12, 13]. The octahedron recurrence has received much attention over the last decade forits combinatorial interpretation in terms of domino tilings of the Aztec diamond [4] andgeneralizations thereof [15, 11, 6, 7, 8, 9]. More generally, the A type T -systems possessthe positive Laurent property: their solutions may be expressed as Laurent polynomials ofany admissible initial data, with non-negative integer coefficients. In Ref.[5], this propertyof the T -systems was connected to an underlying cluster algebra structure [10], and furtherconfirmed by providing an explicit solution based on some representation using a flat two-dimensional connection [6], leading to expressions as partition function of weighted pathson oriented graphs or networks, or equivalently of weighted dimer coverings of suitablebipartite planar graphs.Our punchline is the following: the polynomials of Bessenrodt and Stanley were shownto obey particular determinantal identities [2], which we interpret as initial conditions forthe half-space octahedron recurrence, for which the partition λ determines the geometry ofthe initial data. Reversing the logic, and fixing the values of these determinants, we mayexpress the solutions of the octahedron recurrence as Laurent polynomials thereof, as pathor dimer partition functions from [6, 8]. This produces a general family of multivariateLaurent polynomials attached to any partition, that reduce to the polynomials of [2], forspecial choices of the initial data.The paper is organized as follows. In Section 2 we describe our new family of multivariateLaurent polynomials attached to a partition λ , and how they reduce to the polynomials of[2]. Section 3 is devoted to a survey of the T -system and its general solutions in terms ofnetwork partition functions. We describe in particular the structure of admissible initialdata, which take the form of initial value assignments along a “stepped surface”, andshow how this data encodes an oriented weighted graph or network. In Section 4, weshow that the situation of [2] corresponds to choosing a particular “steepest” initial datastepped surface, that forms a kind of fixed slope roof above the Young diagram of λ . Thecorresponding network is particularly simple, as it takes the shape of the Young diagramitself. Using the explicit solutions of the T -system, we write the solutions first as networkpartition functions (Theorem 4.2, Sect.4.3) and then as dimer partition functions (Theorem4.4, Sect.4.4). Section 5 is devoted to a 3D generalization of the multivariate polynomials,by simply expressing the other solutions of the T -system “under the roof”. This leads us ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 3 to a 3D object we call the pyramid of λ , to the boxes of which we associate other Laurentpolynomials. The latter reduce to polynomials when we apply the previous restriction ofinitial data. These are first expressed as partition functions for families of non-intersectingpaths on the same networks as before (Sect.5.3), and then shown to restrict to sums overnested sub-partitions of λ with the same weights as before (Theorem 5.5, Sect.5.4). Wegather a few concluding remarks in Section 6, where we present a different generalizationfor other boundary conditions of the octahedron recurrence. Acknowledgments . We would like to thank R. Stanley for an illuminating seminar duringthe conference “Enumerative Combinatorics” at the Mathematisches ForschungsInstitutOberwolfach in March 2014 and A. Sportiello for his great help in the early stage of thiswork. This work is supported by the NSF grant DMS 13-01636 and the Morris and GertrudeFine endowment.2.
A family of Laurent polynomials associated to Young diagrams
We consider partitions/Young diagrams of the form λ = ( λ , ..., λ N ) with λ i boxes in row i , and λ ≥ λ ≥ · · · ≥ λ N ≥
1. We represent the Young diagram λ with rows 1 , , ..., N from top to bottom and justified on the left. The boxes of the diagram λ are labeled bytheir coordinates ( i, j ), where i ∈ [1 , N ] is the row number and j ∈ [1 , λ i ] the horizontalcoordinate within the i -th row. We write ( i, j ) ∈ λ when the box ( i, j ) belongs to λ . Forlater use, for any ( a, b ) ∈ λ , we define the sub-diagram λ a,b ⊂ λ obtained by erasing all theboxes of λ that are above the row a and to the left of the column b (it is the intersectionof the South East corner from ( a, b ) with λ ).We also consider the extended Young diagram λ ∗ associated to λ , obtained by adjoininga border strip from the end of the first row to the end of the first column of λ , namelywith λ ∗ = λ + 1, λ ∗ i = λ i − + 1, i = 2 , ..., N + 1. For instance the extended diagram of λ = (3 ,
2) is λ ∗ = (4 , , λ can be recovered from λ ∗ by removing its firstrow and column. We define the squares S a,b of λ ∗ to be the square arrays of the form S a,b = { ( i, j ) , a ≤ i ≤ n a,b + a − , b ≤ j ≤ n a,b + b − } , and such that S a,b ⊂ λ ∗ and n a,b ≥ a, b ) is called the North West (NW) corner of the square S a,b . In other words, the square S a,b is the largest square array of boxes with NW corner( a, b ) that fits in λ ∗ . The integer n a,b is called the size of S a,b . In particular, each box ( u, v )of the border strip λ ∗ \ λ is itself a square of size 1, S u,v = { ( u, v ) } . Definition 2.1.
We fix arbitrary non-zero parameters { θ i,j } ( i,j ) ∈ λ ∗ attached to the boxes of λ ∗ . We now associate to each box ( i, j ) ∈ λ ∗ a function p i,j ≡ p i,j ( θ · ) of all the parameters θ i,j defined by the identities: (2.1) det (cid:16) ( p a + i − ,b + j − ( θ · )) ≤ i,j ≤ n a,b (cid:17) = θ a,b for all ( a, b ) ∈ λ ∗ PHILIPPE DI FRANCESCO namely we fix the value of the determinant of the array of functions p i,j ( θ · ) on each square S a,b to be the parameter θ a,b . In particular, we have p u,v ( θ · ) = θ u,v for all ( u, v ) ∈ λ ∗ \ λ . That (2.1) determines the p i,j ’s uniquely will be a consequence of the rephrasing of the problem as that of findingthe solution of the T -system or octahedron recurrence, subject to some particular initialcondition. As a consequence of the positive Laurent property of the T -system, we have thefollowing main result: Theorem 2.2.
For all ( a, b ) ∈ λ , the function p a,b ( θ · ) is a Laurent polynomial of the θ i,j ’swith non-negative integer coefficients. Example 2.3.
Let us consider the Young diagram λ = (2 , , with the following θ variables(one per box of λ ∗ = (3 , , ): θ , = a θ , = b θ , = cθ , = d θ , = e θ , = fθ , = g θ , = h The polynomials p a,b ( θ · ) are: p , = afh +( b + ce )( d + eg ) efh p , = b + cef p , = cp , = d + egh p , = e p , = fp , = g p , = h The Laurent polynomials p a,b ( θ · ) attached to the Young diagram λ reduce to the poly-nomials introduced by Bessenrodt and Stanley in [2] when the variables θ i,j are restrictedas follows. Let x i,j , ( i, j ) ∈ λ be new variables attached to the boxes of λ . Theorem 2.4.
Under the restrictions: θ i,j = 1 for all ( i, j ) ∈ λ ∗ \ λ ,θ a,b = n a,b − Y r =0 Y ( i,j ) ∈ λ a + r,b + r x i,j for all ( a, b ) ∈ λ , (2.2) the Laurent polynomials p a,b ( θ · ) of Def.2.1 reduce to the polynomials p a,b ( x · ) of [2] . In particular, the change of variables cancels all denominators and produces a polynomial of the variables x i,j .In Section 4, we shall construct each polynomial p a,b ( θ · ) explicitly, as the partition func-tion for paths on a weighted oriented graph (network) N a,b associated to λ a,b , and alterna-tively as the partition function of the dimer model on a suitable bipartite graph G a,b . Wegive two independent proofs of Theorem 2.4 in Sect.4.5. ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 5 A ∞ / T -system and its solutions T -system and initial data. In the case of A type Lie algebras, the T -system takesthe form (also known as the octahedron recurrence):(3.1) T i,j,k +1 T i,j,k − = T i +1 ,j,k T i − ,j,k + T i,j +1 ,k T i,j − ,k where the indices of the indeterminates T i,j,k are restricted to be vertices of the Face-Centered Cubic (FCC) lattice L F CC = { ( i, j, k ) ∈ Z , i + j + k = 1 mod 2 } . This may beviewed as a 2+1-dimensional evolution in the discrete time variable k , while i, j refer tospace indices.The A r condition consists in further restricting i ∈ [1 , r ], and to impose the additionalboundary conditions(3.2) T ,j,k = T r +1 ,j,k = 1 ( j, k ∈ Z )In other words, the A r T -system solutions are those of the octahedron equation in-betweentwo parallel planes i = 0 and i = r + 1, that take boundary value 1 along the two planes i = 0 and i = r + 1.In the following, we will concentrate on the so-called A ∞ / T -system, where we only keepthe restriction i ≥ T ,j,k = 1In turn, the solutions of the A ∞ / T -system are those of the octahedron equation in thehalf-space i ≥
0, that take boundary values 1 along the plane i = 0.The system (3.1,3.3) must be supplemented by some admissible initial data ( k , t ), con-sisting of: • (1) a “stepped surface” k = { ( i, j, k i,j ) } i ∈ Z + ; j ∈ Z , such that | k i +1 ,j − k i,j | = | k i,j +1 − k i,j | = 1 for all i, j ; • (2) the following assignments of initial values t = { t i,j } i ∈ Z > ; j ∈ Z :(3.4) T i,j,k i,j = t i,j ( i ∈ Z > ; j ∈ Z )3.2. Solution.
In Ref. [6] the A r T -system was solved for arbitrary r and arbitrary initialdata. Note that for any finite ( i, j, k ) above the initial data stepped surface (i.e. k ≥ k i,j ),the solution T i,j,k only depends on finitely many initial values t i,j , hence for r large enoughthe solution is independent of r , so that the solution for the case A ∞ / is trivially obtainedfrom that of A r . We now describe this solution.The solution proceeds in two steps. First, one eliminates all the variables T i,j,k for i ≥ T j,k := T ,j,k by noticing that the relation (3.1) is nothing but theDesnanot-Jacobi (aka Dodgson condensation) formula for the following Hankel or discreteWronskian determinants:(3.5) W i,j,k = det (cid:16) ( T j + b − a,k + i +1 − a − b ) ≤ a,b ≤ i (cid:17) PHILIPPE DI FRANCESCO which, together with the initial condition W ,j,k = 1 allows to identify(3.6) T i,j,k = W i,j,k for all i ≥ T j,k for all k ≥ k ,j , by use ofa formulation of the relation (3.1) as the flatness condition for a suitable GL connection.The solution is best expressed in the following manner. First we associate to the initial datastepped surface k a bi-colored (grey/white) triangulation with the vertices of k , defined viathe following local rules, where we indicate the value of k i,j at each vertex of the projectiononto the i, j plane:(3.7) kk k−1ji k+1 k k k−1 k−1 k k k+1 k+1 k k−1k kk k−1 kkk+1 kk k+1 kkk−1 kk k+1 The last two cases give rise to two choices of triangulations each, due to the tetrahedronambiguity (there are two ways of defining a pair of adjacent triangles with the verticesof a regular tetrahedron, namely the two choices of diagonals of the white/grey square inprojection), but our construction is independent of these choices. We may now decomposethe stepped surface into lozenges made of a grey and a white triangle sharing an edgeperpendicular to the k axis, and we supplement the single triangles of the bottom layer i = 1 , i = 0 plane. Example 3.1.
Let us consider the “flat” initial data surface k with k i,j = i + j + 1 mod 2 ∈{ , } . Picking a particular choice of diagonal in the tetrahedron ambiguities, we maydecompose the surface as follows: . . . . . . . . . i ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 7 where we have represented with solid (resp empty) dots the vertices with k = 1 (resp. k = 0 ). To each lozenge, we associate a 2 × bca → U ( a, b, c ) = (cid:18) cb ab (cid:19) bca → V ( c, a, b ) = (cid:18) ab cb (cid:19) Note that the arguments a, b, c of the matrices are the values of initial data attached to thethree vertices of the grey lozenge. These matrices have the remarkable property that theyform a flat connection on the solutions of the T -system, namely we have(3.9) V ( u, a, b ) U ( b, c, v ) = U ( a, x, v ) V ( u, x, c ) iff xb = ac + uv For some fixed integer r (large enough), we may embed the above matrices in r -dimensionalspace, by defining the ( r + 1) × ( r + 1) matrices U i , V i , equal to the ( r + 1) × ( r + 1) identitymatrix, with the central 2 × i, i +1 replaced by U, V . Theposition i corresponds to the i coordinate of the two middle vertices of the correspondinglozenge. We now associate to each “slice” S = [0 , r + 1] × [ j , j ] of the initial data surfacenamely with vertices { ( i, j, k i,j ) i,j ∈ S } the matrix M ( j , j ) equal to the product over alllozenge matrices of the slice, taken in the order of appearance of the lozenges from left toright. This matrix is independent of the choices pertaining to the tetrahedron ambiguityabove.Given any j, k with k ≥ k ,j , let us define the left (resp. right) projections j (resp. j )onto the stepped surface k to be the largest (resp. smallest) integer ℓ such that j − ℓ = k − k ,ℓ (resp. j − ℓ = k ,ℓ − k ). This is illustrated below: . . . . . . k (j,k)j k j j PHILIPPE DI FRANCESCO where we have represented by empty dots the vertices of the intersection of the steppedsurface k with the i = 1 plane. Finally the solution T j,k may be written as:(3.10) T j,k = M ( j , j ) , t ,j independently of r for r large enough. The proof in [6] relies on the flatness condition(3.9). Note that the positive Laurent property for T j,k as a function of the initial data { t i,j } is manifest, as the entries of the matrices U, V are themselves Laurent monomials of theinitial data with coefficient 1 or 0.3.3.
Network interpretation.
The matrix M ( j , j ) can be interpreted as the weightedadjacency matrix of some oriented graph N ( j , j ) (referred to as a “network”), constructedas follows. First interpret the matrices U i , V i as elementary “chips”:(3.11) U i ( a, b, u ) = ibi ai+1 i+1u V i ( v, a, b ) = va bi ii+1 i+1 with two entrance connectors i, i + 1 on the left and two exit connectors i, i + 1 on theright, and such that the ( x, y ) matrix element is the weight of the oriented edge form entryconnector x to exit connector y (here by convention all edges will be oriented from leftto right). Note we have represented by a dashed line the “trivial” oriented edges withweight 1. The top edge in the U i ( a, b, u ) chip has therefore weight a/b , the diagonal one u/b , etc. The arguments a, b , etc. appear as face variables in the network. Any productof such matrices can be interpreted as a larger network, obtained by concatenating thecorresponding chips. We call N ( j , j ) the network corresponding to the matrix M ( j , j ).Now we may interpret the matrix element M ( j , j ) , as the partition function for paths on N ( j , j ) from the leftmost entry connector 1 to the rightmost exit connector 1. This give anice explicit combinatorial description of the solution T j,k in terms of the initial data { t i,j } ,which form the face labels of the network N ( j , j ). This is summarized in the following: Theorem 3.2.
The solution T ,j,k = T j,k of the A ∞ / T -system with initial data ( k , t ) is t ,j times the partition function of paths from the entry connector to exit connector onthe network N ( j , j ) , where j , j are the left/right projections of ( j, k ) onto the onto thestepped surface k . In [6], it was further shown that T i,j,k for i ≥ i non-intersecting paths on the network N ( j , j ). As such, it enjoys the positive Laurentproperty as well. ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 9
Example 3.3.
The network N corresponding to a sample slice S of flat initial data surface(see Example 3.1) reads: e 1 e111 dcbadca b . . . i where we have represented the face labels in a sample row. Note also that due to the A ∞ / boundary condition (3.3) , all the vertex/face values on the bottom row/lower face are equalto . The quantity T j,k /T j ,k ,j is the partition function for paths from entry connector toexit connector on the network. We show here a sample such path, together with its localstep weights: yx e/jd/ey/dx/y1g/c1a b c d e1f g h i j The total contribution of this path to the partition function is therefore: g gc yx xd de ej = xcj Application: computation of the Laurent polynomials p a,b ( θ · )In this section, we show that the functions p a,b ( θ · ) defined by the system (2.1) are thesolutions T j,k = T ,j,k of the A ∞ / T -system with some particular initial conditions andsome particular mapping of indices ( a, b ) → ( j, k ).4.1. Steepest stepped surface.
In this section we consider stepped surfaces with fixedintersection with the i = 1 plane, say equal to a path π = ( j, k j ) with | k j +1 − k j | = 1 and j + k j = 0 mod 2 for all j ∈ Z . We define the “steepest stepped surface” k π to be theunique stepped surface k with k ,j = k j included in the union of all the planes with normalvector (1 , − , −
1) or (1 , , −
1) and passing through pairs of distinct points of π . It is easyto see that such a surface is piecewise-linear. Let ( a j , b j = k a j ), j ∈ Z be the vertices where π changes direction, with say a minimum when j is even, and a maximum when j is odd.The steepest stepped surface k = k i,j is defined by the following equations for m ∈ Z : i − j − k i,j = 1 − a m − b m ( i ≥ a m − ≤ j ≤ a m ) i + j − k i,j = 1 + a m − b m ( i ≥ a m ≤ j ≤ a m +1 )Concretely, the steepest stepped surface is a sort of roof of fixed slope above the infiniteYoung diagram delimited by the path π in the i = 1 plane (see Fig.1 for a 3D view inperspective).4.2. Connection with the Laurent polynomials p a,b ( θ · ) . Let us represent the centersof boxes of the Young diagram λ ∗ as vertices of the i = 1 plane of the lattice L F CC (seeFig. 1 for an illustration). This allows to identify the strip λ ∗ \ λ as a path p from (0 , N + λ , N − λ ), while the NW corner box of the diagram has coordinates ( N, N ) (seesketch below): k j λ λ λ N p π (0,0) (N,N) (N+ ,N− ) ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 11 i=1 i=3 h ta b cdg ei f jki
Figure 1.
The FCC lattice representation of the Young diagram λ = (5 , , , , i = 1 plane) and its augmented tableau λ ∗ (extra green verticesin the i = 1 plane), together with the initial data steepest stepped surface (greenvertices, including λ ∗ \ λ ). We have indicated a particular vertex of the surface inthe plane i = 3, with assigned initial value t , and displayed the pyramid of whichit is the apex, together with its base in the i = 1 plane, identified as a squareof size 3 of λ ∗ . The initial data assignment amounts to imposing that the 3 × T in this square is equal to the value t at the apex. Let us extend arbitrarily the path p into a path π on the entire plane, and consider thesolutions T j,k of the A ∞ / T -system with initial data stepped surface equal to the steepestsurface k π associated to π , and with initial values t i,j . To avoid confusion, note that thereare distinct yet natural frames used so far for expressing the coordinates of the centers ofthe boxes of λ ∗ . On one hand, we have the original frame, in which the coordinate ( a, b )refers to the box in row a ∈ [1 , N ] and column b ∈ [1 , λ a ]. On the other hand we have the T -system (or FCC lattice) frame in the plane i = 1, in which the coordinate ( j, k ) of thecenter of a box is related to that in the original frame by:(4.1) j = λ + a − b, k = λ + 2 − a − b In the following, we will use letters a, b for the original coordinates, and j, k for the FCClattice coordinates.We have the following:
Theorem 4.1.
The system (2.1) has a unique solution p a,b ( θ · ) , ( a, b ) ∈ λ . Moreover, wehave (4.2) p a,b ( θ · ) = T a − b + λ ,λ +2 − a − b (cid:0) ( a, b ) ∈ λ ∗ (cid:1) where T j,k = T ,j,k is the solution of the A ∞ / T -system with steepest initial data steppedsurface k π and initial values t such that (4.3) θ a,b = t a − b + λ , λ − a − b + ka − b + λ (cid:0) ( a, b ) ∈ λ ∗ (cid:1) where ( j, k j ) are the vertices of the path p , for j = 0 , , ..., λ + N .Proof. Consider the solution T i,j,k of the A ∞ / T -system with steepest initial data steppedsurface k π and initial values t . Let us restrict our attention to the solutions T j,k = T ,j,k at the points ( j, k ) with left and right projections on the interval [0 , λ + N ]. These areexactly the centers of the boxes of λ ∗ in the above representation (see Fig.1 for the case λ = (5 , , , , j + b − a, k + i + 1 − a − b ) for 1 ≤ a, b ≤ i are the coordinates in the plane x = 1 of the base of thepyramid Π i,j,k with apex ( i, j, k ), defined as Π i,j,k = { ( x, y, z ) ∈ L F CC , | y − j | + | z − k | ≤| x − i |} . As illustrated in Fig.1, each vertex ( i, j, k i,j ) of the steepest surface k π is the apexof such a pyramid Π i,j,k i,j . By definition of the steepest stepped surface, the base of thepyramid Π i,j,k i,j in the x = 1 plane is the square S a,b of λ ∗ , with NW corner at position( j, k − k j ) = ( a − b + λ , λ + 2 − a − b ) (and SE corner at position ( j, k j ) on the path p ),hence a = 1 − k i,j + j + k j and b = λ + 1 + k i,j + k j − j . We conclude that T i,j,k i,j = t i,j is thedeterminant of the array T ℓ,m for ( ℓ, m ) in the square S a,b . In the example of Fig.1 (left),this amounts to the identity t = (cid:12)(cid:12)(cid:12)(cid:12) a b cd e fg h i (cid:12)(cid:12)(cid:12)(cid:12) .Let us identify t i,j ≡ T i,j,k i,j with θ a,b , for all ( a, b ) ∈ λ ∗ , with a = 1 − k i,j + j + k j and b = λ + 1 + k i,j + k j − j . Then the system of equations (2.1) for p a,b becomes the same asthat for T j,k , with j = a − b + λ and k = λ + 2 − a − b . As all T j,k are uniquely determinedby the initial data, then so are the p a,b , and the theorem follows. (cid:3) Network interpretation.
We may now specialize the general solution of Sect.3 tothe case of the steepest stepped surface. Let us describe the extended Young tableau λ ∗ via the sequence of integers n , m , n , m , ..., n k , m k corresponding to the length of straight ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 13 portions of the path p delimiting the diagram, namely: N+1 n m n k m k λ +1 ...... n m or in the notations of Sect. 4.1: n i = a i − a i − , m i = a i +1 − a i , i = 1 , , ..., k , where a , a , ..., a k are the j coordinates in the FCC lattice of the changes of slope of p .A drastic simplification occurs in the case of the steepest stepped surface: along eachsteepest plane, only one type of lozenge U or V occurs, namely planes orthogonal to(1 , − , −
1) have a lozenge decomposition using only V type lozenges, while those orthog-onal to (1 , , −
1) have a lozenge decomposition using only U type lozenges. Moreoverthe slice of surface corresponding to T i,j,k with lower/upper projections j , j always startswith a (1 , − , − ⊥ plane and ends with a (1 , , − ⊥ one. The corresponding lozenge decomposition reads typically like: n n k m m k j j n ... (j,k) jj where we have only included the minimal number of V or U type lozenges in each slice(any extra U, V would have no effect on the corresponding matrix element M ( j , j ) , ).Recall that the vertices of the surface carry initial data assignments t i,j , in bijection withthe θ a,b parameters, while the bottom layer at i = 0 carries values all equal to 1. ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 15
We may now construct the network associated to the lozenge decomposition above. Forour running example, it reads: cbabc ba where we have indicated how to deform the original network graph to bring it to thesquare lattice with oriented edges. This latter graph is denoted by L ( j , j ) in FCC latticelanguage. Assuming that the top inner face of L ( j , j ) corresponds to the box ( a, b ) of λ , we denote alternatively this network by N a,b ≡ L ( j , j ). We have indicated the facevariables of this network by green dots (the top vertex carries the variable θ a,b ), while allvariables on the (bottom-most) red dots are equal to 1. The medallion summarizes theweighting rule for the edges of L ( j , j ) = N a,b in terms of the face variables t i,j = θ a,b (atthe indicated vertices). This is just a rephrasing of the weights of U and V type chips afterthe above deformation, namely:(4.4) U = a abcbbc a c b , V = b abcbc ba ca The graph L ( j , j ) = N a,b is nothing but the actual initial Young diagram λ a,b (SE cornerat ( a, b )) represented tilted by 45 ◦ , and with all box edges oriented from left to right. Theface variables, expressed indifferently as t i,j ’s or θ a,b ’s, are actually at the centers of boxesof λ ∗ a,b but displaced by a global translation of (2 ,
0) in the ( j, k ) plane (or by one columnto the left and one row up in the original frame). For convenience we denote by λ a,b thislatter displaced array. Note that extra variables equal to 1 occupy the centers of boxes of λ ∗ a,b \ λ a,b . In the above depiction, we have λ a,b = (5 , , , ,
2) and λ ∗ a,b = (6 , , , , , By use of Theorems 3.2 and 4.1, the value of p a,b = T j,k in the top box is given by thepartition function for paths on L ( j , j ) = N a,b from the leftmost vertex to the rightmostone, multiplied by the bottom right initial value t ,j = θ a,λ ∗ a = θ a,λ a +1 . We summarize thisresult in the following: Theorem 4.2.
The solution p a,b to the system (2.1) is the partition function of paths on theweighted graph N a,b associated to λ a,b , from the leftmost to the rightmost vertex, multipliedby the variable θ a,λ a +1 of the top right box of λ ∗ a,b . Note that Theorem 2.2 follows from this result, as the sum over paths produces a mani-festly positive Laurent polynomial of the initial parameters θ i,j . Example 4.3.
Let us revisit Example 2.3. The graph L ( j , j ) = N , for the calculationof p , is associated to λ , = λ = (2 , , and reads: efab bc
11 11 1 1 1 1ag hd e fb c1 1
1h 1fgh hede where we have indicated the edge weights. The Laurent polynomial p , is c times thepartition function for the 5 paths → : path :weight : gefhc dfh gbfhc bdhefc ace wich yields p , = af h + ( b + ce )( d + ge ) hef in agreement with the expression of Example 2.3. Dimer interpretation.
The network formulation for the solution of the T -systemdescribed in Sect.3 can be rephrased in terms of a statistical model of dimers on a planarbipartite graph G with face variables, made only of square, hexagonal and octagonal inner ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 17 faces, and with open outer faces adjacent to 1 or 2 edges (see Ref.[8]). This graph isconstructed as the dual of the lozenge decomposition, essentially by substituting: bc ba ca and bbc caa
Note that the initial data assignments become face variables in the dimer graph. Theconfigurations of the dimer model on a bipartite graph are obtained by covering singleedges of the graph by “dimers” in such a way that each vertex is covered exactly once. Theweight of a given configuration is the product of local outer/inner face weights expressedin terms of the attached face variable. The weight of an inner face is a v − − D where a isthe face variable, v the degree of the face ( v ∈ { , , } ), and D the total number of dimersoccupying edges bordering the face. The weight of an outer face is b − D , where b is theface variable and D the total number of dimers occupying edges of the graph adjacent tothe face. Then we have: Theorem 4.4. [8]
The solution T ,j,k ≡ T j,k of the A ∞ / T -system is the partition functionof the dimer model on the dimer graph dual to the lozenge decomposition of the correspond-ing network. For the particular case of the steepest stepped surface, the dimer graph G a,b for thecomputation of p a,b is particularly simple. Its inner faces occupy a domain of the hexag-onal (honeycomb) lattice with the shape of the young diagram λ a,b , while its outer facescorrespond respectively to λ ∗ a,b \ λ a,b with face variables all equal to 1, and to λ a,b \ λ a,b .The other face variables are the variables θ α,β on λ a,b . For the case λ a,b = (5 , , , ,
2) ofprevious section, the dimer graph G a,b reads: where we have represented in red the centers of boxes of λ ∗ a,b \ λ a,b (all with assigned facevalues 1), and in green the centers of boxes of λ a,b (with the θ α,β ’s or t i,j ’s as assigned facesvalues).Applying Theorem 4.4, we finally get: Theorem 4.5.
The Laurent polynomial p a,b ( θ · ) is the partition function for the dimer modelon the graph G a,b with face variables θ α,β on the faces corresponding to the boxes of λ a,b andface variables on those corresponding to the boxes of λ ∗ a,b \ λ a,b . Example 4.6.
Let us revisit the example 2.3. The graph G , for computing p , reads:
1a b cg dh e f1 1 1 1 and the partition function for dimers on G , is the sum over the following five configura-tions: dimerconfiguration :
1a b cg dh e f1 1 1 1 1a b cg dh e f1 1 1 1 1a b cg dh e f1 1 1 1 1a b cg dh e f1 1 1 1 1a b cg dh e f1 1 1 1 weight : gefh dcfh gbfh bdefh ae
We note that Theorems 4.2 and 4.4 may be connected more directly by showing thatthe path and dimer configurations are in bijection with each-other. To best see this, recallthat the dimer configurations on a domain of the hexagonal lattice are in bijection withrhombus tilings of the dual (triangular) lattice, by means of three types of rhombi obtainedby gluing two adjacent triangles along their common edge. For the above example of
ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 19 λ a,b = (5 , , , , T a,b of the triangular lattice dual to G a,b reads:Notice that generically the domain T a,b only has two vertical boundary edges, dual to theonly two horizontal external edges of G a,b . Dimer configurations on G a,b are in bijectionwith rhombus tilings of T a,b . Moreover, such tilings are uniquely determined by either ofthree sets of non-intersecting paths of rhombi (the so-called De Bruijn lines) defined asfollows. Each boundary edge of T a,b has either of three orientations (vertical, +30 ◦ , or − ◦ ). Starting from any boundary edge, let us construct the chain of consecutive rhombithat share only edges of the same orientation. Such a chain is a path connecting twoopposite boundary edges. For a given orientation of the boundary edge, all such pathsare non-intersecting, and form one of the above-mentioned three families. Any single suchfamily determines the tiling entirely. In the present case, the “vertical” family is particularlysimple, as it is made of a single path of rhombi:This path determines the tiling entirely. We have therefore associated to each dimer con-figuration on G a,b a unique path with up and down steps (say from left to right), whichcan be represented on the network L ( j , j ) = N a,b as a path from the leftmost vertex tothe rightmost one. This gives a bijection between the dimer configurations of Theorem 4.4and the paths on networks of Theorem 4.2. It is easy to show how the local weights of thepath model can be redistributed into those of the dimer model, thus establishing directlythe equivalence between the two theorems. Back to the polynomials of Bessenrodt and Stanley.
In this section we givetwo independent proofs of Theorem 2.4.As pointed out earlier, the polynomials of [2] may be recovered by specializing the vari-ables θ i,j according to (2.2). More precisely, the (restricted) polynomials p a,b ( x · ) of [2] aredefined by the formula:(4.5) p a,b ( x · ) = a, b ) ∈ λ ∗ \ λ P µ ⊂ λ a,b Q ( r,s ) ∈ λ a,b \ µ x r,s if ( a, b ) ∈ λ where in the second formula the sum extends over all sub-diagrams µ of λ a,b . In [2], thedeterminant of the array p a + i − ,b + j − ( x · ), 1 ≤ i, j ≤ n a,b pertaining to the square S a,b of λ with NW corner ( a, b ), was computed to be equal to the “leading term”:(4.6) Z a,b = n a,b Y i =1 Y ( α,β ) ∈ λ a + i − ,b + i − x α,β , equal to the product of leading terms of p a + i − ,b + i − ( x · ) along the first diagonal. Withthe choice of restrictions (2.2) which identify θ a,b with Z a,b , and by uniqueness of the T -system solution, we deduce that the polynomials p a,b ( θ · ) defined by the T -system solutionare identical to the polynomials p a,b ( x · ) of [2], and Theorem 2.4 follows.Let us now give an alternative direct proof of this result, by comparing the expression(4.5) to the restriction of the network expression of Theorem 4.2 for the solution of the T -system with steepest initial data stepped surface. The first part of the formula (4.5) is clearfrom the choice θ a,b = 1 for ( a, b ) ∈ λ ∗ \ λ . To recover the second part, first note that there isa bijection between the sub-diagrams µ ⊂ λ a,b and the paths from the leftmost vertex to therightmost vertex on the network N a,b . To identify the polynomials, we simply have to checkthat the weight of each path, multiplied by the rightmost face variable ( t ,j ≡ θ a,λ ∗ a = 1here) reduces to Q ( r,s ) ∈ λ a,b \ µ x r,s , namely the product of x variables under the path in λ a,b .To best compare the two settings, let us translate the network L ( j , j ) globally by thevector by (0 , , −
2) in the L F CC representation (namely k → k − λ a,b . This changes the local edge weightrules accordingly (by moving the face variables by two steps downwards). In this newrepresentation, the local edge weights of the path model can now be written in terms of ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 21 the x variables as: b b (b) a (a) a where we have shown the two possible cases of an up or down-pointing edge of the path,both with weight a/b , and represented the weight in terms of the box variables x as follows:in case (a) (up step), the weight a/b is the inverse of the product of the x ’s in the dashed(blue) domain, while in case (b) (down step), the weight a/b is the product of the x ’s inthe solid (green) domain. We deduce that only boxes below the path delimiting µ in λ a,b contribute. Moreover, a given such box with variable x receives a contribution x N down − N up ,where N up and N down denote the total number of up/down steps of the path that respectivelybelong to the left/right sector seen from the box as depicted below: x It is clear that we always have N down − N up = 1 as the path always goes up one step less inthe left sector than it goes down in the right sector. The total weight of the path delimiting µ in λ a,b is therefore the product over all the boxes below µ of the box variables x , and theTheorem follows. 5.
3D Generalization
The general solution of the T -system. So far we have concentrated on the solution T j,k = T ,j,k of the T -system in the i = 1 plane. The general solution of the T -system givesaccess more generally to values of T i,j,k in other planes i ≥ i non-intersecting paths on the same type of network as for i = 1. More precisely, we first consider the base of thepyramid Π i,j,k , which is a square array of T i + b − a,j + i +2 − a − b , a, b = 1 , ..., i . We then constructall left and right projections of the points in this array, say ℓ , ..., ℓ i and r i , r i − , ..., r fromleft to right. Then the solution T i,j,k is given by the following: Theorem 5.1. [6]
The solution T i,j,k of the A ∞ / T -system with initial data ( k , t ) is equalto the partition function of i non-intersecting paths on the network N ( ℓ , r ) that start fromthe points ( ℓ a , k ℓ a ) on the stepped surface k and end at the points ( r a , k r a ) on the steppedsurface k , multiplied by the boundary term: Q ia =2 t − ,ℓ a Q ib =1 t ,r a . Pyramid of a partition.
Starting from a partition/Young diagram λ = ( λ , ..., λ N ),we define its pyramid P λ as the family of partitions/Young diagrams λ ( i ) , i = 1 , , ..., k ,such that (i) λ (1) = λ (ii) λ ( i +1) is obtained from λ ( i ) by removing its first row and column(iii) λ ( k ) = ∅ .The centers of boxes of the pyramid P λ may be represented as the set of vertices ( i, j, k )in L F CC such that the base of the pyramid Π i,j,k in the plane i = 1 is entirely contained in λ . In this correspondence, λ ( m ) is simply the intersection of the plane i = m with this setof vertices. Another representation of the pyramid P λ is as a strip decomposition of λ , bysuperimposing all the diagrams λ ( m ) , with their (1 ,
1) box in the same position. Finally,we define the extended pyramid P ∗ λ to be the pyramid of the extended Young diagram λ ∗ ,namely P ∗ λ = P λ ∗ . Example 5.2.
The pyramid of the partition λ = (4 , , is λ (1) = (4 , , , λ (2) = (3 , , λ (3) = (1) , λ (4) = ∅ . The representation of P λ in L F CC and as a strip decomposition of λ are respectively: and5.3. The family of Laurent polynomials for a pyramid.
The solution of the A ∞ / T -system within a pyramid P λ defined as above is entirely fixed by the assignment of initialdata on the “roof” of the pyramid, defined by P ∗ λ \ P λ . Note that this roof is nothing butthe portion of the steepest stepped surface that determines the polynomials p a,b entirely.This leads to a natural extension of the family of polynomials { p a,b } ( a,b ) ∈ λ into a pyramidfamily { p a,b,m } with 1 ≤ m ≤ k and ( a, b ) ∈ λ ( m ) , obtained by identification of the solution T i,j,k at the corresponding vertex of P λ ⊂ L F CC .As a consequence of this definition, we may rewrite (3.6) as:
ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 23
Theorem 5.3.
The pyramid polynomials p a,b,m ( θ · ) associated to a Young diagram λ areentirely determined by the determinant identity: (5.1) p a,b,m ( θ · ) = det (cid:16) ( p a + i − ,b + j − ( θ · )) ≤ i,j, ≤ m (cid:17) where the array of points in the determinant corresponds to the base in the plane i = 1 ofthe pyramid with apex at the center of the box ( a, b ) of λ ( m ) in the L F CC representation.
The network interpretation of the solution of the T -system [6] allows to immediatelyinterpret the pyramid polynomial p a,b,m as the partition function for m non-intersectingpaths on the network N a,b associated to λ a,b and the steepest stepped surface, up to a mul-tiplicative boundary factor, by direct application of the Lindstr¨om Gessel-Viennot Theorem[14, 16]. These paths start/end at the points of k that correspond to the left/right pro-jections of the array of points in the determinant (5.1). On N a,b , these are the left/rightprojections of the top vertex of each box in the corresponding square of size m with topbox ( a, b ). Example 5.4.
Let us consider λ = (5 , , , , and the network L ( j , j ) = N , ofSect.4.3. The pyramid polynomial p , , is equal to the determinant (cid:12)(cid:12)(cid:12)(cid:12) p , p , p , p , p , p , p , p , p , (cid:12)(cid:12)(cid:12)(cid:12) . Itis also proportional to the partition function of paths from the entry to the exit verticesmarked , , below: ecb da
32 31 32 1223 11 where we have also represented a sample configuration of these three non-intersecting paths.The entry/exit vertices of N a,b are the left/right projections of the vertices at the top of theboxes in the corresponding square (here shaded in blue). The proportionality factor is simply a − b − cde , where the corresponding face variablesare immediately above the entry points , and exit points , , . Generalized Bessenrodt-Stanley polynomials for a pyramid.
We may nowrestrict the pyramid Laurent polynomials p a,b,m ( θ · ) attached to a partition λ via the samechange of variables (2.2) to box weights x i,j . This leads us to the definition of the quantities:(5.2) p a,b,m ( x · ) = det (cid:16) ( p a + i − ,b + j − ( x · )) ≤ i,j, ≤ m (cid:17) We have:
Theorem 5.5.
The pyramid polynomials p a,b,m ( x · ) (5.2) have non-negative integer coeffi-cients, and moreover p a,b,m ( x · ) is the partition function for m non-intersecting paths on thenetwork N a,b associated to λ a,b , from the m left projections to the m right projections of thevertices of the square of size m with NW corner at ( a, b ) . Alternatively, p a,b,m ( x · ) is thepartition function for m strictly nested partitions ∅ ⊂ µ m ⊂ µ m − ⊂ · · · ⊂ µ ⊂ λ a,b with µ i ⊂ λ ( i ) a,b inside λ a,b , with the usual weights: p a,b,m ( x · ) = X µi ⊂ λ ( i ) a,bi =1 , ,...,m m Y i =1 Y ( α,β ) ∈ λ a,b \ µ i x α,β Conclusion/Discussion
In this paper, we have expressed the polynomials of [2] as particular solutions of theoctahedron recurrence in a half-space with “steepest” initial data surface attached to afixed partition λ . This connection has allowed us to generalize these polynomials to thefull pyramid P λ , and to find alternative expressions involving paths on networks or dimerson bipartite graphs.More generally, we may consider different initial data surfaces associated to the partition λ , and use the solutions of the corresponding T -system to define different classes of poly-nomials attached to the boxes of λ . Another natural choice is to pick the stepped surfaceto be made of “vertical walls” along the boundary of λ , namely with vertices alternatingbetween two parallel planes with normal vector (0 , ,
1) or (0 , − ,
1) in L F CC . In the case of λ = (5 , , , , ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 25 decomposition look like: j j Here the vertical wall stepped surface is represented in front, with green vertices. Thecenters of the boxes of λ are the blue vertices in the bottom plane ( i = 1). In the lozengedecomposition of the stepped surface, we have only represented the lozenges that willcontribute to the solution within λ , and added as before triangles in the bottom row, withtheir bottom-most vertex assigned value 1 (the A ∞ / boundary condition (3.3)). Note alsothat the boundary vertical walls intersect the plane i = 1 along the boxes of λ ∗ \ λ .The difference with the situation of the steepest stepped surface is subtle: in both cases,the walls are made uniquely of either U type of V type lozenges, but their arrangement(the order in which they come from left to right) is different, due to the rules (3.7). Wemay now translate the lozenge decomposition into a network, which we deform in the same way as before to finally get: where we have indicated the correspondence between a sample row of face weights (greendots in general) with assigned values t i,j = a, b, c, ... , and the bottom outer faces by red dots(with assigned values 1). The edge weights in the network are related to the face variablesin the usual way (4.4), and all the horizontal edges receive the weight 1. The constructionof the network looks more complicated, but again the solution at the box (1 ,
1) of λ is thepartition function for paths on this network, from the leftmost to the rightmost vertex. Infact, it is possible to deform the network to make it match the shape of the Young diagram λ , at the expense of adding some extra edges as follows: a b c d e f g h i a ba b ab where the up/down edge weights are related to the face variables in the usual way, whilethe new horizontal edges have weights as indicated in the medallion. The additional arrows(in red) split up each box of the strip decomposition { λ ( i ) \ λ ( i +1) } i ∈ [2 ,k − of λ , where λ ( i ) is the i -th layer of the pyramid P λ , by connecting centers of box edges (we have shaded the ESSENRODT-STANLEY POLYNOMIALS AND THE OCTAHEDRON RECURRENCE 27 strip λ (2) \ λ (3) in the above example). In particular all paths of red arrows are parallel.Note also that the first strip λ (1) \ λ (2) is not split. Let us denote by ˜ N a,b the network thusconstructed out of the partition λ a,b .Having assigned fixed parameters t i,j to each inner and outer face of the resulting network,we may associate a Laurent polynomial q a,b,m ( t · ) to the ( a, b ) box of the m -th layer λ ( m ) ofthe pyramid of λ , equal to the corresponding T -system solution with vertical wall initialdata stepped surface. Then we have: Theorem 6.1.
The Laurent polynomial q a,b,m ( t · ) is the partition function of m non-intersectingpaths on ˜ N a,b , starting/ending at the left/right projections of the top vertices of the boxesin the square with top box ( a, b ) and size m , multiplied by the factor Q mα =2 t − ,ℓ α Q mβ =1 t ,r β ,where ℓ α / r α is the j coordinate in the FCC lattice of the α -th left/right projection of thepoints in the square of size m with NW corner at ( a, b ) . Example 6.2.
Let us consider the case λ = (2 , . Let us assign the following initial valueson the vertical walls: T , , = ℓT , − , = i T , , = j T , , = kT , − , = f T , , = g T , , = hT , − , = a T , − , = b T , , = c T , , = d T , , = e Then the network ˜ N , reads, with the face variables or alternatively the edge weights: e1 1111 l kji hgf dcba bcab gj 1 1bg1b ij hekhl cd1c kfg gd Let us first compute q a,b = q a,b, for the boxes for the Young diagram λ . The quantities q , /e, q , /e, q , /d, q , /d are the partition functions for paths on ˜ N , , respectively fromvertices → , → , → and → . This gives: q , = aec + efbd + aegbcd + ahbd + cfhbdg + ty + xzgy q , = bec + egcd + hd q , = adc + fb + agbc q , = bdc + gc Finally the quantity q , , b/ ( de ) is the partition function for pairs of non-intersecting pathsfrom , to , : q , , = f hg + bdtcy + gtcy + xzcy + bdxzcgy References [1] E. Berlekamp,
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