Partial domain wall partition functions

We consider six-vertex model configurations on an n-by-N lattice, n =<N, that satisfy a variation on domain wall boundary conditions that we define and call"partial domain wall boundary conditions". We obtain two expressions for the corresponding"partial domain wall partition function", as an (N-by-N)-determinant and as an (n-by-n)-determinant. The latter was first obtained by I Kostov. We show that the two determinants are equal, as expected from the fact that they are partition functions of the same object, that each is a discrete KP tau-function, and, recalling that these determinants represent tree-level structure constants in N=4 SYM, we show that introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.


Introduction
The discovery of classical and quantum integrable structures on both sides of the anti-de Sitter/conformal field theory correspondence, AdS/CFT, in the late 1990's and early 2000's, culminating in [1,2] and in the intensive rapid developments that followed these seminal works, has been beneficial to all subjects involved 1 . On the one hand, integrability is widely considered to be a viable approach to proving the AdS/CFT correspondence. On the other, ideas and insights from AdS/CFT will continue to enrich the subject of integrability. 0.1. Partial domain wall boundary conditions. The purpose of this note is to study six-vertex model configurations on a rectangular lattice with n horizontal and N vertical lines, n N , as in Figure 1, that satisfy (n×N ) partial domain wall boundary conditions, pDWBC's. In our conventions 2 , these are defined as follows.
1. All arrows on the left and right boundaries point inwards, 2. n u (n l ) arrows on the upper (lower) boundary, such that n u + n l = N − n, also point inwards, 3. The remaining n + N arrows on the upper and lower boundaries point outwards, 4. The locations of the inward-pointing arrows on the upper and lower boundaries, with n u and n l fixed, are summed over. The corresponding partition function is an (n×N ) partial domain wall partition function, pDWPF. For n = N , and n u = n l = 0, we recover Korepin's domain wall boundary conditions, DWBC's [4], and the pDWPF reduces to Izergin's domain wall partition function, DWPF [5,6]. 0.2. A brief history of partial domain wall configurations. The configurations considered in this work were first introduced in the work of Bogoliubov, Pronko and Zvonarev in their study of boundary correlation functions in the presence of DWBC's [7], and subsequent works [8,9,10,11]. However, in these works, they were camouflaged by the fact that they were paired with complementary configurations to produce (N×N ) configurations with conventional DWBC's, and that only a subset of the positions that the inverted n u (n l ) arrows on the upper (lower) boundaries can take were included in the statistical sum, as one does in computations of boundary correlation functions.
In [12,13], Escobedo, Gromov, Sever and Vieira studied 3-point functions of three gaugeinvariant single-trace length-N i operators, O i (x i ), i ∈ {1, 2, 3}, that are composed of elementary scalars in SU (2) subsectors of N = 4 supersymmetric Yang-Mills theory 3 . Using the connection with quantum integrable models, Escobedo et al. obtained a sum expression for the studied structure constants. In [14], Gromov, Sever and Vieira considered the same structure constants as in [12,13] in the special case where O 1 and O 3 , in the conventions of [14], are BPS operators. That is, the rapidity variables {x 1 } and {x 3 } that characterize these two operators are taken to infinity. In this special case, the sum expression of [12] simplifies to a sum expression for a quantity that they refer to as A{x 2 }.
In [15], the sum expression of [12,13] was evaluated in determinant form. This determinant is essentially Slavnov's determinant expression for the scalar product of a Bethe eigenstate and a generic state in a periodic spin-1 2 XXX chain. In vertex model terms, Slavnov's determinant is the partition function of rational six-vertex configurations with 2n horizontal lines (auxiliary spaces), N vertical lines (quantum spaces), and boundary conditions that are specified in [16,15]. The limit used in [14] to produce A{x 2 } is achieved in six-vertex model terms by simply deleting the n horizontal lines that represent the Bethe eigenstate, as explained in the sequel. The resulting configurations are the partial domain wall configurations discussed in this note. 0.3. Outline of contents. In Section 1, we recall basic definitions related to the six-vertex model. In Section 2, we start from (N ×N ) domain wall configurations and delete N − n horizontal lines to obtain (n × N ) partial domain wall configurations. For simplicity we consider the case n u = N − n and n l = 0. Once this case is understood, the general case is straightforward to obtain. The corresponding (n×N ) pDWPF is obtained starting from Izergin's (N ×N ) determinant expression for the (N ×N ) DWPF of the initial domain wall configuration, taking the rapidity variables of the lines that are deleted to infinity, and normalizing appropriately to obtain the pDWPF Z N×N in (N ×N ) determinant form.
In Section 3, we start from the (2n×N ) configurations that describe the scalar product of an n-magnon Bethe eigenstate and an n-magnon generic state, on an N -site periodic spin-1 2 chain. We delete the n horizontal lines that describe the Bethe eigenstate to obtain (n×N ) partial domain wall configurations with n u = N − n and n l = 0. The corresponding (n×N ) pDWPF is obtained starting from Slavnov's (n× n) determinant expression for the scalar product, taking the rapidity variables of the lines that we deleted (which are the Bethe roots) to infinity, and normalizing appropriately to obtain the pDWPF Z n×n in (n×n) determinant form. Z n×n was first derived by Kostov [17,18,19]. Expanding Z n×n , one obtains the sum expression of Gromov et al. [12]. Starting from the trigonometric Slavnov scalar product, we also derive the trigonometric version of Z n×n .
Each determinant, Z N×N and Z n×n , is a function of the set {x} of cardinality n, associated with the n horizontal lines, and the set {y} of cardinality N , associated with the N vertical lines. In Section 4, as an independent check of the correctness of our expressions for Z N×N and Z n×n , we show that they can be written as polynomials in each of their variables x i , with the same bound on their degree, and that they satisfy the same recursion relations and initial condition. This proves that they are equal, as expected from the fact that they are different expressions for the same partition function. In Section 5, we recall basic facts regarding Casorati determinants (the discrete analogues of Wronskians) and discrete KP τ -functions, then we show that pDWPF's are discrete KP τ -functions in the {x} as well as in the {y} variables.
In Section 6, we recall a mapping that Gromov and Vieira use in [20,21] to introduce 1-loop corrections into the 0-loop expressions of certain structure constants in N = 4 supersymmetric Yang-Mills theory, and show that the (n × N ) pDWPF remains a determinant under this mapping. In Section 7, we include remarks on recent developments. 0.4. Glossary of frequently used notation. {x} ({y}) is a set of rapidity variables that do not satisfy Bethe equations and that flow along horizontal (vertical) lines. We always take {x} and {y} to be free variables. {b} is a set of rapidity variables that do satisfy the Bethe equations and that flow along horizontal lines. When a set {x} has cardinality N , we sometimes indicate this by writing {x} N . At times we also use the notation for Vandermonde determinants in the variables {x} N . [x − y] = x − y in the rational case, and [x − y] = sinh(x − y) in the trigonometric case.

Six-vertex model configurations
In this section we recall basic definitions related to the six-vertex model on an (n × N ) square lattice, n N , including vertex model descriptions of Korepin's domain wall configurations on an (N×N ) lattice [4], Slavnov's scalar product configurations on a (2n×N ) lattice [16], and the determinant expressions for these objects [5,22]. Finally, we define the partial domain wall configurations and the corresponding partial domain wall partition functions.
1.1. Lines, orientations and rapidity variables. Consider a square lattice with n horizontal lines and N vertical lines that intersect at (n × N ) points, n N . We order the horizontal lines from bottom to top and assign the i-th line an orientation from left to right and a rapidity variable x i . We order the vertical lines from left to right and assign the j-th line an orientation from bottom to top and a rapidity variable y j . See Figure 1. The orientations that we assign to the lattice lines are matters of convention and are meant to make the vertices of the six-vertex model, that we introduce shortly, unambiguous.  1.3. Weights, configurations and partition functions. Assign every vertex v ij a weight w ij that depends on the specific orientations of its arrows, and the rapidities x i and y j that flow through it. Any lattice configuration with a definite assignment of arrows is assigned a weight equal to the product of the weights of its vertices. The partition function of the lattice in Figure 1 is the sum of the weights of all lattice configurations which respect the boundary conditions that we impose.
1.4. Six vertices that conserve arrow flow. Since every arrow can point in either direction, there are 2 4 = 16 possible (types of) vertices. We are interested in models with 'conservation of arrow flow'. That is, the only vertices with non-zero weights are those such that the number of arrows that point toward the intersection point of the vertex is equal to the number of arrows that point away from it. These are six such vertices shown in Figure  2. The remaining vertices have zero weights. In this work, we study the rational and the trigonometric six-vertex model. The former is a special case of the latter. For the rational six-vertex model, we use the weights For the trigonometric six-vertex model, we use the weights where [x] ≡ sinh(x). The weights in Equations (2) and (3) satisfy the Yang-Baxter equations and unitarity. The parametrization in the trigonometric case is not unique. The reason for using the parametrization in Equation (3) is explained in Subsection 2.6. (2) and (3), as x → ∞, the rational weights become

Limiting form of the weights. From Equations
while the trigonometric weights become   Izergin's determinant. Following [5], in the rational parametrization of Equation (2) the DWPF is given by In the trigonometric parametrization of Equation (3), the DWPF is given by where we use the notation |x| = N k=1 x k .
1.8. The scalar product. This is another standard object that is defined in this work in six-vertex model terms 4 as the partition function of the configuration in Figure 4, [22,23,16].   1.9. Slavnov's determinant. Following [22] we assume that one set of variables in Figure  4, {b} n , obeys the Bethe equations 5 . In the rational and trigonometric parametrizations they are given by , ∀ 1 i n respectively. Assuming that the Bethe equations hold, in the rational parametrization the scalar product has the determinant representation In the trigonometric parametrization, it is given by 4 The scalar product is usually defined as a vacuum expectation value of algebraic Bethe Ansatz operators.
For our purposes, this formalism in unnecessary. 5 In this work, all rapidity variables denoted by b i are assumed to obey Bethe equations, while all rapidity variables denoted by x i or y j are free.
The partial domain wall partition function, pDWPF. Let n be an integer satisfying 1 n N . Consider the partition function generated by deleting the top (N − n) rows from the lattice in Figure 3, or the top n rows from the lattice in Figure 4, and whose top boundary is summed over all arrow configurations. We denote these objects by Z 1 ({x} n |{y} N ) and Z 2 ({x} n |{y} N ) respectively, and represent them by the lattices in Figure  5.  We emphasize that, unlike the usual domain wall configurations, the top boundary segments in Figure 5 are not fixed to definite arrow configurations but are summed over just as the bulk segments.
As we will see in Subsection 2.2, up to a numerical coefficient, Z 1 ({x} n |{y} N ) is the leading term in Z({x} N |{y} N ) as x N , . . . , x n+1 → ∞. In this limit, the contribution from the top (N − n) rows of Figure 3 becomes trivial, and we are left with the lattice shown in Figure 5. For this reason, Z 1 ({x} n |{y} N ) is a partial domain wall partition function, pDWPF.
One can also calculate the pDWPF Z 2 ({x} n |{y} N ) as the leading term of the scalar product S({x} n , {b} n |{y} N ) as b n , . . . , b 1 → ∞. In this limit, the contribution from the top n rows of Figure 4 becomes trivial, and we are left with the lattice shown in Figure 5. This is discussed in Subsection 3.1.
In this paper we calculate Z 1 ({x} n |{y} N ) and Z 2 ({x} n |{y} N ) by taking the two limits described above. The starting points for these calculations are, respectively, Izergin's determinant formula for Z({x} N |{y} N ) and Slavnov's determinant formula for S({x} n , {b} n |{y} N ). In the case of the rational six-vertex model, whose vertex weights are invariant under the reversal of all arrows, the two quantities Z 1 ({x} n |{y} N ) and Z 2 ({x} n |{y} N ) are in fact equal (which is easily verified by comparing the two lattices in Figure 5). Therefore in the rational parametrization we obtain two different determinant expressions for the same object.
1.11. Deleting lines from opposite boundaries. By symmetry of Z({x} N |{y} N ) in the variables {x} N , we are free to distribute the rapidities x N , . . . , x n+1 over the horizontal lines of the lattice in any way we wish, prior to taking the limit x N , . . . , x n+1 → ∞.
For example, we can choose to place the variables x N , . . . , x m+1 on the lowest lines of the lattice and x m , . . . , x n+1 on the highest, where m is some integer satisfying n m N . In the limit x N , . . . , x n+1 → ∞, the bottom (N − m) and the top (m − n) rows become trivial, and we obtain the lattice shown in Figure 6. The lattice sum in Figure 6 is equal to the one on the left of Figure 5, up to multiplication by an overall factor. In the rational six-vertex model, this factor is the binomial coefficient

Domain wall partition function in the infinite-rapidity limit
In this section, we obtain a determinant expression for the pDWPF starting from Izergin's formula for the DWPF, Equation (6), and taking appropriate limits. The (N × N ) determinant that we obtain is 'hybrid' in the sense that it contains n rows of the type in Izergin's formula, and (N − n) rows of Vandermonde determinant-type.
2.1. One rapidity becomes infinite. Consider the rational DWPF. Due to the domain wall boundary conditions, and the conservation of arrow flow, the top row of the lattice always contains precisely one c + vertex, while all remaining vertices in that row are of the type a + or b + . Using the asymptotic behaviour of the vertex weights in Equation (4) and the definition of Z 1 ({x} N −1 |{y} N ), it is easy to see that and the pDWPF Z 1 ({x} N −1 |{y} N ) can be computed from the DWPF as

(N −n) rapidities become infinite. Consider the lattice representation of the pDWPF
The top boundary of this lattice consists of a sum over N i possible arrow configurations. From now on, we make remarks which apply to the internal part of this lattice, assuming that the boundary is fixed to any one of these N i configurations.
Consider the x i -row of vertices 6 . Any configuration that this row takes must contain at least one c + vertex and m pairs of {c + , c − } vertices, m = 0, 1, 2, · · · In the large x i limit, the leading contribution to Z 1 ({x} i |{y} N ) corresponds to m = 0. Taking multiple counting into consideration, we obtain Iterating this result through i = {N, . . . , n + 1} we obtain where the limits are to be taken sequentially, starting with x N .

2.3.
Limit of Izergin's determinant as one rapidity becomes infinite. Starting from the expression in Equation (6) for Z 1 ({x} N |{y} N ), it is simple to take the limit specified in Equation (12). Absorbing the factor (6), writing x N = 1/ǫ and taking ǫ → 0, we get and setȳ j = y j − 1 for convenience. In the limit, every entry of the final row goes to 1, hence · · · 1 6 All vertices through which the x i rapidity variable flows.
is the i-th complete symmetric function in two variables y j ,ȳ j , given by the generating series Proof. Let P N −n denote the proposition that Equation (19) is true. Based on Equation (17) for the one-rapidity case, we see that P 1 is true. Let us assume that P N −n is true and show that this implies P N −n+1 . Multiplying Equation (19) by x n , making the change of variables x n = 1/ǫ and taking the limit ǫ → 0 gives Consider the functions f (n) j (ǫ) in the n-th row of the above determinant, which is defined in Equation (16). Since ǫ is small, they can be expanded in powers of ǫ using the definition of the elementary and complete symmetric functions where e k (y 1 , . . . , y N ) and h k (x 1 , . . . , x n−1 ) are elementary and complete symmetric functions, given respectively by the generating functions Using the series expression in Equation (21) for the row of entries f (n) j (ǫ), one can see that all terms in the first sum with 0 k (N − n − 1) give no contribution to the determinant since they are linear combinations of the lower (N − n) rows, and the first sum starts at k = (N − n). Taking the limit ǫ → 0, all higher order terms in this sum vanish.
Studying the k = (N − n) term in the series in Equation (21), it is clear that many of its sub-terms do not contribute to the determinant either. In fact, only the sub-term corresponding to l = m = (N − n) survives, and this is identically h N −n (y j ,ȳ j ). Therefore we obtain This completes the proof of Equation (19) for all 0 n N − 1, by induction.

2.5.
A 'partial Vandermonde' way to write the determinant. A simple check shows that the highest order term in Using row operations to cancel all terms of lower order and extracting an overall factor of (N − n)! from the determinant in Equation (19), we obtain (24) is our (N ×N ) determinant expression for the pDWPF. As previously mentioned, the top n rows are of Izergin determinant-type, whereas the lower (N − n) rows are of Vandermonde determinant-type.
2.6. Towards the trigonometric pDWPF. Using the asymptotic behaviour of the trigonometric weights given in Equation (5), we can repeat the procedure of Subsection 2.2 to derive the relation between trigonometric partial domain wall partition functions. Iterating this result through i = {N, . . . , n + 1}, we obtain Equation (26) is the trigonometric analogue of Equation (14).

2.7.
The trigonometric pDWPF as an (N ×N )-determinant. Starting from Izergin's trigonometric determinant, Equation (7), and taking the limits in Equation (26), it is straightforward to show that We omit the details since they are similar to those in the rational case.

Slavnov scalar product in the infinite-rapidity limit
In this section we obtain an alternative expression for the pDWPF, by starting from Slavnov's formula for the scalar product (Equation (9)) and taking appropriate limits. The resulting expression is an (n×n) determinant.
3.1. n rapidities become infinite. Using the six-vertex model representation of the scalar product, Figure 4, it is possible calculate the partial domain wall partition function in the alternative way where the limits are sequentially, starting with b n . The argument which underlies Equation (29) is the same as the one that underlies Equation (14).
3.2. The infinite rapidity limit of Slavnov's determinant. To obtain an alternative determinant expression for the pDWPF, we start from Slavnov's determinant in Equation (9) and perform the limits specified in Equation (29). We do this using induction, by proving the following result.
Proof. Let Equation (30) be a proposition, P n−m . We begin with the proof of P 1 . Using Equation (9) for the scalar product, we have In the limit being considered, we have t which proves P 1 . Now we assume that P n−m is true and show that this implies P n−m+1 . Taking Equation (30) where ǫ is small in view of the limit being taken. Then we can write where we abbreviate det(·) 1 i n by | · |, and where the first (m − 1) columns are as before, so we do not write them. Now we can move all terms in the prefactor which depend on b m inside the m-th column, and take the limit. This gives us   (29) and (40), we see that the partial domain wall partition function can be written as where we have simultaneously reversed the order of variables in the Vandermonde and the order of columns in the determinant. In contrast to the determinant in Equation (24), which is (N ×N ), the determinant in Equation (41) is (n×n). Equation (41) is due to I Kostov [17,18,19].
3.4. Writing Z 2 ({x} n |{y} N ) as a sum over partitions. To conclude the section, we show that the determinant in Equation (41) can be expanded as a certain sum, which is the precise form in which it appears in [14]. Let us define the functions Using Laplace's formula for the determinant of a sum of two matrices, we write the expression in Equation (43) as a sum over all partitions of the integers {1, . . . , n} into disjoint sets {α} n−m = {α 1 < · · · < α n−m }, {β} m = {β 1 < · · · < β m }. The result is where we have abbreviated 1 i<j n (x j − x i ) = ∆{x} n and sgn(P ) denotes the sign of the permutation P {1, . . . , n} = {α 1 , . . . , α n−m , β 1 , . . . , β m }. It is possible to extract common factors from the determinant in the sum in Equation (44), and write it as Putting this expression back into Equation (44), we get Up to simple changes in variables, this is the sum expression in [14].

The trigonometric version of Kostov's determinant.
For completeness, we give the trigonometric version of the pDWPF obtained taking limits of Slavnov's scalar product in Equation (10). The starting point in the calculation is the relation which arises from the lattice version of the scalar product in Figure 4 and that of the pDWPF on the right of Figure 5, and the asymptotic behaviour of the weights in Equation (5).
Repeating the ideas already developed in this section and working from Slavnov's determinant in Equation (10), Equation (46) ultimately leads us to the expression for the trigonometric pDWPF.

Equivalence of determinants
In this section we show directly that the determinants in Equations (24) and (41) are equal. We do this for completeness and as a check of our limit calculations, because on the surface it is not apparent that the two expressions coincide. We restrict our attention to the determinants obtained from the rational six-vertex model, because as we have already mentioned the two lattice sums in Figure 5 are not equivalent in the trigonometric parametrization.
Our approach is to convert both Equations (24) and (41) to polynomials, by multiplying them by an overall factor (precisely the factor present in the denominator of the b and c weights). We distinguish the resulting expressions by calling them Z N×N ({x} n |{y} N ) and Z n×n ({x} n |{y} N ), in reference to the size of the determinants in question, and show that both Z N×N ({x} n |{y} N ) and Z n×n ({x} n |{y} N ) satisfy a list of conditions which the pDWPF must itself obey. This proves that they are equal, since the conditions only admit a unique solution.
4.1. Set of properties which characterize the pDWPF. Let Z({x} n |{y} N ) be the partition function of either of the lattices in Figure 5, but whose weights are given by C. For all n 2, Z({x} n |{y} N ) satisfies D. Z(x 1 |{y} N ) is known explicitly for all N 1, and is given by The second equality in Equation (51) evaluates the sum over l, and can be established by induction on N . These four properties uniquely determine the functions Z({x} n |{y} N ), for all 1 n N . This is because, from A, Z({x} n |{y} N ) is a polynomial in x n , and from B and C, it is known at more points than its degree.
Combining results, we see that property C is satisfied. Finally, when n = 1, we have Laplace expanding this determinant along the first row and using the Vandermonde determinant identity, we obtain Comparing this polynomial in x 1 with the polynomial in Equation (51) at the points x 1 = {y 1 ,ȳ 1 , . . . , y N ,ȳ N }, we find that they are equal, and property D is satisfied. N j=1 (x i − y j + 1). We denote this by Z n×n ({x} n |{y} N ). Usingx = x + 1,ȳ = y − 1, it is given by Clearly Z n×n ({x} n |{y} N ) is a polynomial in x n , and the highest possible degree it can obtain in this variable is N . So property A is satisfied. Since all {y} N dependence is in the products Rearranging the entries of the top (n − 1) rows to write them as Extracting factors which are common to each row of the determinant, we obtain where we have defined the matrix entries Subtracting (column j + 1)/y N from (column j) for all 1 j < n, it is easy to show that Substituting this into Equation (60), we have verified that property C is satisfied. Finally, when n = 1, observe that Z n×n (x 1 |{y} N ) is identically equal to the right hand side of Equation (51)

Useful identities for h i {x}. From Equation (64), it follows that
Note that by applying ∆ m to a degree i complete symmetric function, h i {x}, one obtains a complete symmetric function h i−1 {x} of degree i − 1, in the same set of variables {x}.
where 3 n N , and In words, if τ {x} has m i copies of the variable x i , then τ +i {x} has (m i + 1) copies of x i and the multiplicities of all other variables remain the same, while τ −i {x} has one more copy of each variable except x i . In the simpler notation  If Ω is a Casoratian matrix, then det Ω is a Casoratian determinant. Casoratian determinants are discrete analogues of Wronskian determinants. 5.7. Notation for column vectors and determinants. We introduce the column vector notation and for the corresponding column vector where the multiplicity of the subset of variables x k1 ,. . . , x kn is increased by 1. We introduce the determinant notation for the determinant with shifted multiplicities. 5.8. Identities for Casoratian determinants. Following [24], Equations (74) and (75) can be used to perform column operations in the determinant expressions for τ [1] and τ [1,...,n] , to obtain the two identities 5.9. Casoratian determinants are discrete KP τ -functions. Following [25], consider which is identically zero. For notational clarity, we have used subscripts to label the position of columns of zeros. Laplace expanding the left hand side of Equation (82) in (N×N ) minors along the top (N ×2N ) block, we obtain with 1 x k · · · x n−2 k denoting the omission of the k-th row of the matrix, we see that Equation (84) (52), is a Casoratian determinant. The discrete derivatives are taken with respect to any of the variables y j . From the above discussion, this is sufficient to show that Z N×N ({x} n |{y} N ) is a τ -function of discrete KP in {y} N .
The first step is to rearrange Equation (52) by bringing the numerator of the prefactor P({x} n , {y} N ) inside the determinant. We do this by multiplying the j-th column of the determinant by n k=1 (x k − y j )(x k −ȳ j ), for all 1 j N . The j-th column of the resulting determinant has entries which are polynomial in y j . After a routine calculation, we obtain where the coefficients c ik {x} depend on the row of the matrix and are given by It remains to take the Vandermonde ∆{−y} N inside the determinant of Equation (86). This is essentially the same as proving the Jacobi-Trudi identity for Schur functions, see [26]. The final result is Up to the Vandermonde factor in the denominator, which is a constant in {y} N , this is clearly a Casoratian determinant.
5.11. Z n×n ({x} n |{y} N ) is a discrete KP τ -function in {x} n . We can repeat the above procedure to write Z n×n ({x} n |{y} N ) as a Casoratian determinant, whose discrete derivatives are with respect to any of the variables x i . Starting from Equation (57), we already have Z n×n ({x} n |{y} N ) as a determinant whose i-th row entries are polynomials in x i . Expanding these polynomials in powers of x i , we obtain where the coefficients d kj {y} depend on the column of the matrix and are given by Taking the Vandermonde ∆{x} n inside the determinant of Equation (89), we have Hence Z n×n ({x} n |{y} N ) is a Casoratian determinant, and satisfies the discrete KP equations in {x} n .

The Gromov-Vieira polynomial version of partial domain wall partition functions
Following [14], partial domain wall configurations are (the essential part of) 3-point functions of tree-level single-trace operators in the SU (2) sector of SYM 4 , with two BPS and one non-BPS operators. In [20,21], Gromov and Vieira showed that 1-loop corrections can be introduced using the mapping discussed in this section. In the sequel, we show that the determinant form of these objects at tree-level is preserved under the GV mapping, thus the corresponding 1-loop corrected objects in SYM 4 can also be expressed as determinants.
6.5. Remarks on notation. Henceforth we reserve i and j for the row and column indices of a determinant, respectively, and assume that they range over all values 1 i, j N . For example, we write the determinant in Equation (88) as where the first equality follows from the invariance of the determinant under matrix transposition. In the second equality we suppress arguments and the summation symbol, but show the j-th row explicitly. In the rest of this section, all calculations will change determinant on a row-by-row basis.
6.6. Degree-2 terms in the determinant. Since the only terms which survive under the action of H 2 and G 2 are degree-2 monomials in the complete symmetric functions, we focus on these terms by expanding our determinant as follows where we maintain the symbol h 0 for clarity, despite the fact that h 0 = 1.
The first two terms come from the sum N l=1 in Equation (101), while the final term comes from the sum 1 l1<l2 N . All other terms vanish under the action of H 2 , either because they have the wrong degree or give rise to a determinant with two equivalent rows.
Combining the first and third determinant in Equation (102), which are the same up to the ordering of their rows, we obtain 6.8. Action of G 2 on Equation (101). Acting on Equation (101) with G 2 and using Equation (99), we find The first two terms come from the sum N l=1 in Equation (101), while the final term comes from the sum 1 l1<l2 N . All other terms vanish under the action of G 2 , either because they have the wrong degree or give rise to a determinant with two equivalent rows.
Combining the first and third determinant in Equation (104), which are the same up to the ordering of their rows, we obtain where the column index j ranges over all values 1 j N , as usual. For the second part of the GV mapping in Equation (92), we wish to calculate Putting together the results of the previous subsections, namely Equations (103) and (105), we find that Using Equations (106-108) we obtain The first three terms of Equation (109) can actually be combined into a single determinant, which is correct up to O(g 2 ). Our final result is 1 . . . c j,N −2 c j,N −1 + g 2 N c j,N +1 c j,N + g 2 N c j,N +2 The result in Equation (110) is such a simple modification of the original expression, obtained by setting g 2 → 0, that we expect that higher derivative versions of the GV mapping will also preserve the determinant form of the pDWPF 7 .
Since the GV mapping is an expansion around the homogeneous limit at which all variables y i = 0, we cannot consider the determinant in Equation (110) to be a discrete KP τ -function in the {y} variables. On the other hand, according to the methods of Section 5, the determinant in Equation (110) is not in Casorati form, hence we cannot conclude that it is a discrete KP τ -function in the {x} variables. 7. Remarks 7.1. Summary of results. Rational and trigonometric partial domain wall partition functions, pDWPF's, are partition functions of six-vertex model configurations on lattices with unequal numbers of horizontal lines L h and vertical lines L v . They can be regarded as less restrictive variations on Korepin's rational and trigonometric domain wall partition functions, DWPF's, which require L h = L v , but can be deduced from them, as well as from configurations that describe scalar products, by taking some of the rapidities to infinity.
In this work, we gave explicit derivations of the determinant expressions for pDWPF's as limits of Izergin's DWPF determinant, as well as of Slavnov's determinant for the scalar product of a Bethe eigenstate and a generic state, in the rational and trigonometric cases, and studied some of their properties. The rational pDWPF was first derived from Slavnov's determinant by I Kostov [17]. We showed how the two determinants obtained as limits of Izergin's determinant and of Slavnov's determinant are different (one is (N ×N ) while the other is (n×n), where n < N ), but can be directly related, that they are KP τ -functions in each of two sets of variables, and that they remain determinants under the mapping of Gromov and Vieira 8 .

7.2.
Taking the free variables to infinity in Slavnov's determinant. In Section 3, following Kostov [17], we derived pDWPF's from Slavnov's scalar products. We kept the free rapidity variables {x} finite, and took the rapidity variables that satisfy Bethe equations, {b}, to infinity. The result is finite and non-trivial.
If we would have kept the Bethe roots {b} finite and took {x} to infinity, the result would have been zero. The reason is that this limit corresponds to the scalar product of a Bethe eigenstate, labeled by {b}, and a descendant of the reference state (the result of the action of spin-lowering operators on the reference state, that lower the net spin but do not introduce Bethe roots [12]). Since the scalar product of the Bethe eigenstate |{b} and the reference state vanishes, the scalar product of |{b} with a descendant of the reference state also vanishes. In other words, a pDWPF with auxiliary space (horizontal line) rapidities that obey Bethe equations vanishes. 7.3. Asymptotics. In [14,17,18,19], pDWPF's were studied in the thermodynamic limit L v → ∞, such that the ratios L h /L v and x i /L v , i ∈ {1, . . . , L h }, remain finite, where L v (L h ) is the number of vertical (horizontal) lattice lines, L h < L v , and {x} are the rapidities of the horizontal lines 9 . While, strictly speaking, the variables {x} are free, in applications, such as computations of 3-point functions of two BPS and one non-BPS operators in the scalar sector of SYM 4 , they are restricted to obey the Bethe equations of a spin chain of 7 An earlier draft of this work contained an incorrect version of Equation (94) that led to a more complicated version of Equation (110). We thank D Serban for pointing this out. 8 It is likely that the determinant expression is preserved under the action of higher derivative versions of the GV mapping. We did not pursue this since, at this stage, the relation between the higher derivative versions and the inclusion of higher loop corrections to the 3-point functions is not clear. However, in [27], D Serban argued that this is indeed the case, at least in the limit L i → ∞, i ∈ {1, 2, 3}. That is, when all three operators are represented by asymptotically long spin chain states. 9 Because of the condition that x i /Lv , i ∈ {1, . . . , L h }, remains finite, this limit is also known as the 'Sutherland limit' [28,29].
length L, such that L = L v . For that reason, Bethe Ansatz asymptotics apply, but the pDWPF is nonetheless non-vanishing. This is the set-up used in [14,18,19]. Following [14,18], in the above thermodynamic limit, the variables {x}, which are solutions of Bethe equations of a spin chain of length L > L v , L ∼ L v , condense on a set of contours Γ = k Γ k , with linear density ρ{x}, ρ ∼ O(1), x i ∼ O(L v ). In the homogeneous limit, y i = 0, i ∈ {1, . . . , L v }, the asymptotic pDWPF can be expressed as an exponential of a contour integral over a dilogarithm function where C encircles Γ counter-clockwise, and The point we wish to mention here is that in the same limit, the Slavnov scalar product factorizes into a product of terms that are either the asymptotic pDWPF in Equation (111), or simple variations of it [18,19]. Thus, at least asymptotically, pDWPF's are building blocks of scalar products.