Quantum Cohomology via Vicious and Osculating Walkers

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u^(n)k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{\hat{u}}(n)_{k}}$$\end{document}-WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra.


Introduction
Let Gr n,n+k be the Grassmannian of n-planes in C n+k and consider its small quantum cohomology ring q H * (Gr n,n+k ). The latter has the following presentation [36] q H * (Gr n,n+k ) ∼ = Z[q][e 1 , . . . , e n , h 1 , . . . , h k ]/I, (1.1) where the two-sided ideal I is generated by the coefficients of the following polynomial in the auxiliary variable x, Denoting by (n, k) the set of all partitions whose Young diagram fits into the n × k bounding box, a vector space basis of q H * (Gr n,n+k ) is given by the finite set {s λ } λ∈(n,k) of Schubert classes s λ = det(h λ i −i+ j ) 1≤i, j≤n = det(e λ i −i+ j ) 1≤i, j≤k , where λ is the conjugate partition of λ. Note that this definition includes the special cases s (r ) = h r and s (1 r ) = e r .

TORIC SCHUR POLYNOMIALS AND FROBENIUS STRUCTURES
Quantum cohomology had its origin in mathematical physics and appeared first in works of Gepner [17], Intriligator [23], Vafa [37] and Witten [38] in connection with the fusion ring F n,k of the gauged u(n) k Wess-Zumino-Novikov-Witten (WZNW) model. It has apparently been proved in the no longer publicly available work [1] that F n,k ∼ = q H * (Gr n,n+k )/ q − 1 . F C n,k = F n,k ⊗ Z C is referred to as Verlinde algebra in the literature.
Denote by λ ∨ the partition obtained by taking the complement of the Young diagram of λ in the n × k bounding box and recall the intersection pairing of Schubert classes, η(s λ , s μ ) := Gr n,n+k s λ · s μ = δ λ ∨ μ . PROPOSITION 1.1. F C n,k with bilinear form η is a commutative Frobenius algebra and its coproduct is given by n,k s ν = d≥0,μ∈(n,k) q d s ν/d/μ ⊗ s μ , s ν/d/μ := λ∈(n,k) C ν,d λμ s λ . (1.6) Setting e r = 1≤i 1 <···<i r ≤n x i 1 · · · x i r and h r = 1≤i 1 ≤···≤i r ≤n x i 1 · · · x i r where x = (x 1 , . . . , x n ) are some commuting indeterminates the Schubert classes can be identified with Schur functions. In light of (1.6) it is then natural to consider so-called toric Schur functions [33] s ν/d/μ (x) = λ∈(n,k) The last identity can be seen as a combinatorial definition of s ν/d/μ in terms of monomial symmetric functions; it is the weighted sum over all toric skew tableaux, which is the subset of the cylindric skew tableaux having at most k boxes in each row. This generalises the notion of an ordinary skew Schur function, s ν/μ = λ K ν/μ,λ m λ , where the sum is over all skew (semi-standard) tableaux and K ν/μ,λ is the ordinary Kostka number; see, e.g. [29]. Since for vanishing degree d the Gromov-Witten invariants equal the Littlewood-Richardson coefficients, one has s ν/0/μ = s ν/μ when d = 0. In the case of infinitely many variables one obtains cylindric Schur functions which have been investigated in [18,31] and in [28] as special case of affine Stanley symmetric functions; see also [7] for a formulation of a random process on cylindric partitions. For a generalisation of cylindric or toric Schur functions to special cases of Macdonald functions in the context of the su(n) k fusion ring, see [27].

EXACTLY SOLVABLE LATTICE MODELS: VICIOUS AND OSCULATING WALKERS
In this article, we identify the toric Schur polynomial (1.7) with the partition function of exactly solvable lattice models in statistical mechanics [3], the lock-step vicious and osculating walker models which have appeared in connection with problems such as percolation in physical systems [12] and the counting of alternating sign matrices [8]. Both models can be formulated in terms of special nonintersecting paths on a square lattice which in the present context we choose to have dimensions n × (n + k) and k × (n + k); see Figure 1 for examples. Fixing start and end positions of the walkers in terms of partitions μ, ν ∈ (n, k) and identifying the left with the right lattice edge, we show that there is a bijection between toric tableaux of shape ν/d/μ and non-intersecting paths of the mentioned mod- Figure 1. Examples of vicious and osculating walks on a square lattice with N = n + k = 4 + 5 = 9. The paths of two osculating walkers can touch but they do not intersect. els on the cylinder. The degree d ∈ Z ≥0 is the number of walkers crossing the boundary, where the horizontal strip is glued together to obtain the cylinder. The weight λ of a toric tableau fixes the number of horizontal path edges in each lattice row.
Denote by H = (H ν,μ ) and E = (E ν,μ ) the transfer matrices whose elements are the partition functions of a single lattice row for the vicious and osculating walker models on the cylinder with fixed start and end configurations μ, ν ∈ (n, k). Then the matrix elements of the powers H n and E k give the partition functions for the lattices with n and k rows mentioned above.
Given a square s = i, j in the Young diagram of a partition λ denote by c(s) = j − i its content and by h(s) = λ i + λ j − i − j + 1 its hook length. THEOREM 1.2. For fixed start and end points μ, ν ∈ (n, k), the number of vicious and osculating walker configurations on the cylinder is given by where the sums run over all integers d ≥ 0 and λ ∈ (n, k) such that |λ| + |μ| − |ν| = d(n + k).
Note that (1.9) provides a linear system of equations for Gromov-Witten invariants.
It is not obvious from their definition, but we will show that the row-to-row transfer matrices E and H commute and possess a common eigenbasis {ê λ } λ∈(n,k) , the so-called Bethe vectors. They yield the idempotents of the Verlinde algebra F C n,k of the gaugedû(n) k -WZNW model. THEOREM 1.3 (idempotents of the Verlinde algebra). Let V n,k be the complex linear span of the Bethe vectors {ê λ } λ∈(n,k) . The generalised matrix algebra (V n,k , ) obtained by settingê λ ê μ = δ λμêλ is isomorphic to F C n,k .
The same basis has been employed in [24,Thm 10.11] to provide an alternative derivation of the presentation (1.1). They are identical with the Bethe vectors of the so-called XX-Heisenberg spin chain [24,Rm 10.3] and, thus, one can identify the ring (1.1) with the conserved quantities or "quantum integrals of motion" of this quantum spin-chain. There are close parallels with recent developments regarding the link between topological field theories and quantum integrable models [32] as well as the quantum cohomology of the cotangent bundles of Grassmann varieties [19,35].
The new result in this article is the connection of the quantum cohomology of the Grassmann varieties themselves with the mentioned statistical lattice models which allows one to relate the counting of lattice paths on the cylinder to Gromov-Witten invariants and to reveal a deeper underlying algebraic structure which we now explain.

QUANTUM GROUP STRUCTURES
The combinatorial connection with the mentioned exactly solvable lattice models is underpinned by an algebraic description known as the quantum inverse scattering method which is the name of a general procedure based on the works of the Faddeev School; see, e.g. [6] and references therein. Here, we show that this method can be applied to the quantum cohomology ring. Our starting point is a solution to the Yang-Baxter equation is the so-called monodromy matrix. The matrix entries of M(x) can be interpreted as vertex-type operators whose commutation relations are encoded in the matrix R. The latter generate the so-called Yang-Baxter algebra which has the structure of a graded bi-algebra. In particular, one can interpret (1.10), (1.11) as the defining relations of a "quantum group"; this is similar to the construction of a Yangian symmetry on the quantum cohomologies of cotangent bundles of Nakajima varieties in a recent preprint by Maulik and Okounkov [30]. Set N = n + k and for simplicity denote q H * (Gr n,k ) ⊗ Z C by q H * n,k . Then there exists a vector space isomorphism are of polynomial degree n and k, respectively and on each preimage of q H * n,k one has the functional identity In particular, the map given by E i = A i + q D i → e i and H j = A j + q D j → h j with i = 1, . . . , n and j = 1, . . . , k yields an algebra isomorphism A n,k ∼ → q H * n,k , where A n,k ⊂ End(q H * n,k ) is the commutative subalgebra generated by the coefficients of the Yang-Baxter algebra elements E(x) and H (x).
The combinatorial results (1.8) and (1.9) then follow from the fact that one recovers the row-to-row vicious and osculating walker transfer matrices at x = 1, i.e. H = H (1) and E = E(1). An alternative way to express the relation between the Yang-Baxter algebras and q H * n,k is to use the coproduct of the Frobenius algebra.
This last results implies in particular that one can use the commutation relations of the Yang-Baxter algebra encoded in (1.10) to compute the toric Schur functions (1.7) by moving the H -operators past the B-operators. We will thus derive the following formula in terms of divided difference operators (also called Demazure or Bernstein-Gelfand-Gelfand operators). Set with s i being the transposition which acts by switching x i with x i+1 and define ∇ i = ∂ n−1+i · · · ∂ i+1 ∂ i . COROLLARY 1.6. Choose y i = x i+n in (1.14) and introduce the "generating function" Then we have the following formula in terms of Demazure operators, where the notation s μ (x)|| · · · denotes the coefficient of the Schur function s μ (x).
We will demonstrate on a simple example how toric Schur functions can be explicitly computed by invoking the last formula in Section 6.
Another identity for toric Schur functions-which is a direct consequence of the quantum group structure and does not seem to have appeared previously in the literature-is the following sum rule. where λ is any partition in the n × n square.
This identity is a true "quantum relation" as it becomes trivial for q = 0, i.e. there exists no analogue of this relation for skew Schur functions where d, d = 0. Similar identities hold also for N / ∈ 2N but look more complicated. They will be stated in Section 6.

OUTLINE OF THE ARTICLE
In Section 2, we introduce some preliminary combinatorial notions regarding 01words and partitions.
In Section 3, we discuss in detail the vicious and osculating walker models. While these have been introduced in the literature previously, our conventions differ from the usual ones by rotating the lattice 45 • and choosing a special set of weights. We also analyse in depth the related Yang-Baxter algebras and show that both models are related via level-rank duality. We derive a matrix functional equation relating the transfer matrices of both models. Section 4 contains the algebraic Bethe ansatz construction of the idempotents of the Verlinde algebra. As a byproduct of this construction, we obtain novel expressions for Schur functions as matrix elements of the above mentioned Yang-Baxter algebras.
Section 5 states explicit bijections between vicious and osculating walker configurations on the cylinder and toric tableaux, which can be interpreted as a special subset of semi-standard tableau of skew shape. As a corollary, we obtain that Postnikov's toric Schur polynomials are the partition functions of the specialised vicious and osculating walker models. The sum rule (1.9), relating Gromov-Witten invariants to the counting of vicious and osculating walker configurations on the cylinder, is then an immediate consequence.
We will conclude with stating the proofs of the various identities for toric Schur functions arising from the quantum group structure.

Preliminaries
Throughout this article we consider non-negative integers N , n, k ∈ Z ≥0 such that N = n + k and set I := {1, . . . , N }. Let V = Cv 0 ⊕ Cv 1 and denote by V * its dual. Consider the tensor product V ⊗N . We identify the standard basis B = {v w 1 ⊗ · · · ⊗ v w N : w i = 0, 1} ⊂ V ⊗N with the set of 01-words of length N , W = {w = w 1 w 2 . . . w N : (2.1) For convenience, we are employing the Dirac notation and denote by w| the dual basis with w|w = N i=1 δ w i ,w i . Furthermore, we shall denote by W n = {w ∈ W : |w| = N i=1 w i = n} the subset of all 01-words with n 1-letters, by B n ⊂ V ⊗N its image under the above map (2.1) and by V n ⊂ V ⊗N the subspace spanned by the corresponding basis elements in B n . As (2.1) is a bijection, we can also introduce the inverse map whose image we denote by There are alternative descriptions of the elements in B n which will be useful for our discussion. Namely, consider the set of partitions λ whose Young diagram fits into a bounding box of height n and width k; we shall denote it by (n, k). Define a bijection (n, k) → W n via where (λ) = ( 1 , . . . , n ) with 1 ≤ 1 < · · · < n ≤ N denote the positions of 1-letters in w(λ) from left to right. We assume the latter to be periodic, that is we set We shall denote the image of the inverse of the map (2.2) by λ(w) and by |λ the corresponding ket vector in B n . Note that the correspondence (2.2) can be easily understood graphically: the Young diagram of the partition λ traces out a path in the n × k rectangle which is encoded in w. Starting from the left bottom corner in the n × k rectangle go one box right for each letter 0 and one box up for each letter 1; see Figure 2 for an example. For later purposes, we introduce the notation for ∈ I and set n +N (λ) = n (λ) + n for any ∈ Z.
Exploiting the bijection (2.2) there are two operations on 01-words which are induced by taking the complement of λ ∈ (n, k) in the bounding box, λ → λ ∨ := (k − λ n , . . . , k − λ 2 , k − λ 1 ), and by considering the conjugate partition λ ∈ (k, n). The corresponding 01-words w(λ ∨ ), w(λ ) are obtained from w(λ) via the maps (2.4) respectively. Note that these maps yield bijections W n → W n and W n → W k . We will also make use of the combined map w → w # := (w ∨ ) = (w ) ∨ which is simply the exchange of 0 and 1-letters.
There is one additional map Rot : W n → W n which we require for our discussion: set w → Rot(w) := w 2 w 3 . . . w N w 1 which translates via ( 2.2) to the action Exploiting the last expression one then derives the following formula with |λ| = Note that obviously we have Rot N (λ) = λ. For obvious reasons we will refer to Rot as the rotation operator. The maps (2.4) and (2.5) are significant for our discussion as they constitute symmetries of Gromov-Witten invariants.

Vicious and Osculating Walkers
We recall the definition of the lock step vicious walker model originally introduced by Fisher [12] and show that this statistical mechanics model with the correct choice of weights and boundary conditions is closely related to the small quantum cohomology ring of the Grassmannian: its partition function can be interpreted as generating function of 3-point, genus zero Gromov-Witten invariants.
There is another statistical model introduced by Brak [8], called osculating walkers, 1 which in our setting turns out to be dual or complementary to the vicious walker model. Namely, we will show that the transfer matrices of the vicious and osculating walker models are given in terms of analogues of complete and elementary symmetric functions in certain noncommutative variables: the generators of the nil affine Temperley-Lieb algebra.

VICIOUS WALKERS: VERTEX AND LATTICE CONFIGURATIONS
We start with the 5-vertex formulation of the vicious walker model. Fix two integers N > 0 and 0 ≤ n ≤ N and consider the square lattice  The weight of a configuration C is defined as the product over its vertex weights, where v i, j denotes the vertex obtained by intersecting the ith horizontal lattice line with the jth vertical one. That is, a vertex configuration is a 4- Connecting the 1-letters in each vertex configuration as shown in Figure 3, it is easy to see that each lattice configuration corresponds to a configuration of 1 We note that Brak's model is a six-vertex model, i.e. has different Boltzmann weights from the one discussed here and in particular has one more allowed vertex configuration. However, the crucial vertex configuration with two paths approaching each other arbitarily close is also present here and we therefore adopt his nomenclature; see Figure 4. n non-intersecting paths, where a path γ = ( p 1 , . . . , p l ) is a sequence of points p r = (i r , j r ) ∈ L such that either p r +1 = (i r + 1, j r ) or (i r , j r + 1), i.e. a connected sequence of horizontal and vertical edges as depicted in Figure 8.

TRANSFER MATRIX AND NIL TEMPERLEY-LIEB POLYNOMIALS
We now interpret the possible vertex configuration in the i th row and j th column as a map L(x i ) : We will therefore drop the row and column labels and, in addition, often suppress the dependence on the indeterminate x i in the notation. Thus, the values of the horizontal edges label the basis vectors in V i while the values of the vertical edges label the basis vectors in V j . The mapping is from the NW to the SE direction through the vertex. That is, label with a, b, c, d = 0, 1 the values of the edges in Figure 3 in clockwise direction starting from the W edge. Interpret the corresponding weight L ab cd = wt(v i, j ) as the matrix element of the map L, where we set L ab cd = 0 whenever the vertex configuration is not allowed. We then obtain

5)
where the matrix R is given by Note that the matrix R(x, y) is non-singular for generic x, y, since det R(x, y) = −y 2 /x 2 . The solution L to the Yang-Baxter equation can be used to define an algebra in End(V ) ⊗N ∼ = End(V ⊗N ), called the Yang-Baxter algebra which plays a central role in the quantum inverse scattering method; see, e.g. [6] for a textbook and references therein. In fact, the Yang-Baxter algebra comes naturally equipped with a coproduct. First rewrite the L-matrix in the form where the matrix elements are polynomials in the indeterminate x with coefficients in End V . The following is a known result how to introduce a bi-algebra structure on solutions of the Yang-Baxter equation; we therefore omit the proof. Here the maps ε, act on the coefficients when expanding with respect to the spectral variable x. The set of solutions of (3.5) equipped with , ε forms a bialgebra, so in particular L is again a solution of (3.5).
Note that we do not have a Hopf algebra structure as the L-operator is not invertible.
Repeatedly applying and the isomorphism End From the Yang-Baxter equation one then deduces-among others -the following commutation relations for the entries of the monodromy matrix, We now describe these commutation relations in terms of divided difference operators. This will allow us in a subsequent section to derive the generating function (1.15) for toric Schur functions and formula (1.16) mentioned in the introduction. We will also use these relations below to construct eigenvectors of the vicious walker transfer matrix.
where the x i 's are some commuting indeterminates. There is a natural action of the symmetric group S m on R m by permuting the x i 's where we denote by Despite first appearance, the latter map polynomials into polynomials: because of linearity it suffices to consider the following action on a monomial The difference operators yield a representation of the nil Hecke algebra H m (0), that is they obey the relations Remark 3.4. The action (3.11) of the difference operators ∂ i is the familiar action of the Hecke algebra H m (q) on the ring of polynomials via Demazure or Bernstein-Gelfand-Gelfand operators in the limit q → 0. In this limit, the algebra H m (0) is known as the nil-Hecke algebra.
We now have the following simple but important lemma.

Then we have the commutation relations
Proof. This is a direct computation using (3.5), (3.6) and the definition (3.8).
To describe the action of the matrix elements A, B, C, D in combinatorial terms we now relate them to a particular representation of the affine nil Temperley-Lieb algebra. The latter is the unital, associative algebra generated by {u 1 , . . . , u N } and relations (3.14) We will refer to the subalgebra generated by {u 1 , . . . , u N −1 } as the finite nil Temperley-Lieb algebra. Note that the latter is a quotient of the nil Hecke algebra.
PROPOSITION 3.6 hopping operators. The map yields a faithful representation of the finite nil Temperley-Lieb algebra over V n . If we further set we obtain a faithful representation of the affine nil Temperley-Lieb algebra over Remark 3.7. In [24] a free fermion description of the small quantum cohomology ring was presented. The relationship between the current description of the affine nil Temperley-Lieb algebra and that in loc. cit. is given via the following formulae In particular, one easily verifies that The last proposition is then an obvious reformulation of [24, Prop 9.1] and we therefore omit the proof.
The action (3.16) suggest to introduce the following quasi-periodic boundary conditions, σ ± i+N = −q ±1 σ ± i and σ z i+N = σ z i . We also introduce the adjoint endo- where q * = q −1 . For ease of notation, we will henceforth simply write V n,q := C[q, q −1 ] ⊗ V n and V ⊗N One easily deduces the following identities which we state without proof; compare with [24, Section 8.2 and Lemma 9.3].

by the maps in (2.4) and (2.5).
Then where all indices are understood modulo N .
We have the following expressions for the Yang-Baxter algebra in terms of the f i 's: Proof. From the definition of the monodromy matrix one easily derives the expression where the sum runs over all compositions α = (α 1 , . . . , α N −1 ) with α i = 0, 1. The assertion is now immediate.
The action of the operators A r in V n ⊂ V ⊗N is easily described using the wellknown bijections between 01-words and partitions explained in Section 2. LEMMA 3.10 (horizontal strips). Let μ ∈ (n, k), and A(x) = r ≥0 x r A r . Then the polynomials act on the basis vector |μ by adding all possible horizontal r -strips to the Young diagram of μ such that the result λ lies within the n × k bounding box, A r |μ = λ−μ=(r ) |λ . Remark 3.11. Polynomials in noncommutative variables such as (3.20) have been considered in e.g. [13] and their affine extensions in [33] and [24,25]. Here the new aspect is that we show that they are part of the Yang-Baxter algebra of the vicious walker model.
Proof. Using the bijection (2.2) one readily verifies that either f i |μ = |λ , where λ is obtained by adding a box in the (i − n)th diagonal of the Young diagram of μ, or if this is not possible f i |μ = 0. Consider now a consecutive string f i+r · · · f i+1 f i |μ = |λ with r ≤ r and suppose w i (μ) = 1, w j (μ) = 0 for i < j ≤ i + r ; otherwise the action is trivial. Then the 1-letter at position i in w(μ) is moved past r 0letters whose position each decreases by one. Since μ k+1− j = j (μ ) + j, where N + 1 − k+1− j (μ ) are the positions of 0-letters in w(μ), we find λ k+1− j − μ k+1− j = 1. Thus, λ is obtained from μ by adding a horizontal strip of length r . This proves the assertion.
From the last lemma, the action of the remaining Yang-Baxter algebra generators is obtained by observing that σ + 1 |λ = |λ 1 − 1, . . . , λ n − 1 if λ n > 0 or σ + 1 |λ = 0 if λ n = 0. In contrast, σ − N |λ =|μ if μ can be obtained by adding a column of maximal height to the Young diagram of λ and then subtracting a boundary ribbon of length N starting in the first row. Otherwise, we have σ − N |λ = 0.
Proof. The first assertion is immediate from the commutation relation (3.9 ). The operator H : is the so-called row-to-row transfer matrix of the vicious walker model on the cylinder, i.e. its matrix elements are the partition functions of one lattice row when imposing quasi-periodic boundary conditions in the horizontal direction of the square lattice. The transfer matrix can be written as the following partial trace, For completeness, we summarise our previous findings on the combinatorial action of the Yang-Baxter algebra (3.18) in the following formula for each r = 0, 1, . . . , N . LEMMA 3.14. Let μ ∈ (n, k) then

23)
where in the second sum λ [1] denotes the partition obtained from λ by adding a boundary ribbon of length N starting in the first and ending in the n th row. For r > k, we have H r |μ = 0.
Proof. We postpone a detailed proof to Section 5 where we discuss the bijection between row configurations of the vicious walker model and toric tableaux; see the proof of Proposition 5.8.

OSCULATING WALKERS: VERTEX AND LATTICE CONFIGURATIONS
Define another 5-vertex model but this time on a k × N lattice with k = N − n, Denote by E the set of its horizontal and vertical edges. As before, we define the weight of a lattice configuration C :  minates (x 1 , . . . , x k ). The corresponding L -matrix reads Via a straightforward computation, which we omit, one arrives at the following result.
PROPOSITION 3.15. We have the identity Note that the matrix R (x/y) is non-singular for generic x, y.
In complete analogy with our previous discussion of the vicious walker model, we can define also here a monodromy matrix and Yang-Baxter algebra.
Similar like before, one now verifies the following combinatorial action of the Yang-Baxter algebra.
act on |μ with μ ∈ (n, k) a partition by adding all possible vertical r -strips to the Young diagram of μ such that the resulting diagram lies still within the n × k bounding box, A r |μ = λ/μ=(1 r ) |λ .
We now impose again quasi-periodic boundary conditions in the horizontal direction of the square lattice and introduce the corresponding transfer matrix E : As in the case of the vicious walkers model one now exploits the Yang-Baxter equation to arrive at the following set of commuting polynomials in the f i 's.

COROLLARY 3.18 (integrability). We have the expansion
The following lemma shows that vicious and osculating walkers are related via level-rank duality. Proof. This is an easy consequence of the second relation in (3.17) and the formulae (3.21), (3.34).
An immediate consequence is the following combinatorial action of the E r 's which simply follows from the combinatorial action of the vicious walker transfer matrix discussed previously. LEMMA 3.20. Let μ ∈ (n, k) then where in the second sum λ [1] again denotes the partition obtained from λ by adding a boundary ribbon of length n + k starting in the first and ending in the n th row. For r > n, we have E r |μ = 0.
We can also determine the commutation relations between the Yang-Baxter algebras of the vicious and osculating walker models via a third and final Yang-Baxter relation, which-again-is obtained by a tedious but direct computation which we omit.

PROPOSITION 3.21. We have the additional identity
where R is the singular matrix Exploiting this last result one now proves in a similar manner as before that the transfer matrices H (x) and E(y) commute for arbitrary x, y.

COROLLARY 3.22. We have the following commutation relations
and In particular, finite and affine nil Temperley-Lieb polynomials H r , E r pairwise commute, i.e. H r E r = E r H r for all r, r = 0, 1, . . . , N .
We have the following functional relation between the transfer matrices H, E which generalises the known relation of the generating functions for elementary and complete symmetric polynomials in the ring of symmetric functions.

PROPOSITION 3.23. We have the operator identity
with respect to W ⊕ W ⊥ and one only needs to verify that M, M yield the asserted terms on the right hand side of (3.40) using that We leave the details of this last step to the reader since it is a simple computation. COROLLARY 3.24. Let A n,k ⊂ End(V n,q ) be the commutative algebra generated by {H j } k j=0 and {E i } n i=0 . The map E i → e i and H j → h j provides a canonical algebra isomorphism A n,k ∼ = q H * (Gr n,n+k ) ⊗ Z C.
Proof. This is clear from our previous results: the combinatorial actions (3.23) and (3.35). In particular, one has that H r |V n,q = 0 for r > k and E r |V n,q = 0 for r > n. The Yang-Baxter relations (3.5), (3.26), (3.36) provided us with a proof that the E i 's and H j 's commute among themselves and with each other. Finally, the last result (3.40) gives the desired algebraic dependence expressed in (1.2).

Algebraic Bethe Ansatz and Idempotents
We show in this section that the eigenvectors of the transfer matrices H, E for the vicious and osculating walker models are the idempotents of the fusion ring of the gauged WZNW model. The eigenbasis of the affine nil Temperley-Lieb polynomials H λ := H λ 1 H λ 2 . . . and E λ := E λ 1 E λ 2 . . . with λ i < N has been previously constructed in [24, Section 10] using the free fermion formalism mentioned earlier.
Here, we obtain a new result: we show that the same eigenbasis can be obtained from the Yang-Baxter algebras (3.18), (3.30) of the vicious and osculating walker models by a procedure known as algebraic Bethe ansatz; see, e.g. [6] for a textbook reference. In particular, this construction will furnish us with the vertex-type operator formulae ( 4.3), (4.4) for Schur functions which differ from the known expressions; see, e.g. [29, Chapter I, Section 5, Ex 29] and references therein.

4)
where ρ k = (k, . . . , 2, 1). So, in particular, we have the following identities between matrix elements and Schur functions, Proof. The proof is graphical. Draw a diagonal line across the square lattice as indicated in Figure 5. We postpone the proof of this claim to the next section where we establish a bijection between non-intersecting paths on the cylinder and toric Young tableaux. The claim will then follow as a special case by considering only those vertex weights which lie below the dotted line and noting that the vertices above the line contribute the total weight factor x n 1 x n−1 2 · · · x n . Thus, redrawing the lattice paths as indicated in Figure 5 and using the known sum formula s λ (x) = |T |=λ x T (see, e.g. [29, Chapter I]), we arrive at the desired expression The second identity simply uses the fact that σ + i = σ − N −i , which is easily verified. Thus, from A(x) = A (x) (see Lemma 3.19 for q = 0) and (3.18), (3.30) it follows that B(x) = xC (x). Applying the involution on both sides of the previous identity and swapping n and k afterwards, the second assertion is proved. PROPOSITION 4.4. Fix 0 ≤ n < N . For each λ ∈ (n, k), the vector e λ is an eigenvector of H = A + q D and E = A + q D with y r |0 · · · 01 r 0 · · · 0 is an eigenvector of H (x) with eigenvalue (1 + qx N )/(1 − x/y) provided that y N q = 1. This computation generalises to n > 1, using an induction argument one finds with the help of the equations (3.10), Employing these identities, one deduces that for e(y) to be an eigenvector the socalled Bethe roots y i have to satisfy the following set of constraints, The explicit solution to these equations is easily obtained and can be found in [24,Prop 10.4].
To arrive at the eigenvalue equation for E one can either perform a similar computation using the operators C introduced earlier and employing level-rank duality or employ Proposition 3.23.
The statement for n = N is obvious and follows from the definition of H, E.
Let P be the linear operator V ⊗N → V ⊗N defined by P|λ = |λ ∨ and T the linear operator defined by T |λ = |λ and T q = q −1 T for each λ ∈ (n, k) with N = n + k. We introduce the operators (4.9) Note that it follows from the definition that we have the identity An analogous formula holds for E * (x).

LEMMA 4.5. We have the dual affine Pieri rules
and |λ , (4.11) where the notation μ [1] in the second sum in both formulae stands for the partition obtained by adding a boundary rim hook of length n + k to μ.

Proof. Simply note that Pσ
The rest is then a straightforward computation which follows along similar lines as in the case of H r and E r using the explicit polynomial expressions in the f i 's given in (3.21), (3.34).
The last result motivates us to define an isomorphism (V ⊗N q ) * → V ⊗N q by identifying the dual basis vector λ| with P|λ = |λ ∨ , and more generally any vector In other words, the pairing v|w between the vector space and its dual is given by the bilinear form η(v, w) := (v, PT w), where (|λ , |μ ) = δ λμ . LEMMA 4.6. The left or dual eigenvectors e * λ defined by e * λ |e μ = δ λμ are given by the expansion Proof. This is a direct consequence of our previous discussion, applying PT to (4.3) and then using part (i) of Theorem 4.1 .
Remark 4.7. In [24] the operatorĤ N = −q N j=1 σ z j was defined, which differs from H N = q|0 0|, which simply is the projection onto the unique basis vector in V 0,q .
In loc. cit. the decomposition V ⊗N q = N n=0 V n,q has been employed to define via the eigenbasis {e λ } λ∈(n,k) the following set of operatorŝ Clearly, we have H r =Ĥ r for 0 ≤r < N . The new insight here is that these operators originate from a Yang-Baxter algebra which has a local description in terms of a statistical vertex model on a square lattice.

THE VERLINDE ALGEBRA
Since the q-dependence can be removed via a simple rescaling of the Bethe roots, y → q − 1 N y (compare with (4.8)) we now set for simplicity q = 1. Interpret the eigenbasis {e λ } λ∈(n,k) as a complete set of orthogonal idempotents of an associative, unital and commutative algebra. Next, we show that the resulting generalised matrix algebra is isomorphic to the Verlinde algebra F C n,k , where we recall from the introduction that F n,k ∼ = q H * (Gr n,n+k )/ q − 1 . THEOREM 4.8 (idempotents of the Verlinde algebra). Endow V n with the following product, e λ e μ := δ λμ ||e λ || 2 e λ and bilinear form η(|λ , |μ ) = δ λ ∨ μ . Then (1) (V n , , η) is a commutative Frobenius algebra.
Remark 4.9. An analogous statement holds true for the su(n) k -WZNW fusion ring using so-called ∞-friendly walkers; see [24]. In [26, Section 5] the role of the Bethe vectors as idempotents has been highlighted and Section 7 of loc. cit. explains how the construction might generalise to other integrable models.
Our main motivation to emphasise the Frobenius structure is the connection with the toric Schur polynomials mentioned after Proposition 1.1 in the introduction.

Bijections Between Walks and Toric Tableaux
In this section, we prove that the lattice configurations of the vicious and osculating walker models are in bijection with toric tableaux which were used in [5,33]. This will provide a combinatorial link between these statistical mechanics models and the quantum cohomology ring: the partition functions of vicious and osculating walkers on the cylinder are toric Schur functions. To keep this article self-contained we start by recalling the definition of toric tableaux following [33]. As mentioned earlier, toric tableaux are a particular subset of cylindric tableaux; see [18] for the original definition of cylindric (plane) partitions, lattice paths and cylindric Schur functions. For r = 0, the cylindric loop can be visualised as a path in Z × Z determined by the outline of the Young diagram of λ which is periodically continued with respect to the vector (n, −k) . For r = 0, this line is shifted r times in the direction of the lattice vector (1, 1); see Figure 6 for an illustration.
We shall refer to λ/d/μ as a cylindric skew diagram of degree d.
A cylindric skew diagram ν/d/μ which has at most one box in each column will be called a (cylindric) horizontal strip and one which has at most one box in each row a (cylindric) vertical strip. The length of such strips will be the number of boxes within the skew diagram.  Figure 7 for an example of a toric skew diagram. Note that for d = 0, we recover the familiar skew diagram of two partitions, i.e. λ/0/μ = λ/μ. DEFINITION 5.4 (cylindric tableaux). A (semi-standard) tableau T of cylindric shape λ/d/μ is a mapping T : λ/d/μ → N of the squares of the associated diagram such that in each row and column of connected squares the numbers are, respectively, weakly increasing (left to right) and strictly increasing (top to bottom). See Figure 7 for an example. As for ordinary skew diagrams, we define the weight vector α(T ) = (α 1 , α 2 , . . .) by setting α i to be the number of i-entries in T . DEFINITION 5.5 (Quantum Kostka numbers). The cardinality of the set of all cylindric tableaux T of shape ν/d/μ and weight α is denoted by K ν/d/μ,α .
As already pointed out in the introduction the quantum Kostka number K ν/d/μ,α in (1.3) equals the number of semi-standard cylindric tableaux of weight α, and specialises for d = 0 to the ordinary Kostka number K ν/μ,α .
We will also make use of K ν /d/μ ,α , the number of conjugate cylindric and toric diagrams ν /d/μ , which are obtained by interchanging n and k. Here, ν , μ ∈ (k, n) denote the conjugate partitions of ν, μ. DEFINITION 5.6 (Cylindric Schur functions). Introduce the following generalisation of a skew Schur function, where m α are the monomial symmetric functions in an infinite set of variables x i and the sum runs over all cylindric tableaux of fixed shape ν/d/μ.
Note that the cylindric Schur function (5.1) specialises to an ordinary skew Schur function for d = 0.

LATTICE CONFIGURATIONS AND QUANTUM KOSTKA NUMBERS
Throughout this section, we assume ν, μ ∈ (n, k). Denote by ν,μ , ν,μ the sets of all allowed lattice configurations C, C on the cylinder for the vicious and osculating walker models where the values of the lower and upper vertical lattice edges are fixed by the 01-words w(ν) and w(μ), respectively. It will also be convenient to consider the subsets ν/d/μ ⊂ ν,μ of configurations which do have a fixed number of 2d outer horizontal edges with value one (they come in pairs due to the quasiperiodic boundary conditions in the horizontal direction), including in particular the special case of d = 0 (q = 0) when there are no outer horizontal edges ν/0/μ . Finally, we introduce for each α = (α 1 , . . . , α n ) ∈ Z n ≥0 the subsets Analogously, we define for β = (β 1 , . . . , β k ) ∈ Z k ≥0 the set ν/d/μ (β) as the lattice configurations C of the osculating walker model which have weight wt (C ) = x β . PROPOSITION 5.8. The set of allowed lattice configurations ν/d/μ and ν/d/μ are in bijection with the sets of toric skew tableaux of shape ν/d/μ and ν /d/μ , respectively. In particular, where α = (α 1 , . . . , α n ) is some weight vector with non-negative integer entries.
Note that for d = 0 and μ = ∅ the statement specialises to Claim 4.3 which we used earlier to prove Lemma 4.2. Proof. We concentrate on the vicious walker model, the generalisation to the osculating walker model will then be obvious. First, we state the bijection. Recall from Figure 3 that each lattice configuration C ∈ ν/d/μ defines an n-tuple of nonintersecting paths γ = (γ 1 , . . . , γ n ) of which d cross the boundary. Draw the Young diagram Y (μ) of μ in the bounding box (n, k). Reading the 01-word of μ from left to right take the path which originates at 1 (μ), that is from the first 1-letter in w(μ), and note down the lattice rows of each of its horizontal edges starting from the top, say i 1 ≤ · · · ≤ i r . Add a corresponding row of boxes with entries i 1 ≤ · · · ≤ i r to the bottom row of Y (μ). If the path has no horizontal edges do not add any boxes. Continue with the path originating from the second 1-letter in w(μ) at 2 (μ) and write the corresponding filled boxes in the row above the bottom row of Y (μ) starting at the first square which does not lie in μ. Continue until you have reached the last 1-letter in w(μ); see Figure 8 for an example.
That the described map is indeed a bijection follows from the following lemma.
LEMMA 5.9. Each toric skew tableau T of shape ν/d/μ can be written as a sequence of toric horizontal strips, i.e. there is a unique sequence of cylindric loops ( Since toric tableaux can be seen as a special subset of ordinary (semi-standard) tableaux of shape ν[d]/μ the proof of this lemma follows along very similar lines as in the case of ordinary tableaux (see, e.g. [29, Chap I]) and we therefore omit it.
Thus, it suffices to derive the assertion for n = 1 in which case d = 0 or 1. Then we have the following generalisation of (2.2) to cylindric loops where i+n (μ) = i (μ) + N , i+n (λ) = i (λ) + N . Analogously, one finds for the conjugate loops, Using these formulae together with the action (3.15), (3.16), one now easily verifies that an allowed row configuration of the vicious walker model, i.e. a non-vanishing matrix element of a monomial in the hopping operators f i appearing in (3.21), defines a toric horizontal strip ν/d/μ with d = 0, 1 and vice versa. In particular, the action of a consecutive string such as f i · · · f 2 f 1 f N · · · f j b μ = b λ can only be nonzero if there are as many consecutive 0-letters in w(μ) starting at j and ending at i as there are hopping operators f l in the string. Thus, the horizontal strip has at most length k. The bijection for the osculating walkers is analogous. Start with the leftmost path originating at 1 (μ) and for each horizontal path edge in lattice row i add a box labelled i in the rightmost column of the k × n bounding box beneath the Young diagram of μ . Continue with the second path placing the boxes now in the second column from the right and so forth. The result is a conjugate toric tableau of shape ν /d/μ ; see Figure 9 for an example.
The proof that the described map is indeed a bijection employs the expansion (3.34) and follows closely along similar lines as in the previous case of vicious walkers. We therefore omit it.
Number the lattice columns of L ( L ) from left to right. That is, the ith column is the collection of horizontal lattice edges ( p, p ) ∈ E ( E ) such that p 1 = i and p 1 = i + 1. Let C : E → {0, 1} (C : E → {0, 1}) be a vicious (osculating) walker configuration with start and end positions (ν) and (μ), ν, μ ∈ (n, k) . The number of horizontal path edges in column i is the sum over the values of the horizontal lattice edges in column i in configuration C (C ).
Remark 5.11. The integer d min (ν ∨ , μ) is the minimal power appearing in the quantum product s μ * s ν ; see [16]. In fact, there exists an interval of integers [d min , d max ] describing all powers occurring in this product. This was first conjectured in [39] and proved in [33].
Proof. Denote by w i ∈ {0, 1) the values of the, respectively, horizontal and vertical lattice edges in column i and row j in the configuration C or C . Since the argument is completely analogous for both, vicious and osculating walkers, we only consider the former for the rest of the proof. By construction, we have v . Note that the allowed configurations shown in Figure 3 preserve the sum of values on the N and W edge and the E and S edge, that is w i . Hence, we compute The assertion now follows by observing that θ N = d, the number of horizontal path edges on the boundary. LEMMA 5.12 Let λ ∈ (n, k). K ν/d/μ,λ = 0 unless |λ| + |μ| − |ν| = d N .

PATH COUNTING AND GROMOV-WITTEN INVARIANTS
We now compute the weighted sums over lattice configurations of the vicious and osculating walker models; these are called partition functions. Consider the following operator products H ( . Then by construction, we have for the vicious walker model and for the osculating walker model, where ν|X |μ is shorthand for b(ν), Xb(μ) and wt(v i, j ), wt (v i, j ) denote the vertex weights of the vicious and osculating walker models in Figures 3, 4.
Let c(s) = j − i be the content and h(s) = λ i + λ j − i − j + 1 the hook-length of a square s = (i, j) ∈ λ. As a special case of the last corollary we obtain the following solution to the counting problem of non-intersecting paths on the cylinder which is a refinement of the one stated in the introduction.

Toric Schur Function Identities
We are now in the position to derive the formula (1.16) and the generating function (1.15) stated in the introduction. We start with proving the identity (1.14).
But according to Proposition 1.1 the right hand side-apart from the monomial factor x δ n -is the image of the Schubert class s λ under the coproduct of the Verlinde algebra. The analogous result holds true for osculating walkers using Lemma 3.19.
We now invoke the commutation relations of the Yang-Baxter algebra (3.18) to provide a closed formula for the above matrix element in terms of the divided difference operators (3.11).
Proof of Corollary 1.6. Recall Lemma 3.5, then it follows from the definition . For our purposes it suffices to make the stronger assumption that f does not depend on x i , x i+1 and, thus, we simply write H (x i+1 ; q)B(x i ) = ∂ i B(x i+1 )H (x i ; −q) as an operator identity. We prove the following formula by induction, H (x n ; q)B(x n−1 ) · · · B(x 1 ) = ∂ n−1 · · · ∂ 2 ∂ 1 B(x n ) · · · B(x 2 )H (x 1 ; (−1) n−1 q) .
Equating powers of q on both sides, we find the claimed identities.