aa r X i v : . [ m a t h . C O ] A ug POSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA
PHILIPPE DI FRANCESCO AND RINAT KEDEM
Abstract.
We give the path model solution for the cluster algebra variables of the A r T -system withgeneric boundary conditions. The solutions are partition functions of (strongly) non-intersecting pathson weighted graphs. The graphs are the same as those constructed for the Q -system in our earlier work,and depend on the seed or initial data in terms of which the solutions are given. The weights are “time-dependent” where “time” is the extra parameter which distinguishes the T -system from the Q -system,usually identified as the spectral parameter in the context of representation theory. The path model isalternatively described on a graph with non-commutative weights, and cluster mutations are interpreted asnon-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurentpolynomial of the seed data. Introduction
In this paper we study solutions of the T -system associated to the Lie algebras A r , which we write inthe following form:(1.1) T α,j,k +1 T α,j,k − = T α,j +1 ,k T α,j − ,k + T α +1 ,j,k T α − ,j,k , where j, k ∈ Z , α ∈ I r = { , ..., r } , and with boundary conditions(1.2) T ,j,k = T r +1 ,j,k = 1 , j, k ∈ Z . We consider these equations to be discrete evolution equations for the commutative variables { T α,j,k } inthe direction of the discrete variable k .Originally, this relation appeared as the fusion relation for the commuting transfer matrices of thegeneralized Heisenberg model [1, 17] associated with a simply-laced Lie algebra g , where it is written inthe form(1.3) T α,j,k +1 T α,j,k − = T α,j +1 ,k T α,j − ,k − Y β = α T − C β,α β,j,k , with appropriate boundary conditions. The matrix C is the symmetric Cartan matrix of one of the Liealgebra of type ADE . Our relation (1.1) is obtained by a rescaling of the variables T α,j,k and specializingto the Cartan matrix of A r .With special initial condition at k = 0, it has been proved that the solutions to (1.3) are the q -characters[10] of the Kirillov-Reshetikhin modules of the affine Lie algebra U q ( b sl r +1 ) [19].The T -system also appears in several other contexts. Of particular relevance here is the fact [18] thatthe system is a discrete integrable equation, the discrete Hirota equation. It is therefore to be expectedthat the system has a complete set of integrals of motion, and that it is exactly solvable. This equationalso appears in a related combinatorial context, as the octahedron equation, which was studied by [16, 21].In this paper, we do not impose any special boundary conditions, but express the general solution ofthe T -system in terms of arbitrary initial conditions. For example, initial conditions can be chosen byspecifying the values of the parameters T α,j,k at k = 0 and k = 1, or a more exotic boundary can bespecified. To solve the system, we use a path model which is a simple generalization of the path model weconstructed for the solutions of the Q - system of A r [5, 6]. Date : November 1, 2018.
In our previous work, we constructed a set of path models, and proved that the solutions of the Q -systemof A r [15], Q α,k +1 Q α,k − = Q α,k + Q α − ,k Q α +1 ,k , Q ,k = Q r +1 ,k = 1; k ∈ Z , α ∈ I r , are the generating functions for paths on a positively weighted graph, where the weights are a function ofthe initial conditions.With special initial conditions at k = 0 and k = 1 (together with a rescaling as in (1.3) which restoresthe minus sign in the second term on the right hand side of the Q -system), the solutions are the charactersthe finite- dimensional, irreducible modules of A r with highest weights which are multiples of one of thefundamental weights.Note that this Q -system is obtained by “forgetting” the spectral parameter j in Equation (1.1). Thusthe T -system can be regarded as an affinization or q -deformation of the Q -system, and the path model wepresent here is therefore a deformation of the path model for the Q -system.Without fixing any special initial conditions, it was shown in [13] that the solutions of the Q -systemare cluster variables in a cluster algebra [8]. We showed in [4] that all Q -systems, corresponding to anysimple Lie algebra, can be formulated as cluster algebras. Thus, the solution of the Q -system in terms ofthe statistical model allowed us to prove the positivity conjecture of [8] for these cluster variables. In fact,as we showed in [6], the solutions are related to the totally positive matrices of [9] corresponding to pairsof coxeter elements.Similarly, we showed in [4] that a large class of equations which we call generalized bipartite T -systemscan be formulated as cluster algebras. Equation (1.1) is perhaps the simplest example of such a system.Motivated by our statistical model introduced in [5], we introduce a path model which provides us with thesolution to the T -system, in terms of a set of initial conditions, as the partition function of a path modelwith time-dependent (or non-commutative) weights. Here, we refer to the variable normally identified asthe spectral parameter as the time parameter, as it is a natural interpretation from the point of view ofpaths.This paper is organized as follows. In Section 2, we review the necessary definition of a cluster algebra.We recall our formulation [4] of T -systems as cluster algebras. We describe the conserved quantities ofthe T -system in terms of discrete Wronskian determinants in Section 3. We define a generalized notionof hard particle models on a graph in Section 4 and identify the conserved quantities as hard particlepartition functions on a specific graph. In Section 5, we use our conserved quantities to write the solutionsof the T -system as the partition functions of paths on a weighted graph. The weight of a step in a pathdepends on the order in which the steps are taken, that is, the weights are time-dependent. The solutionsare written as functions of the fundamental initial data, and the graph is the same as the one used in the Q -system solution. Positivity of the T -system solutions in terms of the fundamental seed variables followsfrom this formulation.To prove the positivity in terms of other seeds, we give a formulation of our model in terms of non-commutative weights in Section 6. We are then able to describe the solutions of the T -system as a functionof other seed data as partition functions on new graphs with weights which depend on the mutated seeds.The key to the construction is an operator version of the fraction rearrangement lemmas used in [4]. Theserearrangements are equivalent to mutations in the case of the Q -system. Here, they are equivalent tocompound mutations. We are thus able to write the T -system solution explicitly in terms of its initialdata, for a subset of cluster seeds.This paper should be considered as a (special case of) non-commutative generalization of our workon the solutions of Q -system [5, 6]. In particular, the graphs on which we build our path models arethe same as for the Q -system, and the only difference is the time-dependence or non-commutativity ofthe weights. The various key properties, such as the rearrangement lemmas for continued fractions andthe generalization of the Lindstr¨om-Gessel-Viennot theorem for strongly non-intersecting paths, all havestraightforward non-commutative counterparts which are used here. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 3
Acknowledgements:
P.D.F.’s research is supported in part by the ANR Grant GranMa, the ENIGMAresearch training network MRTN-CT-2004-5652, and the ESF program MISGAM. R.K.’s research is sup-ported by NSF grant DMS-0802511. R.K. thanks IPhT at CEA/ Saclay for their kind hospitality. Wealso acknowledge the hospitality of the Mathematisches Forschungsinstituts Oberwolfach (RIP program),where this paper was completed. 2. T -systems as cluster algebras Cluster algebras.
We use the following definition of a cluster algebra [8, 22], slightly specialized tosuit our needs in this paper.Let S ⊂ e S be two discrete sets (possibly infinite) and consider the field F of rational functions over Q in a set of independent variables indexed by e S .We define a seed in F to be a pair ( e x , e B ), where e x = { x m : m ∈ e S } is a set of commuting variables, and e B is an integer matrix, with rows indexed by e S and columns indexed by S . The matrix B , which is thesquare submatrix of e B made up of the rows of e B indexed by S , is skew symmetric.The cluster of the seed ( e x, e B ) is the set of variables { x m : m ∈ S } , and the coefficients are the set ofvariables { x m : m ∈ e S \ S } .Next, we define a seed mutation . For any m ∈ S , a mutation in the direction m , µ m : ( e x, e B ) ( e x ′ , f B ′ ),is a discrete evolution of the seed. Explicitly, • The mutation µ m leaves x n with n = m invariant, and updates the variable x m only, via theexchange relation(2.1) x ′ m = x − m Y n ∈ e S x [ e B n,m ] + n + Y n ∈ e S x [ − e B n,m ] + n where [ n ] + = max( n, • The exchange matrix e B ′ has entries(2.2) e B ′ i,j = ( − e B i,j if i = m or j = m ; e B i,j + sign( e B i,m )[ e B i,m e B m,j ] + ) otherwise . Note that we only define mutations for the set S , and not for the coefficient set e S \ S . That is, coefficientsdo not evolve.Fix a seed ( e x , e B ) and consider the orbit X ⊂ F of the cluster variables under all combinations of themutations µ m , m ∈ S . The cluster algebra is the Z [ c ± ]- subalgebra of F generated by X , where c is thecommon coefficient set of the orbit of the seed. Remark 2.1.
The particular system which we solve in this paper does not require us to have a coefficientset, that is, we can set S = e S . However, to make more direct contact with representation theory, it isdesirable to have the coefficient set be enumerated by the roots of the Lie algebra. In this context, we needto set the values of the coefficients to the special points − . Cluster algebras can be considered to be discrete dynamical systems, which is the point of view we adoptin this paper.2.2. bipartite T -systems as cluster algebras. In this section we review some of the definitions ofAppendix B of [4], where generalized bipartite T -systems were shown to have a cluster algebra structure. Definition 2.2.
A generalized bipartite T -system is a recursion relation for the commuting, invertiblevariables { T α,j ; k } , where α ∈ I r and j, k ∈ Z , of the form (2.3) T α,j ; k +1 T α,j ; k − = T α,j +1; k T α,j − k + q α Y j ′ Y α ′ ( T α ′ ,j ′ ; k ) A j ′ ,jα ′ ,α PHILIPPE DI FRANCESCO AND RINAT KEDEM where A is an incidence matrix, that is, a symmetric matrix with positive integer entries. The matrix A is generally of infinite size, unless special boundary conditions are imposed on the systemwhich truncate the range of the variables j . We do not impose such boundary conditions in this paper,although they are clearly of interest [12, 20]. The symmetry of A is required for the bipartite property tohold (see below). T -systems which are not bipartite can also be defined, and in that case, the matrix A isnot symmetric. Example 2.3.
The first example of such a T system is the one described in (1.3) . In that case, we takethe matrix A to be as follows: (2.4) A j,j ′ α,β = I α,β δ j,j ′ , where I α,β = C − I is the incidence matrix of the Dynkin diagram associated with a simply-laced Lie algebra g . The coefficients q α are all set to be − . However, it is always possible to renormalize the variables sothat q α = 1 in these cases [13] , and we use this approach here.In particular, if g = A r , ( I ) α,β = δ α,β +1 + δ α,β − . This is the case we solve in this paper. We note that another example of generalized T -systems appeared in the context of preprojective algebrasand the categorification program of [11]. The explicit connection was made in [4], Example 4.4.Finally, define the (possibly infinite) matrix P with entries(2.5) P j,lα,β = δ α,β ( δ i,j +1 + δ i,j − ) . Then we can rewrite (2.3) as(2.6) T α,j ; k +1 T α,j ; k − = Y α,j T P j ′ ,jβ,α β,j ′ + q α Y j ′ Y α ′ ( T α ′ ,j ′ ; k ) A j ′ ,jα ′ ,α . In the systems considered in [4], we allowed the matrix P to be a matrix with positive integer entries, suchthat it commutes with the matrix A , together with another condition on the sum of its entries (see Lemma2.5 below). Such a system is also a generalized bipartite T - system.2.3. Cluster algebra structure.
We recall the formulation found in Appendix B of [4] of the clusteralgebra associated with generalized (bipartite) T -systems.In the notations of Section 2, let S = ( I r ⊔ I r ) × Z , and e S = S ⊔ I ′ r . Each set I r , I r and I ′ r is just theset with r elements. For convenience, if α ∈ I r , then by α we mean the α th element of I r , etc.We define the fundamental seed ( e x , e B ) as follows. The variables e x are x α,j = T α,j ;0 , ( α ∈ I r , j ∈ Z ); x α,j = T α,j ;1 , ( α ∈ I r , j ∈ Z ); x α ′ = q α , α ′ ∈ I ′ r ;(2.7)(2.8)The elements of the set { x α,j }⊔{ x α,j } are the cluster variables and { x α ′ } are the coefficients. The exchangematrix of the fundamental seed is defined as follows: B α,j ; β,l = 0 , ( α, β ∈ I r , j, l ∈ Z ) , B α,j ; β,l = 0 , ( α, β ∈ I r , j, l ∈ Z ) ,B α,j ; β,l = − P j,lα,β + A j,lα,β = − B β,l ; α,j e B α ′ ; β,j = − e B α ′ ,β,j = − δ α,β . (2.9)The last equation above denotes the entries of the extended B -matrix, corresponding to the coefficients,which do not mutate. The matrices A, P are those of equation (2.6) for the generalized T -system. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 5 jj − j + 1 j + 2 · · · k + 12 k q α Figure 2.1.
A slice of the quiver graph of e B , corresponding to constant α . The nodesin the strip are labeled by ( j, k ) of T α,j ; k . The two subgraphs with even and odd j + k decouple in this slice, so we illustrate the only the connectivity of nodes of the same parityto node q α . The mutation µ reverses all arrows connected to q α . Example 2.4.
In the case of the A r system (1.1) , we have the matrix A as in (2.4) , P as in (2.5) and q α = 1 . In that case we do not need to include the coefficients q α , and the matrix e B is equal to the matrix B . To recover the original T -system (1.3) , we take q α = − . It is clear that each of the mutations µ α,j and µ α,j exchanges one of the cluster variables in e x via oneof the T -system equation relations (2.6). The mutation µ α,j acts on e x as one of the T -system evolutions(2.6), where we specialize to k = 1: µ α,j ( T α,j ;0 ) = T α,j ;2 . Similarly, µ α,j T α,j ;1 = T α,j ; − is a T -systemequation specialized to k = 0.Quite generally, if B a,b = 0 then µ a ◦ µ b = µ b ◦ µ a . Since B α,j ; β,l = 0 for all α, β ∈ I r and j, l ∈ Z , whenacting on the initial seed ( e x , e B ) , the mutations µ α,m commute with each other for all α, m . Similarly themutations µ α,m also commute among themselves.Therefore we can define the compound mutations µ := Y α,m µ α,m , µ := Y α,m µ α,m which act on ( e x , e B ) . More generally, Define ( e x , e B ) k to be the seed with x α,j = T α,j ;2 k , x α,j = T α,j ;2 k +1 and e B k = e B . Define ( e x , e B ) k +1 to be the seed with x α,j = T α,j ;2 k +2 , x α,j = T α,j ;2 k and e B k +1 = − e B .Then it is clear that µ ( e x k ) = e x k +1 : Each mutation µ α,j mutates the variable T α,j ;2 k into the variable T α,j ;2 k +2 . Similarly, it is easy to check that µ ( e x k ) = e x k − , µ ( e x k +1 ) = e x k and µ ( e x k +1 ) = e x k +2 .The following statement is Lemma 4.6 of [4]: Lemma 2.5.
Assume that the matrix A commutes with the matrix P , and that (2.10) X k P kjα,β = 2 δ α,β for any j . Then the cluster algebra X which includes the seed ( e x , e B ) as in (2.7) , (2.9) includes all thesolutions of the T -system (2.6) . All the T -system relations are exchange relations in this cluster algebra. To prove this Lemma, we need
PHILIPPE DI FRANCESCO AND RINAT KEDEM jj − j + 1 q α µ α,j − µ α,j +1 µ α,j Figure 2.2.
The local action of the mutation µ on a section of the quiver graph. Thecompound mutation reverses all arrows connected to q α . Lemma 2.6. µ (cid:16) ( e x , e B ) k (cid:17) = ( e x , e B ) k +1 , µ (cid:16) ( e x , e B ) k (cid:17) = ( e x , e B ) k − . Proof.
In light of the preceding discussion, all that needs to be proved is that µ ( e B ) = µ ( e B ) = − e B . Let e B ′ = µ ( e B ). Then, since B α,j ; β,k = 0, we have • µ α,i ( B β,j ; γ,k ) = sign( B β,j ; α,i )[ B β,j ; α,i B α,i ; γ,k ] + = 0; • µ α,i ( B β,j ; γ,k ) = − B β,j ; γ,k if ( α, i ) = ( γ, k ), and is otherwise unchanged, since if ( α, i ) = ( γ, k ), µ α,i ( B β,j ; γ,k ) = B β,j ; γ,k + sign( B β,j ; α,i )[ B β,j ; α,i B α,i ; γ,k ] + = B β,j ; γ,k . Similarly, µ α,i ( B β,j ; γ,k ) = − B β,j ; γ,k . • Recall the restriction that [
P, A ] = 0. Then µ ( B β,j ; γ,k ) = X α,i sign( B β,j ; α,i )[ B β,j ; α,i B α,i ; γ,k ] + = ( P A − AP ) j,kβ,γ = 0 . • We have µ α,i ( B β ′ ; γ,k ) = − B β ′ ; γ,k , and otherwise, if ( α, i ) = ( γ, k ) then µ α,i µ α,i ( e B β ′ ; γ,k ) = e B β ′ ; γ,k + X α,i sign( e B β ′ ; α,i )[ e B β ′ ; α,i B α,i ; γ,k ] = δ β,γ , so that µ ( e B β ′ ; γ,k ) = − e B β ′ ; γ,k . • Finally, using the restriction (2.10) on the summation of elements of P , µ ( e B β ′ ; γ,k ) = δ β,γ + X α,i sign( e B β ′ ; α,i )[ e B β ′ ; α,i B α,i ; γ,k ] + = δ β,γ − X α δ α,β X i P i,kα,γ = − δ β,γ . In the quiver graph corresponding to e B , the last two statements are about how nodes x α,j = T α, k and x α,j = T α,j ;2 k +1 are connected to node x β ′ = q β . If α = β , they are not connected, and if α = β , theconnectivity is illustrated in Figure 2.3 and the mutations in Figure 2.3. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 7
We have shown that µ ( e B ) = − e B . The proof that µ ( e B ) = − e B is similar. (cid:3) Thus, we have shown that all the variables T α,j ; k appear in the cluster algebra, in fact, within a bipartitegraph composed of the nodes reached from ( e x , e B ) via combinations of the compound mutations µ and µ only.In this paper, we study the A r T -system solutions in terms of the fundamental seed cluster e x . Theresult will be an explicit interpretation of the solutions as partition functions of paths on a graph whoseweights which are positive monomials in the variables e x . This will imply the positivity property [8] forthe cluster variables T α,j ; k : They can be expressed as Laurent polynomials with non-negative coefficientsin terms of the initial data. 3. Basic properties of the T -system From here on, we specialize the discussion to the T -system (1.1). Note that the equation (1.1) is athree-term recursion in the index k , and allows to determine all the { T α,j,k +1 } α ∈ I r ,j ∈ Z in terms of the {{ T α,j,k , T α,j,k − } α ∈ I r ,j ∈ Z . We wish to first study the solution T α,j,k to Equation (1.1) in terms of the“fundamental” initial data x = ( T α,j, , T α,j, ) α ∈ I r ,j ∈ Z , that is, x . The techniques used in this sectionare a straightforward generalization of the methods used for the Q -system in [5]. We therefore present theproofs of the theorems in the Appendix, as they use standard techniques in the theory of determinants.3.1. Discrete Wronskians and conserved quantities.
We can express the subset of variables { T α,j,k : j, k ∈ Z , α > } as polynomials of the variables in the set { T ,j,k : j, k ∈ Z } , cf [17]: Theorem 3.1. (3.1) T α,j,k = det ≤ a,b ≤ α ( T ,j − a + b,k + a + b − α − ) , α ∈ I r , j, k ∈ Z The proof of this theorem uses the standard Pl¨ucker relations, and is similar to the case of the Q -system.We therefore present the details of the proof in the Appendix, Section A.2.If we consider α = r + 1 in Equation (3.1), since T r +1 ,j ; k = 1, we have the polynomial relation amongthe variables { T ,j ; k } :(3.2) ϕ j,k ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ,j,k − r T ,j − ,k +1 − r · · · T ,j − r +1 ,k − T ,j − r,k T ,j +1 ,k +1 − r T ,j,k +2 − r · · · T ,j − r +2 ,k T ,j − r +1 ,k +1 ... ... . . . ... ... T ,j + r − ,k − T ,j + r − ,k · · · T ,j,k + r − T ,j − ,k + r − T ,j + r,k T ,j + r − ,k +1 · · · T ,j +1 ,k + r − T ,j,k + r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1This is the “equation of motion” for the system. Since ϕ j,k is a discrete Wronskian determinant, it remainsconstant for solutions of a difference equation. The difference equation can be found by taking the differenceof two Wronskians and arguing that a non-trivial linear combination of its columns must vanish. Theorem 3.2.
We have the following linear recursion relations (3.3) r +1 X b =0 T ,j − b,k + b ( − b c r +1 − b ( j − k ) = 0 j, k ∈ Z where the coefficients c r +1 − b ( j − k ) depend only on the difference j − k , with c ( m ) = c r +1 ( m ) = 1 for all m ∈ Z , and: (3.4) r +1 X a =0 T ,j + a,k + a ( − a d r +1 − a ( j + k ) = 0 j, k ∈ Z where the coefficients d r +1 − a ( j + k ) depend only on the sum j + k , with d ( m ) = d r +1 ( m ) = 1 for all m ∈ Z . PHILIPPE DI FRANCESCO AND RINAT KEDEM
Such linear recursion relations can be obtained by noting that T r +2 ,j,k = 0 and expanding the corre-sponding Wronskian determinant along the first row or column. The key fact to be proven is that theminors depend only on the difference j − k or the sum j + k . The proof is presented in the Appendix,Section A.3.By analogy with the case of the Q -systems [5, 6], we may still call the variables c b ( k ) and d b ( k ) integralsof motion of the T -system, as they depend on one less variable than T . Moreover, they can be expressedentirely in terms of the fundamental initial data for the T -system, e x . Example 3.3.
In the A case, we have T ,j,k − c ( j − k ) T ,j − ,k +1 + T ,j − ,k +2 = 0 T ,j,k − d ( j + k ) T ,j +1 ,k +1 + T ,j +2 ,k +2 = 0 with the integrals of motion c ( j ) = T ,j, T ,j − , + 1 T ,j − , T ,j − , + T ,j − , T ,j − , d ( j ) = T ,j, T ,j +1 , + 1 T ,j +1 , T ,j +2 , + T ,j +3 , T ,j +2 , An explicit expression for the conserved quantities of Theorem 3.2 is as Wronskian determinants with a“defect”:
Lemma 3.4.
The conserved quantities c m ( j ) ( m = 0 , , ..., r + 1 , j ∈ Z ) of Equation (3.3) are (3.5) c m ( j ) = det ≤ a ≤ r +11 ≤ b ≤ r +2 , b = r +2 − m ( T ,j + n + a − b,n + a + b − ) for any n ∈ Z . Again the proof uses the standard techniques, and is found in Section A.4 of the Appendix.4.
Conserved quantities and hard particles
Recursion relations for conserved quantities.
The conserved quantities (3.5) satisfy linear re-cursion relations, which allow us to express them in terms of the initial data x . We use recursion relationson the size r , so we first relax the boundary conditions T r +1 ,j,k = 1 for all j, k ∈ Z .Consider the A ∞ / T -system: t α,j,k +1 t α,j,k − = t α,j +1 ,k t α,j − ,k + t α +1 ,j,k t α − ,j,k , t ,j,k = 1 , ( j, k ∈ Z , α ∈ Z > ) . (4.1)Solutions of this system are expressible in terms of the initial data ( t α,j, , t α,j, ) α ∈ Z > ,j ∈ Z . By definition, T α,j,k = t α,j,k if we impose the boundary condition t r +1 ,j,k = 1 for all j, k ∈ Z .The proof of Theorem 3.1 does not involve the boundary condition T r +1 ,j ; k = 1, so the determinantexpression for t α,j ; k still holds:(4.2) t α,j,k = det ≤ a,b ≤ α ( t ,j + a − b,k + a + b − α − ) , α > . Define the Wronskians of size N with a defect in position N − m :(4.3) c N,m,j,k = det ≤ a ≤ N ≤ b ≤ N +1 , b = N +1 − m ( t ,j + a − b,k + a + b − N − ) , where c N,m,j,k = 0 if m > N or m < c N, ,j,k = T N,j,k , and c N,N,j,k = T N,j − ,k +1 by Theorem 3.1. If we impose the second boundarycondition of the T -system on the t ’s, then c r +1 ,m,j,k = c m ( j − k + r ). OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 9
Lemma 4.1.
The Wronskians with a defect c α,m,j,k satisfy the following recursion relations: t α − ,j − ,k − c α − ,m,j,k = t α,j − ,k c α − ,m − ,j,k − + t α − ,j,k c α − ,m,j − ,k − (4.4) t α − ,j − ,k c α,m,j,k = t α,j − ,k +1 c α − ,m − ,j,k − + t α,j,k c α − ,m,j − ,k (4.5) for α ≥ and m, j, k ≥ .Proof. The first equation (4.4) follows from the Desnanot-Jacobi relation (A.3), with N = α , i = 1, i = α , j = α − m , j = α , for the matrix M with entries M a,b = T ,j + a − b − ,k + a + b − α − , a, b = 1 , , ..., α .The second equation (4.5) follows from the Pl¨ucker relation (A.2), with N = α , and the N × ( N + 2)matrix P with entries P a, = δ a,α , and P a,b = T ,j + a − b,k + a + b − α − for b = 2 , , ..., α + 2 and a = 1 , , ..., α ,and by further picking a = 1, a = 2, b = α + 2 − m , and b = α + 2. (cid:3) Theorem 4.2.
The Wronskians with a defect c α,m,j,k defined in Equation (4.3) are uniquely determinedby the following recursion relation, for α ≥ : t α − ,j − ,k − t α − ,j,k c α,m,j,k − = t α − ,j − ,k − t α,j − ,k c α − ,m − ,j +1 ,k − + t α,j,k − t α − ,j,k c α − ,m,j − ,k − + t α,j,k − t α,j − ,k c α − ,m − ,j,k − (4.6) and the boundary conditions c ,m,j,k = δ m, , for all j, k ∈ Z and c ,m,j,k = δ m, t ,j,k + δ m, t ,j − ,k +1 forall m, j, k ∈ Z .Proof. Using Equation (4.4), the second line in (4.6) is equal to t α,j,k − t α − ,j − ,k − c α − ,m,j,k . Cancelingthe overall factor t α − ,j − ,k − , we must prove that(4.7) t α − ,j,k c α,m,j,k − − ( t α,j − ,k c α − ,m − ,j +1 ,k − + t α,j,k − c α − ,m,j,k ) = 0 . Multiplying the l.h.s. of (4.7) by t α,j +1 ,k and using (4.1) we have t α,j +1 ,k t α − ,j,k c α,m,j,k − − ( t α,j,k +1 t α,j,k − − t α +1 ,j,k t α − ,j,k ) c α − ,m − ,j +1 ,k − − t α,j +1 ,k t α,j,k − c α − ,m,j,k = t α − ,j,k ( t α,j +1 ,k c α,m,j,k − + t α +1 ,j,k c α − ,m − ,j +1 ,k − ) − t α,j,k − ( t α,j,k +1 c α − ,m − ,j +1 ,k − + t α,j +1 ,k c α − ,m,j,k )= t α,j,k − ( t α − ,j,k c α,m,j +1 ,k − t α,j,k +1 c α − ,m − ,j +1 ,k − − t α,j +1 ,k c α − ,m,j,k ) = 0where we have simplified the third line by use of (4.4), and finally used (4.5). Equation (4.6) follows.Equation (4.6) is a three-term linear recursion relation in the variable α and therefore has a uniquesolution c given the initial conditions at α = 0 ,
1. Moreover, these initial conditions are identical to thosefor (4.3), hence this solution coincides with the definition (4.3) for all α, m, j, k . (cid:3) Let us define:(4.8) C α +1 ,m ( j, k ) = c α +1 ,m,j + k − α,k t α +1 ,j + k − α,k , α ≥ , m, j, k ∈ Z . These satisfy C α +1 , ( j, k ) = 1 and C α +1 ,α +1 ( j, k ) = t α +1 ,j + k − α − ,k +1 /t α +1 ,j + k − α,k . The conserved quan-tities of the A r T -system are obtained by imposing the boundary condition t r +1 ,j,k = 1, in which case: c m ( j ) = C r +1 ,m ( j, k ) for any j ∈ Z , and m = 0 , , , ..., r + 1, independently of k ∈ Z . Corollary 4.3.
The quantities C α +1 ,m ( j, k ) of eq. (4.8) are the solutions of the following linear recursionrelation, for α ≥ : (4.9) C α +1 ,m ( j, k ) = C α,m ( j − , k ) + y α +1 ( j − α, k ) C α,m − ( j, k ) + y α ( j − α − , k ) C α − ,m − ( j − , k ) . . . G r 3 2r+1 Figure 4.1.
The graph G r , with 2 r + 1 vertices labeled i = 1 , , ..., r + 1. with coefficients: y α +1 ( j, k ) = t α +1 ,j + k − ,k +1 t α,j +1+ k,k t α +1 ,j + k,k t α,j + k,k +1 ( α ≥ y α ( j, k ) = t α +1 ,j + k,k +1 t α − ,j + k +1 ,k t α,j + k,k t α,j + k +1 ,k +1 ( α ≥ y ( j, k ) = t ,j + k,k +1 t ,j + k +1 ,k , (4.10) subject to the initial conditions C ,m ( j, k ) = δ m, and C ,m ( j, k ) = δ m, + δ m, y ( j − , k ) . Example 4.4.
We have the following first few values of C α +1 ,m ( j, k ) : α = 0 : C , ( j, k ) = 1 C , ( j, k ) = y ( j − , k ) α = 1 : C , ( j, k ) = 1 C , ( j, k ) = y ( j − , k ) + y ( j − , k ) + y ( j − , k ) C , ( j, k ) = y ( j − , k ) y ( j − , k ) α = 2 : C , ( j, k ) = 1 C , ( j, k ) = y ( j − , k ) + y ( j − , k ) + y ( j − , k ) + y ( j − , k ) + y ( j − , k ) C , ( j, k ) = y ( j − , k ) y ( j − , k ) + y ( j − , k ) y ( j − , k )+ y ( j − , k )( y ( j − , k ) + y ( j − , k ) + y ( j − , k )) C , ( j, k ) = y ( j − , k ) y ( j − , k ) y ( j − , k ) Remark 4.5.
The Corollary 4.3 allows to interpret the conserved quantities of the A r T -system as follows.From the recursion relation (4.9) , we deduce that C r +1 ,m ( j, k ) is a homogeneous polynomial of the weights y , y , ..., y r +1 , themselves ratios of products of some t a,b,c ’s with c only taking the values k and k + 1 . Ifwe impose t r +1 ,j,k = 1 , we see that, as explained above, C r +1 ,m ( j, k ) = c m ( j ) is independent of k . We maytherefore write C r +1 ,m ( j, k ) = C r +1 ,m ( j, , the latter involving only T a,b,c ’s with c = 0 , . These give r conservation laws for m = 1 , , ..., r . For r = 1 , we have for instance C , ( j, k ) = T ,j + k,k T ,j + k − ,k +1 + 1 T ,j + k − ,k +1 T ,j + k − ,k + T ,j + k − ,k +1 T ,j + k − ,k = C , ( j,
0) = T ,j, T ,j − , + 1 T ,j − , T ,j − , + T ,j − , T ,j − , Hard particle interpretation.
In this paper, we introduce a slightly generalized model of hardparticles on a graph.4.2.1.
Definition of the model.
Let G r be the graph of Figure 4.1, with vertices labeled as shown. When r = 1, G is just the chain with 3 vertices, and when r = 0 G is a single vertex. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 11
To each vertex labeled i in G r , we associate a height function h , where h ( i ) = (cid:22) i + 12 (cid:23) , ( i > , h (1) = 0 . A configuration of hard particles on G r is a subset S of I r +1 such that i, j ∈ I implies that vertices i and j are not connected by an edge. We can think of the elements of I as the vertices occupied by particles.The set of all hard particle configurations of cardinality m on G r is called C m . There is a natural orderingon the set I r +1 , and in the generalized hard particle model we define in this paper, the set S is consideredto be an ordered set.In general, a hard particle model on G r associates weights to the occupied vertices which depend onthe vertex label, and possibly also on the total number of occupied particles. The corresponding partitionfunction is the sum over all possible hard-particle configurations of the products of the occupied vertexweights.For the purpose of this work, we define the partition function for m hard particles as(4.11) Z G r m ( j, k ) = X S ∈ C m m Y ℓ =1 y i ℓ ( j − r + ℓ − m ) − h ( i ℓ ) , k )with the weights y i as in (4.10) and S = { i , ..., i m } .4.2.2. Conserved quantities as hard particle partition functions.
We have the following.
Theorem 4.6.
The partition function Z G α m ( j, k ) (4.11) for m -hard particles on G α coincides with thequantity C α +1 ,m ( j, k ) of (4.8) .Proof. Hard particle partition functions on G r satisfy a recursion relation in r . Fix m and consider theconfiguration of particles on vertices (2 r + 1 , r ). There are 3 possible pairs of occupation numbers forthese two neighboring vertices, (0 , ,
0) and (0 , • (0 ,
0) contributes Z G r − m ( j − , k ). • (1 ,
0) contributes y r +1 ( j − r, k ) Z G r − m − ( j, k ). • (0 ,
1) contributes y r ( j − r − , k ) Z G r − m − ( j − , k ).This implies that Z G r m satisfies the recursion relation(4.12) Z G r +1 m ( j, k ) = Z G r m ( j − , k ) + y r +1 ( j − r, k ) Z G r m − ( j, k ) + y r ( j − r − , k ) Z G r − m − ( j − , k ) . But this is the same relation satisfied by C r,m ( j, k ), Equation (4.9), with the same initial conditions, Z G r ( j, k ) = 1 (for any r ) and and Z G ( j, k ) = y ( j − , k ) = C , ( j, k ). The theorem follows. (cid:3) Setting t r +1 ,j,k = 1, we have: Corollary 4.7.
The conserved quantities c m ( j ) of the A r T -system are the partition functions for m -hardparticles on G r , with the weights: y α +1 ( j, k ) = T α +1 ,j + k − ,k +1 T α,j +1+ k,k T α +1 ,j + k,k T α,j + k,k +1 (1 ≤ α ≤ r ) y α ( j, k ) = T α +1 ,j + k,k +1 T α − ,j + k +1 ,k T α,j + k,k T α,j + k +1 ,k +1 (1 ≤ α ≤ r ) y ( j, k ) = T ,j + k,k +1 T ,j + k +1 ,k (4.13) where T ,j,k = T r +1 ,j,k = 1 for all j, k ∈ Z . As the resulting hard-particle partition functions are independent of k , we may set k = 0 in the expressionfor the weights. G s s s s s
13 116 j−5j−6h(v)67320 tj−4j−8j−10j−12j−14 31
Figure 4.2.
A graphical interpretation of a hard-particle configuration on the graph G , with m = 5 particles at positions { , , , , } . The label of the occupied vertex is indicated in acircle, and the time and height coordinates in rectangles. A distinct diagonal stripe correspondsto each particle. The leftmost stripe has x -intercept j − r + 1) = j −
14, and the rightmost isat j − r + 1 − m ) = j −
4. The weight of this configuration is y ( j − , k ) y ( j − , k ) y ( j − , k ) y ( j − , k ) y ( j − , k ). A pictorial representation for the hard particle partition function.
The hard particle con-figurations which give rise to the partition function of the form (4.11) can be represented graphically as inFigure 4.2. • A particle at a spine vertex v ( v ∈ { , , , , ..., r − , r, r + 1 } ) is represented by a diamond onthe two-dimensional lattice, its center at the height of the vertex, at the point ( t, h ( v )) for some t ∈ Z , and its vertices at the four neighboring lattice sites. • A particle a vertex v ∈ { , , , ..., r − } is represented by the lower half of such a diamond.We call t the time coordinate, and h ( v ) the height. Each polygon is at ( t, h ( v )) contained in a diagonalstripe s , bordered by the lines y = x − ( t + 1 − h ( v )) and y = x − ( t − − h ( v )). We denote s by its x -intercepts, s = { t − − h ( v ) , t + 1 − h ( v ) } .Given a configuration S ∈ C m , with S = { i < i < · · · < i m } , the polygon representing the particle i is drawn in the stripe s = { t − , t } ; that of i in the stripe immediately above and to the left, s = { t − , t − } , and the k -th polygon representing i k lies in stripe s k = { t − k, t − k + 2 } . The heightof each polygon is determined by h ( i j ) and its time coordinate by its stripe: t k = h ( i k ) − t − k − t = j − r + 1 − m ), then Equation (4.11) can be written as(4.14) Z G r m ( j, k ) = X S ∈ C m m Y ℓ =1 y i ℓ ( t ℓ , k )5. Path formulation and positivity
We now give an expression for T α,j,k as a function of the initial data x = ( T β,j, , T β,j, ) β ∈ I r ,j ∈ Z . It canbe interpreted as the partition function of weighted paths on a certain graph, with time-dependent weights. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 13 . . . r+2r+1521 430 3’ 4’ 5’ r−1r−1’ rr’r G Figure 5.1.
The graph e G r , with 2 r + 2 vertices. G Figure 5.2.
The planar representation of a typical path in P , , on the graph ˜ G . That is, we generalize the notion of a weighted path, so that the weight of a step in the path depends onthe time at which it is taken.As a corollary of the formulation in this section, we have the positivity Theorem 5.7 for the variables T α,j,k as a function of the initial data.5.1. Definitions.
Let e G r be the graph in Figure 5.1. It has 2 r + 2 vertices, which are ordered as(0 , , , ′ , , ′ , ...r, r ′ , r + 1 , r + 2). Its incidence matrix A is A m,m ′ = A m ′ ,m = 1 , (2 ≤ m ≤ r ); A m,m +1 = A m +1 ,m = 1 , (0 ≤ m ≤ r + 1) . The vertex labelled 0 is called the origin of the graph. We call the vertices i the spine vertices of e G r , andthe edges which connect i → ı ± P a,bt ,t of paths p on the graph ˜ G r , starting at time t and vertex a , and endingat time t ≥ t at vertex b . We take t i ∈ Z , and each step takes one time unit. The path p may berepresented by the succession of visited vertices, p = ( p ( t )) t = t ,t +1 ,...,t , with p ( t ) = a and p ( t ) = b and A p ( s ) ,p ( s +1) = 1 for any s .Let w i,j ( t ) be the weight of a step vertex i to vertex j at time t . We define the weight of a path p ∈ P a,bt ,t to be(5.1) w ( p ) = t − Y s = t w p ( s ) ,p ( s +1) ( s ) , p ( t ) = a, p ( t ) = b. The partition function for weighted paths in P a,bt ,t is(5.2) Z a,bt ,t = X p ∈ P a,bt ,t w ( p ) . For later use, we define Z a,bt ,t = 0 if t > t .Paths can be represented on the lattice Z as in Figure 5.2. We associate a vertical coordinate h ( i ) = h ( i ′ ) = i to each vertex of e G r . The horizontal axis is the time. A step a → b at time t on e G r is a step( t, h ( a )) → ( t + 1 , h ( b )). p’(b)(a)S pS’ p’ S pS’ Figure 5.3.
The involution ϕ between pairs of m - hard particle configurations and paths in P , j − r +1 − m ) ,j +2 k . Case (a): the first stripe traversed by p is absorbed into S ′ , which has m + 1particles, and p ′ ∈ P , j − r +1 − m )+2 ,j +2 k . Case (b): The bottom stripe of S is absorbed into p ′ ,now in P , j − r +1 − m ) − ,j +2 k , while S ′ has only m − We claim (see Theorem 5.5) that there exists a choice of weights w a,b ( s ), as functions of x = ( T α,j, , T α,j, ) α ∈ I r ,j ∈ Z ,such that T ,j,k /T ,j + k, is equal to the partition function Z , j − k,j + k .Dividing a path, which takes place from time t to time t ′ , into a first part from t to t ′ , and a secondpart, from t ′ to t ′′ , we have(5.3) Z a,bt,t ′ = X x ∈ e G r Z a,xt,t ′ Z x,bt ′ ,t ′′ , t ′ ∈ [ t, t ′′ ] . In particular, the matrix of one-step partition functions Z a,bt,t +1 is called the transfer matrix T ( t ), withentries(5.4) ( T ( t )) a,b = Z a,bt,t +1 = w a,b ( t ) A a,b . The transfer matrix is a decorated adjacency matrix. The recursion relation (5.3) implies(5.5) Z a,bt ,t = ( T ( t ) T ( t + 1) · · · T ( t − a,b . We use the following definition for weights w a,b ( s ) of paths on e G r : w m,m ′ ( s ) = 1 , w m ′ ,m ( s ) = y m +1 ( s, , ( m ∈ { , ..., r } ) ,w m,m +1 ( s ) = 1 , w m +1 ,m ( s ) = y m ( s, , ( m ∈ { , ..., r − } ) , (5.6) w , ( s ) = 1 , w , ( s ) = y ( s, , in terms of the weights y i ( s, k ) of Equation (4.13).5.2. An involution on pairs of weights.
We define an involution ϕ on the set C m × P , j − r +1 − m ) ,j +2 k ,consisting of hard-particle configurations on G r and paths on e G r .Let ( S, p ) ∈ C m × P , j − r +1 − m ) ,j +2 k , with m ∈ { , ..., r + 1 } , k ∈ Z + . We refer to the graphicalrepresentations of Figures 4.2 and 5.2, and we draw S and p on the same lattice (see Figure 5.3), where S is represented between the diagonal lines y = x − ( j − r + 1)) and y = x − ( j − r + 1 − m )), and p starts at ( j − r + 1 − m ) , x -intercept of the bottom stripe of S . OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 15
The path p has an initial section p within the diagonal stripe { j − r + 1 − m ) , j − r − m ) } ,consisting of u consecutive up steps and (i) a down step ( s, u ) → ( s + 1 , u −
1) or (ii) two horizontal steps( s, u ) → ( s + 1 , u ) → ( s + 2 , u ), where s = j − r + 1 − m ) + u . p \ p is then to the right of this initialstripe.Let σ be a map from path steps of type (i) or (ii) on e G r to the vertex set of G r . It is defined as follows: σ (( s, u ) → ( s + 1 , u − u − , ≤ u ≤ r + 1;1 , u = 1;2 r + 1 , u = r + 2 .σ (( s, u ) → ( s + 2 , u )) = 2 u − . Remark 5.1.
Graphically, steps of type (i) and (ii) in p can mapped precisely to the polygons representingparticles on G r . A step of type (i) is the NE edge of a diamond (hence a particle on a spine vertex) and astep of type (ii) the upper edge of a half-diamond. The map σ represents this correspondence. Denote by i the image of a step under the map σ . We must now distinguish between two cases. • Case (a): If i < i and S ′ := { i, i , i , ..., i m } ⊂ C m +1 , define p ′ ∈ P , j − r +1 − m )+2 ,j +2 k to be thepath with p ′ ( j − r + 1 − m ) + 2 + x ) = x for x = 0 , , ..., u (case (i)) or x = 0 , , ..., u − p ′ ( x ) = p ( x + 2) otherwise. • Case (b): If i ≥ i or { i, i , ..., i m } / ∈ C m +1 , define S ′ = { i , i , ..., i m } ∈ C m − , the hard particleconfiguration with the right stripe removed. It is now drawn between the diagonal lines y = x − ( j − r + 1)) and y = x − ( j − r + 1 − ( m − p ′ , – If i ∈ { , , ..., r − } , define p ′ ( j − r + 2 − m ) + x ) = x for x = 0 , , ..., h ( i ), p ′ ( j − r +2 − m ) + h ( i ) + y ) = h ( i ) for y = 1 , – Otherwise, p ′ ( j − r + 2 − m ) + x ) = x for x = 0 , , ..., h ( i ) + 1, and p ′ ( j − r + 2 − m ) + h ( i ) + 2) = h ( i ).In both cases, p ′ ( x ) = p ( x −
2) for the remaining times.
Remark 5.2.
Graphically the map can be visualized as follows. If the particle represented by p can beadded to S while keeping the hard-particle condition, then we do this, while changing p so that it consistsonly of up steps, starting two steps to the right of the original starting point of p . Otherwise, perform theopposite operation, changing the first particle to a path segment. In view of the graphical description, the map ϕ is clearly an involution. Moreover it is weight-preserving:In Equation (5.6), only the steps of type (i) or (ii) have a non-trivial weight. Moreover, w ( σ (step)) = w (step) according to Equation (4.13) (setting k = 0). Therefore, w ( S, p ) = w ( S ) w ( p ) = w ( S ′ ) w ( p ′ ) = w ( S ′ , p ′ ) . We have
Lemma 5.3. (5.7) r +1 X m =0 ( − r +1 − m Z G r m ( j, Z , j − r +1 − m ) ,j +2 k = 0 , ( j ∈ Z , k ≥ . Proof.
This is the partition function for pairs (
S, p ) ∈ ∪ r +1 m =0 C m × P , j − r +1 − m ) ,j +2 k , with an extra factor( − r +1 − m which ensures that the contributions of ( S, p ) and ϕ ( S, p ) cancel each other. (cid:3)
We can also consider the sum in Equation (5.7) in the case where k <
0. The sum is non-trivial in thosecases only if k ≥ − r −
1, since Z a,bt,t ′ = 0 if t > t ′ . We extend the definition of ϕ : ϕ ( S, p ) = (
S, p ) if S = ∅ or if the path p has length zero and i > Lemma 5.4. (5.8) r +1 − i X m =0 ( − r +1 − m Z G r m ( j, Z , j − r +1 − m ) ,j − i = ( − i Z G ′ r r +1 − i ( j, where Z G ′ r m ( j, k ) is the partition function of m hard particles on G ′ r , the graph G r with vertex removed(or the contribution to Z G r m in which vertex is unoccupied).Proof. We apply the involution argument in the previous Lemma for k in the range − r − ≤ k <
0. Pairs(
S, p ) which are not invariant under ϕ cancel each other. We are left with the contribution of the invariantpairs. The latter always have p = ∅ and the vertex 1 unoccupied. (cid:3) Equations (5.8) are an expression for the initial conditions of the partition functions Z , j − r +1 − m ) ,j − i with 1 ≤ i ≤ r − The T -system solution T ,j,k as a partition function of paths. Our main result in this sectionis the following.
Theorem 5.5. (5.9) T ,j,k = T ,j + k, Z , j − k,j + k . Proof.
We will show that S j,k = T ,j + k, Z , j − k,j + k satisfies the linear recursion relation (3.3) and coincideswith T ,j,k when k ∈ { , . . . , r } for any j ∈ Z . Given that (3.3) has r + 1 terms, this implies S j,k = T ,j,k for all other k .The sum r +1 X m =0 ( − m c r +1 − m ( j − k ) Z , j − k − m,j + k is equal to the sum in Equation (5.7), since Z G r m ( j,
0) = c m ( j ). Therefore, it vanishes for all j ∈ Z and k ∈ Z + . This implies that S j,k satisfies the same recursion relation (3.3) as T ,j,k .As for the initial conditions, we see from Equation (5.8) that S j,k satisfies(5.10) i X m =0 ( − m Z G r i − m ( j, S j − m − r +1 − i ) ,m = Z G ′ r i ( j, T ,j − r +1 − i ) , . We will show that the variables T ,j,k satisfy the same relations. Let(5.11) W G α i ( j ) = i X m =0 ( − m Z G α i − m ( j, T ,j − α +1 − i ) − m,m T ,j − α +1 − i ) , , (0 ≤ i ≤ α + 1)Using the recursion relations (4.12), we find W G α +1 i ( j ) = W G α i ( j −
2) + y α +1 ( j − α ) W G α i − ( j ) + y α ( j − α − W G α − i − ( j − . This is identical to the recursion relations (4.12) for Z G α i ( j, W G ( j ) = 1 and W G ( j ) = Z G ( j, − T ,j − , T ,j, Z G ( j,
0) = 0, so that W G i ( j ) = Z G i ( j, | y ( j − .Moreover, W G ( j ) = 1 ,W G ( j ) = Z G ( j, − T ,j − , T ,j − , Z G ( j,
0) = Z G ( j, | y ( j − ,W G ( j ) = Z G ( j, − T ,j − , T ,j, Z G ( j,
0) + T ,j − , T ,j, Z G ( j,
0) = 0 , where we have used the identity between Z and C , Example 4.4, the definitions (4.13), and T -systemrelations. In short, we have W G α i ( j ) = Z G α i ( j, | y =0 , valid for all initial data α = 0 ,
1. This implies(5.12) W G α i ( j ) = Z G α i ( j, | y =0 for all α. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 17
Thus, W G α i ( j ) is the partition function for i hard particles on G α with weight 0 on vertex 1, or alternatively,vertex 1 unoccupied. Therefore, W G r i ( j ) = Z G ′ r i ( j, T ,j,k and S j,k satisfy the same recursion relation and have the same boundary conditions.The Theorem follows. (cid:3) The reasoning of Theorem 5.5 can be carried through by considering paths with weights y α ( j ) replacedby the weights y α ( j, k ) of eq.(4.10). We therefore have the following. Corollary 5.6.
If we define the weights w a,b ( s, p ) as follows: w m,m ′ ( s, p ) = 1 , w m ′ ,m ( s, p ) = y m +1 ( s, p ) , ( m ∈ { , ..., r } ) ,w m,m +1 ( s, p ) = 1 , w m +1 ,m ( s, p ) = y m ( s, p ) , ( m ∈ { , ..., r − } ) ,w , ( s, p ) = 1 , w , ( s, p ) = y ( s, p ) , with the y i ( s, p ) as in (4.13) , then the following identity holds: (5.13) T ,j,k = T ,j + p,k − p Z , j − p,j + p ( { y α ( s, k − p ) , j − p ≤ s ≤ j + p, ≤ α ≤ r + 1 } ) . Proof.
We may view this as a particular case of the translational invariance T α,j,k → T α,j,k +1 of the T -system, namely that T α,j,k is expressed as the same function of the initial data { T β,j,k − p , T β,j,k − p +1 } β ∈ I r ,j ∈ Z as T α,j,p is expressed in terms of { T β,j, , T β,j, } β ∈ I r ,j ∈ Z . We deduce that T ,j,p = T ,j + p, Z , j − p,j + p ( { y α ( s, } )has the same expression as T ,j,k = T ,j,p + k − p in terms of y α ( s, k − p ), and the corollary follows, as theprefactor itself comes from the substitution T ,j + p, → T ,j + p,k − p . (cid:3) General T -system solution T α,j,k : families of non- intersecting paths. We may interpret T α,j,k directly in terms of paths by use of the determinant expression of Theorem 3.1 for T α,j,k in terms of the T ,ℓ,m . Indeed, given weighted paths on an acyclic graph Γ, say with partition function Z s,e for pathsstarting at vertex s and ending at vertex e , the Lindstr¨om-Gessel-Viennot formula gives an expressionfor the partition function of α non-intersecting paths on Γ (i.e. such that no to paths share a vertex) as Z s ,...,s α ; e ,...,e α = det ≤ i,j ≤ α Z s i ,e j . We obtain: T α,j,k = det ≤ a,b ≤ α (cid:16) T ,j + k +2 b − α − , Z , j − k + α +1 − a,j + k +2 b − α − (cid:17) = α Y b =1 T ,j + k +2 b − α − , ! Z , s ,...,s α ; e ,...,e α (5.14)where Z , s ,...,s α ; e ,...,e α stands for the partition function of families of α non-intersecting paths in the planerepresentation of Section 5.1, starting at the points s a = ( j − k + 2 a − α − , a = 1 , , ..., α and endingat the points e b = ( j + k + α + 1 − b, b = 1 , , ..., α . Alternatively, one may think of the partitionfunction Z s ,...,s α ; e ,...,e α as that of α “vicious” walkers (i.e. never meeting at a vertex) on ˜ G r , going fromthe root to the root, respectively starting at times j − k + 2 a − α − j + k + α + 1 − a , a = 1 , , ..., α , each step corresponding to a unit of time.Interpreted in this way, the T α,j,k are manifestly positive Laurent polynomials of the initial data, viathe weights y β ( t ) and the prefactor in (5.14). We therefore have the: Theorem 5.7.
The solution T α,j,k of the T -system is expressed as a positive Laurent polynomial of theinitial data x = { T β,j, , T β,j, } β ∈ I r ,j ∈ Z for all α ∈ I r and all j, k ∈ Z .Proof. The statement is clear from the above discussion for k ≥ α + 1 for which all the partition functionsin the determinant (5.14) have the form Z , t,u with t ≤ u , and therefore can be interpreted within the LGVframework. For k ≥ T -system, it is clear that T α,j,k only depends ona finite part of the initial data { T β,ℓ, , T β,ℓ, , | β − α | < k, | ℓ − j | < k } . In particular, if 0 < k < α + 1, thenas only the β ≥ α + 1 − k > T -system to some A r ′ , with r ′ = r − ( α + 1 − k ) < r . Upon renaming the initial data accordingly, we may interpret T α,j,k as T α ′ ,j,k in this new T -system, where α ′ = α − ( α + 1 − k ) = k −
1. For this α ′ ≥ k − T α,j, − k in terms of { T β,j, , T β,j, } is the sameas that of T α,j,k in terms of the reflected initial data { T β,j, , T β,j, } , hence positivity follows for k < (cid:3) Operator formulation and positivity in terms of mutated initial data
Let A be the space of Laurent polynomials in the variables { T α,j,k } . We consider the invertible “shiftoperator” d acting on the infinite-dimensional vector space over A with basis {| t i : t ∈ Z } , with d | t i = | t − i .It acts on the restricted dual space V ∗ , with basis h t | such that h t | t ′ i = δ t,t ′ , as h t | d = h t + 1 | . We considerthe algebra of formal Laurent series in d with coefficients in A acting on V . All operator relations whichwe derive below are considered in the weak sense, as identities between matrix elements. We also adoptthe operator notation for diagonal operators in this basis, for example, w a,b | t i = w a,b ( t ) | t i .6.1. An expression using operator continued fractions.
Theorem 5.5 implies(6.1) T ,j,k = T ,j + k, ( T ( j − k ) T ( j − k + 1) · · · T ( j + k − , where the transfer matrix T ( s ) is defined in Equation (5.6).We define the operator-valued transfer matrix T to be the matrix with entries h t | T a,b = T a,b ( t ) h t + 1 | .We also define operator-valued weights Y α , such that(6.2) h t | Y α = y α ( t, h t + 1 | where the y ’s are defined in (4.13).Using these, we can write(6.3) T ,j,k = T ,j + k, h j − k | (cid:0) T k (cid:1) , | j + k i = T ,j + k, h j − k | (cid:0) ( I − T ) − (cid:1) , | j + k i , where ( I − T ) − := P n ≥ ( T ) n . Therefore, the operator F = (cid:16) ( I − T ) − (cid:17) , generates the variables T ,j,k .To compute F , we row-reduce the matrix I − T . The result can be written as a non-commutativecontinued fraction: F =(6.4) − d (cid:18) − d (cid:16) − d Y − d (cid:0) · · · (1 − d Y r − − d (1 − d Y r +1 ) − Y r ) − · · · (cid:1) − Y (cid:17) − Y (cid:19) − Y ! − . Alternatively, we can write F = F where the operators F i are defined inductively: F r +2 = 0 , F k = (1 − d Y k − − d F k +1 Y k ) − , ( k = r + 1 , r, ..., , , F = (1 − d F Y ) − , F = (1 − d F Y ) − = F , where each term is understood formal power series in d .This expression is easily understood in terms of paths. Note that each time increment corresponds toan insertion of an operator d . An up step at height k followed by a down step contributes a weight d Y k ,while a level step at height k contributes the weight d Y k − .The operator generating function for paths above height k , F k , is obtained by shuffling the two followingpossibilities: (i) a level step pair k → k → k (ii) insertion of a path above height k + 1 between steps k → k + 1 and k + 1 → k (see Figure 6.1). OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 19 ....k F k+1 F k+1 y y d d d d .... k+1 = k k F Figure 6.1.
The enumeration of paths on e G r with time-dependent weights y α ( t ). The pathsfrom and to height k which stay above height k (generated by F k ) are related to those fromheight k + 1 by arranging any number of horizontal step-pairs k → k → k and up-down step-pairs k → k + 1 → k , in-between which we insert any path from and to height k + 1 staying aboveheight k + 1. The operator weights are indicated on the bottom. Mutations and operator continued fractions.
As with the Q -system of [5], we would like tohave expressions for T α,j,k as functions of other possible initial data of cluster seeds in the cluster algebra.Cluster positivity means that they are positive Laurent polynomials in this data, and we can prove thisby giving path generating functions on graphs with positive weights for them. The operator formulationintroduced above was designed to allow us to do this in the case of special seeds of the form(6.5) x M = { T α,j,m α + i | i = 0 , , α ∈ I r , j ∈ Z } , where M is a Motzkin path of length r : M = ( m , ..., m r ) with | m i − m i +1 | ≤ Q -system.The only difference is that we must now use the operator-valued transfer matrix, instead of a scalar, toaccount for time-dependent weights.6.2.1. Compound mutations and restricted initial data.
The cluster seeds in Equation (6.5) are obtainedfrom x by acting on it with a sequence of the compound mutations of the form(6.6) µ α = Y j ∈ Z µ α,j , µ α = Y j ∈ Z µ α,j . Note that the mutation matrix B has the property that B j,j ′ α,α = 0 if j = j ′ , hence µ α,j commutes with µ α,j ′ , so the compound mutations are well-defined.The mutations (6.6) act on initial data x via the simultaneous use of all relations (1.1) for all j ∈ Z totransform T α,j ; k − → T α,j ; k +1 (forward mutation) or T α,j ; k +1 → T α,j ; k − (backward mutation), the actionbeing that of µ α when k is odd and µ α when k is even. Starting from the seed x , and acting only with(6.6) generates a restricted set of cluster seeds. If, moreover, we require that each of the mutations be oneof the T -system equations, we obtain only seeds of the form x M as in (6.5). Remark 6.1.
This is very similar to the situation of Reference [5] , where seeds of type x M consist ofvariables { R α,m α ; R α,m α +1 } . Here, we replace each variable R α,m with the infinite sequence ( T α,j,m ) j ∈ Z . Operator continued fraction rearrangements.
In [5], we have shown that the generating function for R ,n may be expressed in terms of any mutated seed x M via local rearrangements of the initial continuedfraction in terms of the seed x . Here, we give the non-commutative version of the starting point, which isoperator version of the two rearrangement lemmas for fractions used in [5]. The following are operator identities to be understood as identities between matrix elements of Laurentseries in d . They are proved by a simple calculation. Lemma 6.2.
Let A , B be elements in A (( d )) . Then (6.7) 1 + d (1 − A d − d B ) − A = (cid:0) − d (1 − d B ) − A (cid:1) − Lemma 6.3.
Let A , B , C , U be elements in A (( d )) , with A + B invertible. Then (6.8) A + (1 − d (1 − U ) − C ) − B = (cid:0) − U − d (1 − d C ′ ) − B ′ (cid:1) − A ′ where (6.9) A ′ = A + B , B ′ = CB ( A + B ) − , C ′ = d − CA ( A + B ) − d Remark 6.4.
Lemma 6.2 has a path interpretation. The r.h.s. of equation (6.7) is the generating functionfor paths on the integer segment [0 , , from vertex to , with operator-valued weights: w (0 →
1) = w (1 →
2) = d, w (2 →
1) = B , w (1 →
0) = A The l.h.s. of equation (6.7) decomposes these paths into the trivial one (length , contribution ), and allthe others, which start with a step → and end up with a step → , with respective weights d and A (in this order). In-between, we have the generating function for “rerooted” paths, from vertex to vertex , which consist of arbitrary sequences of either steps → → (with weight A d ) or steps → → (with weight d B ). We call the rearrangement of this Lemma a “rerooting”. General case: mutations as rearrangements.
For each Motzkin path M , the solution of the T -systemis can be expressed in terms of the initial data at x M as(6.10) T ,j,k = T ,j + k − m ,m h j − k + m | F M | j + k − m i for some operator continued fraction F M . Our main claim is that this fraction is obtained from F (6.4) viaa succesion of applications of Lemmas 6.2 and 6.3.For each Motzkin path we will define weights Y i ( M ), ( i ∈ I r +1 ), which are monomials in x M and d .The fraction F M is a function of these. As in [5], we find that the effect of mutations on F M F M ′ is thefollowing: • If α = 1, use Lemma 6.2 to write(6.11) F M = 1 + d F ′ M Y ( M ) . where h t | Y ( m ) = T ,t,m +1 /T ,t +1 ,m h t + 1 | . Then Lemma 6.3 enables us to rewrite F ′ M as F M ′ ,a function of x M ′ . • If α >
1, apply Lemma 6.3 to the part of F M involving the weights Y β ( M ) with β ≥ α − Y β ( M ) are transformed into weights Y β ( M ′ ).We will provide the precise construction and proof in Section 6.3, but for clarity, we refer the reader toAppendix B, where the example of A is worked out completely.6.3. Paths on graphs with non-commutative weights.
In this section, we define graphs with weightsin A [ d, d − ]. The generating functions F M are path partition functions on these graphs. The graphs areidentical to those introduced in [5], and the weights contain exactly the same information contained inthe two-dimensional representation of paths on these graphs introduced in [5]. The construction presentedhere is therefore a rephrasing of these paths in terms of operators. Remark 6.5.
Although the two-dimensional representation of paths used here is identical to the one weused in [5] , we did not, in the earlier paper, have use for the full information contained in this pathrepresentation. In particular, the horizontal coordinate (“time” in our language) had no interpretation inthe context of Q -systems. Here, it corresponds to what is known as the spectral parameter in the T -systemequations. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 21 m (a) α (c)(b) Figure 6.2.
The construction of the graph Γ M (c) for some sample Motzkin path M (a) for A . All the pieces glued (b) are represented vertically. The skeleton edges in (c) are labelled 1 to19, the spine vertices 0 to 13. Since one is used to reading lattice paths from left to right, we have chosen to act on the space V ∗ instead of V . In that way, the order in which they act on the space is the same as the order the path istraversed.6.4. The target graphs Γ m . Let M be a Motzkin path. We decompose it into pieces which do notchange direction: M = M ∪ M ∪ · · · , where M i = ( a, a + k, a + 2 k, · · · , a + ( l i − k ) with k = 0 , − type of subpath is called k . All the graphs used below must be drawn vertically (see Fig.6.2), whichmakes unambiguous the notion of top and bottom edges.We construct a graph Γ M i for each i as follows: • If k = 0 then Γ M i = e G ′′ l i , the graph e G l i of Figure 5.1 (represented vertically), with its bottom andtop edges removed. • If k = 1, then Γ M i is a simple (vertical) chain with 2 l i vertices. • If k = −
1, then Γ M i is the graph e G ′′ l i (represented vertically) decorated with additional oriented“descending” edges b → a with l i + 1 ≥ b > a + 1 > M i | Γ M i +1 is the graph obtained by identifying the top edge of Γ M i with thebottom edge of Γ M i +1 . Define Γ ′′ M = Γ M | Γ M | · · · , and Γ M is Γ ′′ M together with one additional bottomand top edge and vertex. The graph Γ M is rooted at its bottom vertex.Each graph thus constructed has a spine , namely a maximal vertical chain of vertices consisting of theunprimed vertices of the various pieces glued. We label the spine vertices consecutively starting from 0 atthe bottom (see Fig.6.2(c)). The vertices off the spine, which are attached only to a vertex i are labeled i ′ . We define the skeleton of Γ M as the graph with all edges i → j removed where i > j + 1. The edges ofthe skeleton, referred to as skeleton edges are labeled 1 , , ..., r + 1 from bottom to top (see Fig.6.2(c)). A path traverses each edge of a graph in one direction or another, and in our formulation, we weight stepsin each direction differently. Therefore, we now consider the non-oriented edges in Γ M as doubly-orientededges, each orientation corresponding to a different weight.Assign a weight Y a,b ( M ) to the edge a → b . The weight of any edge away from the root, Y i,i +1 = Y j,j ′ = d . Skeleton edges Y α ( M ) pointing towards the root are independent weights, which we definebelow. Weights on edges i → i − k with k > Y i,i − k = i − k +1 Y a = i − Y a +1 ,a ( Y a,a ′ ) − ( Y a ′ ,a ) − ! Y i − k +1 ,i − k . The ordered product is taken over edges from top to bottom, along a path from vertex i to i − k . Notethat Y a,a ′ = d . By inspection of (6.12) we see that h t | Y i,i − k ∝ h t − k + 2 | , hence Y i,i − k “goes back in time”by k − The positivity theorems for { T α,j,k } . We now write T ,j,k as the partition function for paths onΓ M . The values of the skeleton weights are determined by considering the effect of a mutation on the seeddata – They are determined by a recursion relation, which can be solved explicitly.6.5.1. Transfer matrices and mutations.
As we illustrated in [5], any Motzkin path has a unique expressionas a sequence of forward mutations, m β m ′ β = m β + δ β,α where M = ( m , ..., m r ) and M ′ = ( m ′ , ..., m ′ r ).We restrict the mutations to those which increase m α by +1 only in the following two cases: • Case (i): m α − = m α = m α +1 − • Case (ii): m α − = m α = m α +1 ,(together with their boundary versions). This restricted set of mutations is sufficient to construct allMotzkin paths in the fundamental domain.The initial step in the induction is the Motzkin path M . The path interpretation on Γ M = e G r wasgiven in Section 5. The operator transfer matrix T M = T and the operator generating function F M = F are expressed entirely in terms of the d operator and the skeleton weights Y α ( M ) = Y α (6.2).The inductive step is as follows. Given Γ M and its operator weights, consider a forward mutation µ α or µ α : M M ′ . These have associated transfer matrices T M and T M ′ corresponding to the graphs Γ M andΓ M ′ . We compare the associated generating functions F M = ( I − T M ) − , and F M ′ = ( I − T M ′ ) − , usingthe row reduction process. Both are operator continued fractions, which differ locally due to the structionof the graphs. We find that the two operator continued fractions are equal to each other if and only if theweights of the graph Γ M ′ are related to those of Γ M as follows. Theorem 6.6.
Let Y ′ = Y ( M ′ ) and Y = Y ( M ) , where M ′ = µ α ( M ) or µ α ( M ) . If α = 1 , then, • Case (i): Y ′ α − = Y α − + Y α Y ′ α = Y α +1 Y α ( Y ′ α − ) − (6.13) Y ′ α +1 = d − Y α +1 Y α − ( Y ′ α − ) − d • Case (ii): in addition to the previous, we have (6.14) Y ′ α +2 = d − Y α +2 Y α − ( Y ′ α − ) − d, and Y ′ β = d − Y β d, ∀ β ≥ α + 3 If α = 1 , we simply have to substitute Y → d − Y d in the above formulas.Proof. The proof is by Gaussian elimination as in [5]. The case α = 1 is special, as it requires a rerootingof the generating function. The transformation of weights must be applied on F ′ M = ( I − T M ) − , as in(6.11), which induces the substitution Y → d − Y d . (cid:3) OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 23
Let(6.15) λ α,t,m = T α,t,m +1 T α,t +1 ,m , µ α,t,m = T α,t,m T α − ,t +1 ,m Corollary 6.7.
The skeleton weights obeying the recursions of Theorem 6.6, subject to the initial condition (6.2) are the operators Y β ( M ) , acting as h t | Y β ( M ) = y β ( M ; t ) h t + 1 | , with: y α − ( M ; t ) = λ α,t + m α − m − ,m α λ α − ,t + m α − − m ,m α − (6.16) y α ( M ; t ) = µ α +1 ,t + m α − m ,m α +1 µ α,t + m α − m ,m α × ( λ α +1 ,t + mα +1 − m ,mα +1 λ α +1 ,t + mα − m ,mα if m α = m α +1 + 11 otherwise ) (6.17) × ( λ α − ,t + mα − m ,mα λ α − ,t + mα − − m ,mα − if m α = m α − −
11 otherwise ) (6.18) Proof.
By direct check of the recursion relations (6.13-6.14). (cid:3)
Thus, we have two expressions for the generating function of T ,j,k , one in terms of the seed data x M andthe other in terms of the seed data x M ′ . We call the transition between the two expressions a mutation: Itacts on the graph Γ M and on its weights. Alternatively, it acts on the operator continued fraction expressonfor F M as a rearrangement.6.5.2. Positivity of T ,j,k . We note that the weights (6.16) y α ( M ; t ) are positive Laurent monomials of theinitial data at x M . We therefore have a positivity result: Theorem 6.8. T ,j,k + m /T ,j + k,m is the partition function for paths on the rooted graph Γ M with theweights of Theorem 6.7, starting from the root at time j − k and ending at the root at time j + k . As suchit is a positive Laurent polynomial of the mutated data at x M . General solution and strongly non-intersecting paths.
We now turn to the expression of T α,j,k interms of the mutated initial data x M . We will interpret the determinant formula Theorem 3.1 for T α,j,k `ala Gessel-Viennot, in terms of the strongly non-intersecting paths on the graph Γ M introduced in [5].Let us briefly recall the two-dimensional Γ M -lattice paths used to represent paths on Γ M , given in [5].There are fundamentally three kinds of oriented edges in Γ M : the horizontal and vertical “skeleton” edges,and down-pointing long edges, with weights which depend on the skeleton weights. The steps taken alongthese edges on Γ M are represented in Z as follows (see Fig.6.3 for an illustration): • A skeleton step i → i + ǫ , ǫ = ±
1, at time t becomes the segment from ( t, i ) to ( t + 1 , i + ǫ ) • A skeleton step i → i ′ or i ′ → i at time t becomes the segment from ( t, i ) to ( t + 1 , i ) • A long step j → i , j > i + 1 at time t becomes the segment from ( t, j ) to ( t + j − i − , i )Note that the increment of x -coordinate for each step coincides with the time shift we have associated witheach step. Indeed, all steps advance by one unit of time, except the long ones, which go back in time by j − i − ≤
0. The only difference with [5] is that we now attach time-dependent weights to the steps namelya weight y a,b ( t ) for a step a → b starting at time t (In the operator language, we have operator weights Y a,b that act as h t | Y a,b = y a,b ( t ) h t + h | , where h is the time-shift of the corresponding step, h = +1 for allsteps except the long ones, for which h = a − b − x -coordinate is nothing but the time-coordinate.Let us consider T α,j,k as a function of the initial data at x M . Writing(6.19) T α,j,k + m Q αb =1 T ,j + k +2 b − α,m = det ≤ a,b ≤ α T ,j − a + b,k + a + b − α − m T ,j + k +2 b − α − ,m Using Theorem 6.8, we may interpret T ,j − a + b,k + a + b − α − m /T ,j + k +2 b − α − ,m as the partition functionfor Γ M -lattice paths from s a = ( j − k + α + 1 − a,
0) to e b = ( j + k + 2 b − α − , down stepsup stepstarget path Figure 6.3.
Two-dimensional lattice path representation on the graph Γ M , M = (2 , ,
0) ofthe A case. We have indicated the “up” steps (i.e. away from the root) and the “down” steps(towards the root). h+v,v y i+k+v,i+v y h+v,i+v y i+k+v,v y (a) (b) (u,v) (u+h−2,v+h)(u+i,v+i)(u+i+k−2,v+i+k) 1 P P’ P P’ Figure 6.4.
A typical edge intersection of Γ M -paths (a) and the result of the flipping operationon it (b). We have indicated the weights of the steps. The paths in (b) are said to be “too close”to each other. simply a signed sum of products of such path partition functions, corresponding in turn to the partitionfunction for families of paths starting at { s a } αa =1 and ending at { e b } αb =1 , with the usual weights times thesignature of the permutation of endpoints induced by the configuration.In the standard Gessel-Viennot case, these signs produce the necessary cancellations to only leave uswith the contribution of non-intersecting paths, namely families in which no two paths share a vertex. Thisis best proved by introducing a sign-reversing involution that pairs up and cancels all the unwanted termsin the expansion of the determinant. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 25
In the case of Γ M -paths, the situation is more subtle, as paths may intersect without sharing a vertex.In [5], we have produced an involution, which allows to interpret an analogue of the determinant (6.19)as the partition function of strongly non-intersecting Γ M -lattice paths. This involution consists in flippingpaths as follows. We consider the first intersection between two paths within a family. If the intersection isat a common vertex, we interchange the portions of paths before the intersection. If it is not at a commonvertex, we flip the two paths as indicated in Fig.6.4, by switching their beginnings until the crossing.We make then the following crucial observation: Lemma 6.9.
In the generic flipping situation of Fig.6.4, the flipped pair of paths has the same (time-dependent) weight as the original one, up to the sign of the permutation of starting points, due to thefollowing relation: y h + v,v ( u + h − y i + k + v,i + v ( u + i + k −
2) = y h + v,i + v ( u + h − y i + k + v,v ( u + i + k − Proof.
By direct application of the formula (6.12) for the long edge weights. (cid:3)
The only invariant families under this involution are those where the paths do not lie “too close” toeach-other, as otherwise they get cancelled by applying a flip.Therefore all the conclusions of [5] still hold in the present case, and we have:
Theorem 6.10. T α,j,k + m / Q αb =1 T α,j + k +2 b − α − ,m is the partition function for configurations of α stronglynon-intersecting Γ M -lattice paths, with the weights of Theorem 6.7. As such, T α,j,k + m is a positive Laurentpolynomial of the mutated data at x M . Conclusion
The T -system equations are a special case of a non-commutative Q -system. That is, one can write a Q -system for non-commutative variables such that its matrix elements coincide with the T -system equations.One can think of the non-commutative Q -system as a “non-commutative” cluster algebra. Special casesof non-commutative cluster algebras have been considered in several contexts, for example the quantumcluster algebras of Berenstein and Zelevinsky [3], or the more general recursion relation introduced byKontsevich [14] (in rank 2), with similar Laurent properties, and which can be solved in some special“affine” cases using our methods [7]. The main idea is that the path formulation seems to be particularlywell adapted to the explicit solution of such problems, and makes Laurentness and positivity manifest.This will be discussed in a future publication.We should mention that there has been a great deal of interest in T -systems with more restrictiveboundary conditions [20, 12]. We hope that the construction introduced in this paper will provide a simpleway of treating such boundary conditions and their consequences. Appendix A. Discrete Wronskians
A.1.
Pl¨ucker relations.
Let P be an N × ( N + k )-matrix. Let | P b ,...,b k | be the determinant of the matrixobtained by deleting the k columns b , ..., b k of P , times the signature of the permutation that reordersthese column indices in increasing order. Then we have:(A.1) | P a ,...,a k | | P b ,...,b k | = k X p =1 | P b p ,a ...,a k | | P b ,...,b p − ,a ,b p +1 ,...,b k | . for any choice of 2 k columns a , ..., a k and b , ..., b k of P . In particular, when k = 2, we have(A.2) | P a ,a | | P b ,b | = | P b ,a | | P a ,b | + | P b ,a | | P b ,a | . for any N × ( N + 2) matrix P .Equation (A.2) implies the Desnanot-Jacobi relation. Let M be an N × N matrix, and let | M | , M ji , | M j ,j i ,i | denote the determinants of M , the minor obtained by erasing row i and column j of M , and the double minor obtained by erasing rows i , i and columns j , j of M , respectively. Let 1 ≤ i < i ≤ N and 1 ≤ j < j ≤ N, then(A.3) | M | | M j ,j i ,i | = | M j i | | M j i | − | M j i | | M j i | It is easily obtained as a particular case of eq.(A.2), for a = j , b = j , P i,a = δ i,i , P i,b = δ i,i , and M is the matrix P with columns a and b erased. Indeed, one checks directly that: | P a ,b | = | M | , | P a ,b | = | M ,N ,N | , | P a ,a | = | M N | , | P b ,b | = −| M N | , | P a ,b | = | M NN | , and | P b ,a | = −| M | .A.2. T -system as discrete Wronskians. Here present the proof of Theorem 3.1 which uses the relationsin the previous subsection.
Proof.
Consider Equation (A.3). Let N = α + 1, i = j = 1, i = j = N and choose the matrix M withentries M a,b = T ,j + a − b,k + a + b − α − for a, b = 1 , , ..., α +1. We denote by W α +1 ,j,k = | M | the corresponding“discrete Wronskian” determinant. Substituting this definition into eq. (A.3), we have(A.4) W α +1 ,j,k W α − ,j,k = W α,j,k − W α,j,k +1 − W α,j − ,k W α,j +1 ,k valid for α ∈ I r , provided we set W ,j,k = 1. Note that W ,j,k = T ,j,k by definition. Comparing eq.(A.4)with the T -system (1.1), we deduce that the T ’s and W ’s obey the same recursion relations and share thesame initial conditions at α = 0 and 1. As the system is a three-term recursion in α this determines thesolution uniquely and therefore we have W α,j,k = T α,j,k for all α ∈ I r , j, k ∈ Z . (cid:3) A.3.
Linear recursion relations.
Here, we present a proof of Theorem 3.2, that the variables T α,j ; k satisfy linear recursion relations, with constant coefficients which are the conserved quantities. Proof.
We perform the discrete analog of differentiating the Wronskian, and compute ϕ j − ,k +1 − ϕ j,k = 0.Denoting by ϕ j,k = | g , g , · · · , g r +1 | and ϕ j − ,k +1 = | f , f , · · · , f r +1 | as the determinants of columnvectors g i , f i , we note that g i +1 = f i for i = 1 , , ..., r . We may therefore rewrite(A.5) ϕ j − ,k +1 − ϕ j,k = 0 = | f , f , · · · f r , f r +1 − ( − r g | with ( f b ) a = T ,j − a − b,k + a − r − b and ( g ) a = T ,j − a,k + a − r − . Therefore, there must exist a non-triviallinear combination of the columns of the matrix which vanishes. We write it as(A.6) r X b =1 ( − b c r +1 − b ( j, k ) f b + c ( j, k ) ( f r +1 − ( − r g ) = 0 . Recall that the entries of the vectors f b depend on j, k in a very particular way, namely ( f b ( j, k )) a +1 =( f b ( j + 1 , k + 1)) a , and similarly for g . In order for the above linear combination to be non-trivial, wemust therefore have c b ( j, k ) = c b ( j − , k −
1) = · · · = c b ( j − k,
0) for all j, k ∈ Z , hence the coefficients c b only depend on the difference j − k . Finally, we may normalize the coefficients in such a way that c = 1identically, and the first part of the Theorem follows.The second part is treated analogously, by considering the difference of Wronskians ϕ j +1 ,k +1 − ϕ j,k = 0and reasoning on the rows of the corresponding matrices. (cid:3) A.4.
Conserved quantities as Wronskian determinants with defect.
Here, we give the proof ofLemma 3.4 expressing the conserved quantities of the T -system as Wronskian determinants with defects. Proof.
Let γ m ( j, n ) denote the right hand side of of eq.(3.5).It is clear that that γ ( j, n ) = T r +1 ,j + n,n + r = 1 and γ r +1 ( j, n ) = T r +1 ,j + n − ,n + r +1 = 1 as consequencesof Theorem 3.1 and of the A r boundary condition.Let p ∈ Z and define the ( r +2) × ( r +2) matrix D to be the matrix with entries D ,b = T ,j + p +1 − b,p + b − ,and D a,b = T ,j + n + a − b,n + a + b − for a = 2 , , ..., r + 2 and b = 1 , , ..., r + 2. The identity (3.3) may be recastinto a vanishing non-trivial linear combination of the columns of D , with coefficients c r +2 − b ( j )( − b − , b = 1 , , ..., r + 2, hence the determinant of D vanishes. OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 27
Expanding the determinant along the first row, we find that(A.7) 0 = det( D ) = r +2 X b =1 ( − b +1 D ,b | D b | = r +2 X b =1 ( − b − γ r +2 − b ( j, n ) T ,j + p +1 − b,p + b − , as the determinants γ m ( j, n ) are the minors | P r +2 − m | .Since the Wronskian determinant T r +1 ,j,k = 1 is non-zero, there exist no other non-trivial linear recursionrelation than Equation (3.3) with strictly fewer terms, hence the coefficients in Equation (A.7) must beproportional to those in Equation (3.3). As c ( j ) = γ ( j, n ) = 1, we deduce that γ m ( j, n ) = c m ( j ) for all m = 0 , , ..., r + 1, and the Lemma follows. (cid:3) Appendix B. Example of A : rearrangements, graphs and paths Here, we illustrate the program of Section 6.2.3 in the case r = 2. We first present the rearrangementsof the operator continued fraction F , which make positivity of R ,n manifest in all three cases. Next, weinterpret these in terms of partition functions for operator-weighted paths on graphs, to illustrate Section6.3.B.0.1. Rearrangements.
The fundamental domain for the action of mutations on the fundamental seed e x is coded by the following three Motzkin paths with 2 vertices: m = (0 , m = µ ( m ) = (1 ,
0) and m = µ ( m ) = (0 , Seed x : The continued fraction F ( y ) reads for the fundamental seed corresponding to the Motzkimpath m is:(B.1) F ( y ) = (cid:18) − d (cid:16) − d (cid:0) − d Y − d (1 − d Y ) − Y (cid:1) − Y (cid:17) − Y (cid:19) − with operators Y i , i ∈ I , acting as h t | Y i = y i ( t ) h t + 1 | , and: y ( t ) = T ,t, T ,t +1 , , y ( t ) = T ,t, T ,t, T ,t +1 , , y ( t ) = T ,t +1 , T ,t − , T ,t, T ,t, y ( t ) = T ,t +1 , T ,t, T ,t +1 , , y ( t ) = T ,t +1 , T ,t, Seed x = µ ( x ) : Following Section 6.2.3, we apply the Lemma 6.3 to F , with A = Y , B = Y , C = Y and U = 0: This yields F ( y ) = F ( w ), where(B.2) F ( w ) = − d − d (cid:16) − d (cid:0) − d (1 − d W ) − W (cid:1) − W (cid:17) − W ! − W − with operators W i , i ∈ I , acting as h t | W i = w i ( t ) h t + 1 | , with: w ( t ) = T ,t, T ,t +1 , , w ( t ) = T ,t, T ,t, T ,t +1 , , w ( t ) = T ,t +1 , T ,t, T ,t, T ,t +1 , w ( t ) = T ,t +1 , T ,t, T ,t +1 , , w ( t ) = T ,t +1 , T ,t, To obtain this, we have written W = Y , W = Y , and W = Y + Y , while W = Y Y W − and W = d − Y Y W − d , and used the T -system to simplify the expressions. Seed x = µ ( x ) : Following Section 6.2.3, we first apply the rerooting Lemma 6.2, with A = Y and B = VY , where V = (cid:0) − d Y − d (1 − d Y ) − Y (cid:1) −
18 PHILIPPE DI FRANCESCO AND RINAT KEDEM
03 4215 5423 15 6 mmm µ µ Figure B.1.
The fundamental domain for A , coded by the Motzkin paths m , m , m ,and the corresponding target graphs for the path interpretation, with their edge labels.Mutations are indicated by arrows.This allows to rewrite F ( y ) = 1 + dF ′ ( y ) Y , with F ′ ( y ) = (1 − Y d − d UY ) − . We may now apply therearrangement Lemma 6.3, with A = d − Y d , B = Y , C = Y + (1 − d Y ) − Y , and U = 0: this yields F ( y ) = 1 + dF ( z ) Y , where:(B.3) F ( z ) = − d − d (cid:16) − d Z − d (1 − d Z ) − Z (cid:17) − (cid:16) Z + (1 − d Z ) − Z (cid:17)! − Z − where Z i , i ∈ I act as h t | Z i = z i ( t ) h t + 1 | , with: z ( t ) = T ,t, T ,t +1 , , z ( t ) = T ,t − , T ,t +1 , T ,t, T ,t +1 , T ,t, , z ( t ) = T ,t +1 , T ,t − , T ,t, T ,t − , z ( t ) = T ,t − , T ,t +1 , T ,t − , T ,t, T ,t, , z ( t ) = T ,t, T ,t − , and Z is a long step weight, expressed in terms of the skeleton weights as: Z = Z ( d Z ) − Z (a particularcase of Eq.(6.12)). Note that it is diagonal, namely: h t | z = z ( t ) h t | , where z ( t ) = z ( t ) z ( t ) z ( t −
1) = 1 T ,t, T ,t − , The above weights follow from the identifications: Z = d − Y d + Y , Z = Y Y Z − , Z = d − Y Y Z − , Z = d − Y d − Y d Z − , Z = d − Y d − Y d Z − and Z = d − Y d , and the use of the T -system to simplifythe expressions.B.0.2. Paths on graphs with operator weights.
Recall first that the continued fractions F ( y ) , F ( w ) aresuch that T ,j,k /T ,j + k, = h j − k | F i | j + k i , i = 0 ,
2, while, due to the re-rooting, we have T ,j,k /T ,j + k − , = h j − k + 1 | F | j + k − i . OSITIVITY OF THE T-SYSTEM CLUSTER ALGEBRA 29
The three above operator continued fractions may be interpreted in terms of path counting as follows.We have represented in Figure B.1 the three rooted target graphs Γ m , attached to the three Motzkin paths m = m , m , m , together with their edge labelings. We have the following Theorem B.1.
For i = 0 , , , the quantities h t | F i | t ′ i are the partition functions for paths on the graphs Γ m i , from and to the root, starting at time t and ending at time t ′ , and with operator weights defined asthe product over the operator weights for each successive step of the path, in the same order. The weightsare d per step away form the root, and respectively Y i , Z i and W i per step towards the root, along the edgelabeled i .Proof. The proof is a straightforward adaptation of the argument of Section 6.1: it uses operator trans-fer matrices T i , and amounts to performing the Gaussian elimination of I − T i , in order to compute (cid:0) ( I − T i ) − (cid:1) , , where 0 indexes the root vertex on Γ m i . We give explicit expressions below. (cid:3) We now list the transfer matrices for the three cases above. In all cases, we have F i = (cid:0) ( I − T i ) − (cid:1) , . T = d Y d Y d Y d d Y Y T = d Z d Z d Z d d Z Z Z T = d W d W d W d
00 0 0 W d W Remark B.2.
Note that, as opposed to the two other cases, the transfer matrix T is not made of diagonaloperators times d , as Z is diagonal, hence goes back one step in time compared to the other operators Z i , i = 1 , , ..., . This necessity for the longer descending steps to go back in time was already observed in [5] in the two-dimensional representation of the Γ m -paths. References [1] V. Bazhanov and N. Reshetikhin,
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