# Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures

- 148 Downloads
- 9 Citations

## Abstract

We prove two identities of Hall–Littlewood polynomials, which appeared recently in Betea and Wheeler (2014). We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in Betea and Wheeler (2014), via the introduction of additional parameters. The left-hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right-hand side of each identity is (one of the two factors present in) the partition function of the six-vertex model on a relevant domain.

### Keywords

Cauchy and Littlewood identities Symmetric functions Alternating sign matrices Six-vertex model## 1 Introduction

Equation (1) is due to Warnaar [17], based on earlier results of Kirillov and Noumi [4]. In [2], we exposed a combinatorial interpretation of (1): The left- hand side can be viewed as a generating series of *path-weighted* plane partitions [16], while the right-hand side is the partition function of the six-vertex model under domain wall boundary conditions [3, 6], and thus a generating series of alternating sign matrices (ASMs) [7, 8].

Equations (2) and (3), both conjectured in [2], are further examples of such a relationship. In both of these equations, the right-hand side is the partition function of the six-vertex model on a certain lattice (in (2), the underlying lattice has off-diagonal symmetry [8]; the partition function in (3) arises from reflecting domain wall boundary conditions [14]) and may be viewed as a multi-parameter generating series of a symmetry class of ASMs [off-diagonally symmetric ASMs in the case of (2); U-turn ASMs in the case of (3)]. Although we are able to view the left-hand side of (2) as a generating series of path-weighted symmetric plane partitions [2], for the moment there is no known combinatorial interpretation of the left-hand side of (3) in terms of plane partitions or other tableaux-related objects.

The goals of the present work are as follows. Firstly, we provide a new proof of (1) by applying the *Izergin–Korepin technique*^{1} [3, 6] to the left-hand side of the equation, before adapting this method to prove (2). We remain unable to prove (3) by such methods, due to the absence of a simple combinatorial (tableau) formula for the \(BC_n\)-symmetric Hall–Littlewood polynomials.

Secondly, we shall generalize all three identities by the introduction of additional parameters. It was already demonstrated in [17] that (1) may be refined by two extra parameters \(q\) and \(u\), with the original identity being recovered when \(q=0\) and \(u=t\). The introduction of the \(q\) parameter elevates the participating symmetric functions to Macdonald polynomials, and the equation itself comes from acting on the Cauchy identity with a generating series of Macdonald difference operators [9] (where \(u\) is the indeterminate of the generating series). We prove that even in the presence of the two extra parameters, the right- hand side of the identity remains a determinant (a fact which was not explicit in either [17] or [4]). In a similar vein, we find that it is possible to refine (2) by the introduction of the parameters \(q\) and \(u\). To round off, we conjecture a deformed version of (3) involving \(u\) and four parameters \(t_0,t_1,t_2,t_3\) which elevates it to the level of *lifted Koornwinder polynomials* [11].

Finally, we investigate the meaning of the deformation parameters thus introduced in the setting of the six-vertex model. Surprisingly, the inclusion of the indeterminate \(u\) in our equations does not break the correspondence with partition functions of the six-vertex model: The \(u\)-deformed versions of (1)–(3) all lead to determinants/Pfaffians which have appeared in [8] in the context of further symmetry classes of ASMs. We will not comment on the role of \(q\) in this scheme, since it appears to play only a trivial role.^{2}

The paper is organized as follows. In Sect. 2, we give proofs of two results: identity (1) (using an independent method to that of [17]) and (2) (conjectured in [2]). In Sect. 3, we discuss the generalization of (1) to Macdonald polynomials (obtained in [17]), and conjecture a companion generalization of (2) to this level. \(u\)-deformations of the Cauchy-, Littlewood- and \(BC\)-type Cauchy identities are discussed in Sects. 4, 5 and 6, and their connection with partition functions of ASM symmetry classes is exposed. The main result in Sect. 4 is that a \(u\)-deformed version of (1) is closely related to the partition function of half-turn symmetric ASMs (for a particular value of \(u\)). The main result in Sect. 5 is Theorem 7, a \(u\)-generalization of Eq. (2). In this case, for an appropriate value of \(u\), we obtain a close connection with the partition function of off-diagonally/off-anti-diagonally symmetric ASMs. In Sect. 6, we conjecture a \(u\)-generalization of (3) (Conjecture 2). We prove a simpler, companion identity involving symplectic Schur polynomials (Theorem 9) but are unable to prove the conjecture (due to the lack of a suitable branching rule for the lifted Koornwinder polynomials which participate). The conjecture has been verified for small partitions using Mathematica and Sage. Once again, a certain value of \(u\) leads to a correspondence with a six-vertex model partition function (in this case, the partition function of double U-turn ASMs). Finally, following Rains [11], in the Appendix, we present a few results on \(BC\)-type interpolation and Koornwinder polynomials (and their symmetric function analogs) that we use.

*part*and \(\ell (\lambda ):=k\) the

*length*(number of nonzero parts) of \(\lambda \). If all parts of \(\lambda \) are even, we call the partition

*even*. \(m_i(\lambda )\) stands for the number of parts in \(\lambda \) equal to \(i\). If for some prespecified \(n\) we have \(\ell (\lambda ) \leqslant n\), we abuse notation and define \(m_0(\lambda ) = n-\ell (\lambda )\) to be the number of zeros we need to append to \(\lambda \) to get a vector of length \(n\). Moreover, we call \(|\lambda |:=\sum _{i=1}^{\ell (\lambda )} \lambda _i\) the

*weight*of the partition. For any \(\lambda \), we have a

*conjugate partition*\(\lambda '\) whose parts are defined as \(\lambda '_i := |\{j: \lambda _j \geqslant i\}|\). We finally define the notion of interlacing partitions. Let \(\lambda \) and \(\mu \) be two partitions with \(|\lambda | \geqslant |\mu |\). They are said to be

*interlacing*, and we write \(\lambda \succ \mu \) if and only if

*horizontal strip,*meaning that \(\lambda '_i-\mu '_i \leqslant 1\) for all \(i \geqslant 1\).

## 2 Proofs

The primary goal of this section is to prove Eq. (2), effectively by using the Izergin–Korepin technique familiar from quantum integrable models. As a warm-up, we begin by providing a new proof of (1) along such lines. This approach to proving (1) and (2) may not be the most elegant (indeed, in the case of (1), there is a much simpler proof using Macdonald difference operators—see Sect. 3), but it is powerful since it only assumes two standard properties of Hall–Littlewood polynomials: their branching rule and a Pieri identity.

### 2.1 Branching rule for Hall–Littlewood polynomials

^{3}

### 2.2 A Pieri identity for Hall–Littlewood polynomials

### 2.3 Proof of Equation (1)

In this subsection, we prove Theorem 1, which is an alternative statement of Eq. (1). Our strategy is to show that the left-hand side of (1) satisfies four conditions, which are obvious properties of the right-hand side (they are the usual four properties in the Izergin–Korepin approach to calculating the domain wall partition function). Since these conditions are uniquely determining, it follows that the two sides of (1) must be equal.

**Theorem 1**

**1.**\(\mathcal {F}_n(x_1,\ldots ,x_n)\) is symmetric in \(\{x_1,\ldots ,x_n\}\).

**2.**The renormalized function \(\prod _{i,j=1}^{n} (1-x_i y_j) \mathcal {F}_n(x_1,\ldots ,x_n)\) is a polynomial in \(x_n\) of degree \(n-1\).

**3.**- Setting \(x_n = 1/(t y_n)\), one obtains the recursion$$\begin{aligned} \mathcal {F}_n \Big |_{x_n = 1/(t y_n)} = -t^n \mathcal {F}_{n-1}(x_1,\ldots ,x_{n-1}). \end{aligned}$$
**4.**\(\mathcal {F}_1(x_1) = (1-t)/(1-x_1 y_1)\).

Since \(\mathcal {F}_n(x_1,\ldots ,x_n)\) is a sum of Hall–Littlewood polynomials, each being symmetric in \(\{x_1,\ldots ,x_n\}\), property **1** is immediate. The remaining properties **2**–**4** will be proved in Sects. 2.3.2–2.3.4, after we make a preliminary observation about the function \(\mathcal {F}_n(x_1,\ldots ,x_n)\) in Sect. 2.3.1.

#### 2.3.1 Alternative form for \(\mathcal {F}_n(x_1,\ldots ,x_n)\)

*i.e.,*\(\lambda ^{*} = (\lambda _1-1,\ldots ,\lambda _n-1)\). Then, one has the following identity between Hall–Littlewood polynomials:

**2**and the recursive property

**3**.

#### 2.3.2 Polynomiality

Of course the value of \(m\) is arbitrary, so one can state (12) with no constraint imposed on \(\mu \).

*Remark 1*

^{4}:

**2**, using the identity (10), it is sufficient to show that

**2**is equivalent to proving that:

*Remark 2*

#### 2.3.3 Recursion relation

*i.e.,*\(\mu '_i = \nu '_i\) for all \(i > I\). Then, either \(\delta _{I} = 1, \epsilon _{I} = 0\) or \(\delta _{I} = 0, \epsilon _{I} = 1\), and the summation is constrained by \(\delta _i = \epsilon _i\) for all \(i > I\). Given that the summation indices are forced in this way, it is easy to deduce the recurrences

#### 2.3.4 Initial condition

### 2.4 Proof of Equation (2)

In this subsection, we prove Theorem 2, which is equivalent to proving Eq. (2). Similar to the previous proof, we will show that the left-hand side of (2) satisfies four conditions, which are basic properties of the right-hand side. Since these conditions only admit a unique solution, it follows that the two sides of (2) must be equal.

**Theorem 2**

**1.**\(\mathcal {G}_{N}(x_1,\ldots ,x_N)\) is symmetric in \(\{x_1,\ldots ,x_N\}\).

**2.**The renormalized function \(\prod _{1 \leqslant i<j \leqslant N} (1-x_i x_j) \mathcal {G}_{N}(x_1,\ldots ,x_N)\) is a polynomial in \(x_N\) of degree \(N-2\).

**3.**- Setting \(x_N = 1/(t x_{N-1})\), one obtains the recursion$$\begin{aligned} \mathcal {G}_N \Big |_{x_N = 1/(t x_{N-1})} = -t^{N-1} \mathcal {G}_{N-2}(x_1,\ldots ,x_{N-2}). \end{aligned}$$
**4.**\(\mathcal {G}_2(x_1,x_2) = (1-t)/(1-x_1 x_2)\).

The property **1** is obvious, since \(\mathcal {G}_{N}(x_1,\ldots ,x_N)\) is a sum of Hall–Littlewood polynomials and therefore manifestly symmetric in \(\{x_1,\ldots ,x_N\}\). As we did in the proof of Theorem 1, we begin with an alternative expression for \(\mathcal {G}_{N}(x_1,\ldots ,x_N)\) in Sect. 2.4.1, before proving the remaining properties **2**–**4** in Sects. 2.4.2–2.4.4.

#### 2.4.1 Alternative form for \(\mathcal {G}_N(x_1,\ldots ,x_N)\)

^{5}:

**2**, as we will see below.

#### 2.4.2 Polynomiality property

**2**, because of the identity (28), it suffices to show that

**2**is equivalent to the statement

#### 2.4.3 Recursion relation

#### 2.4.4 Initial condition

## 3 Identities at Macdonald level

The identities (1)–(3) listed at the start of this paper apply at the level of Hall–Littlewood polynomials. Since Hall–Littlewood polynomials are the \(q=0\) specialization of Macdonald polynomials, it is natural to suggest that these equations are special cases of yet more general identities involving extra parameters.

In this section, we show that this is indeed the case, by presenting Macdonald analogs of both equations (1) and (2). It turns out that these equations can be generalized by the introduction of two additional parameters, one being the \(q\) from Macdonald theory. The Macdonald generalization of (1) has been known since the work of Warnaar in [17] and can be proved using Macdonald difference operators. Although we obtain a completely analogous generalization of (2) to Macdonald level, it remains conjectural, since we lack an appropriate family of difference operators to expedite its proof.

As an aside, we remark that we do not know of an appropriate generalization of (3) to Macdonald level, even conjecturally. We are, however, able to deform it by the introduction of certain additional parameters, but we defer this result to Sect. 6 since it does not pertain directly to symmetric polynomials at Macdonald level.

### 3.1 \(u\)-Deformed Macdonald Cauchy identity

*arm*and

*leg length*of the box \(s\), respectively.

^{6}The following theorem can be deduced by acting on the Macdonald Cauchy identity with the generating series

*mutatis mutandis*when \(u\) is generic. For that reason, we attribute this theorem to Warnaar.

**Theorem 3**

*Proof*

### 3.2 \(u\)-Deformed Macdonald Littlewood identity

^{7}is motivated by (44). Although we do not have a proof of this conjecture, it is tempting to suggest that it can be deduced by acting on the Littlewood identity (44) with an appropriate family of difference operators, in much the same way that Theorem 3 follows from the Cauchy identity (38). A preliminary step in this direction is given in the remark following the conjecture.

**Conjecture 1**

*Remark 3*

One can express the Pfaffian on the right-hand side of (46) as a sum over subsets of \(\{1,\ldots ,N\}\), as in the following lemma.

**Lemma 1**

*Proof*

The expression of the Pfaffian appearing on the right-hand side of (46) as a sum over subsets of \(\{1,\ldots ,N\}\), as achieved by Eq. (47), would seem to be an important step toward the proof of (46). Indeed, the analogous result (42) was crucial in the proof of (41), since the type of sum arising in that case was manifestly related to the generating series (40) of difference operators.

Nevertheless, we do not yet know of a family of operators whose action on the right-hand side of the Littlewood identity (44) produces the sum in (47). The discovery of such operators would not only lead to the completed proof of (46), but constitute an important development in the theory of Macdonald polynomials in its own right.

## 4 \(u\)-Deformed Cauchy identity and half-turn symmetric alternating sign matrices

The aim of this section is to study the \(q=0\) specialization of Eq. (41) and its relation with the six-vertex model. In particular, we will study the six-vertex model on a lattice with *half-turn symmetry* and calculate its partition function as a product of two determinants following Kuperberg [8]. One of the determinants is precisely the domain wall partition function (the right-hand side of (1)), while the remaining determinant is equal to the right-hand side of (41) with \(q=0\) and \(u=-\sqrt{t}\).

### 4.1 Six-vertex model in the bulk

*rapidity*. The six types of vertex are assigned Boltzmann weights, which are rational functions depending on the ratio \(x/y\) of the rapidities incident on the vertex:

### 4.2 Partition function on half-turn symmetric lattice

**Lemma 2**

**1.**Multiplying by \(\prod _{i,j=1}^{n} (1-x_i y_j)^2\), it is a polynomial in \(x_n\) of degree \(2n-1\).

**2.**It is symmetric in \(\{y_1,\ldots ,y_n\}\).

**3.**- It obeys the recursion relations$$\begin{aligned} Z^{(n)}_\mathrm{HT} \Big |_{x_n = \bar{y}_n/t}&= -t^{2n-1/2} Z^{(n-1)}_\mathrm{HT}, \end{aligned}$$(49)$$\begin{aligned} \lim _{x_n \rightarrow \bar{y}_n} \Big ( (1-x_n y_n)^2 Z^{(n)}_\mathrm{HT} \Big )&= (1-t)^2 \prod _{i=1}^{n-1} \frac{(1-ty_i \bar{y}_n)^2}{(1-y_i \bar{y}_n)^2} \frac{(1-tx_i y_n)^2}{(1-x_i y_n)^2} Z^{(n-1)}_\mathrm{HT}. \end{aligned}$$(50)
**4.**- When \(n=1\), it is given explicitly by$$\begin{aligned} Z^{(1)}_\mathrm{HT} = \frac{(1-t)(1+\sqrt{t})(1-\sqrt{t} x_1 y_1)}{(1-x_1 y_1)^2}. \end{aligned}$$

*Proof*

**1.**Multiplying the partition function by \(\prod _{i,j=1}^{n} (1-x_i y_j)^2\) is equivalent to multiplying each individual Boltzmann weight by \((1-x_i y_j)\). After this renormalization, it is clear that every Boltzmann weight is a degree-1 polynomial in \(x_i\), with the sole exception of the \(c_{+}\) vertex (which is a constant). Focusing attention on the top and bottom rows of the lattice in Fig. 3, which are the only places which have dependence on \(x_n\), one can easily deduce that exactly one \(c_{+}\) vertex occurs in these two rows. It follows that the renormalized partition function is a polynomial in \(x_n\) of degree \(2n-1\).

**2.**Symmetry in the \(y\) variables is deduced using a standard argument involving the Yang–Baxter equation (see, for example, [5]). Indeed, any two adjacent vertical lines can be exchanged using this procedure.

**3.**- Setting \(x_n = \bar{y}_n/t\) eliminates the possibility that the top-left vertex of the lattice is an \(a_{+}\) vertex. It follows that it must be a \(c_{+}\) vertex. This forces a subset of the vertices into a
*frozen configuration*, shown on the left of Fig. 4. Studying the Boltzmann weights of the frozen region, we find that they contribute the total factor \(-t(\sqrt{t})^{4n-3} = -t^{2n-1/2}\). The remaining (unfrozen) region is just \(Z^{(n-1)}_\mathrm{HT}\). Hence, we recover the first recursion relation (49). Multiplying by \((1-x_n y_n)^2\) and taking \(x_n \rightarrow \bar{y}_n\) eliminates the possibility that the bottom-left vertex of the lattice is a \(b_{+}\) vertex. It must therefore be a \(c_{+}\) vertex. Once again, this forces a subset of the vertices to freeze out, as is shown on the right of Fig. 4. The Boltzmann weights of these frozen vertices contribute the total factorwhile the unfrozen region again represents \(Z^{(n-1)}_\mathrm{HT}\). This yields the second recursion relation (50).$$\begin{aligned} (1-t)^2 \prod _{i=1}^{n-1} \frac{(1-ty_i \bar{y}_n)^2}{(1-y_i \bar{y}_n)^2} \frac{(1-tx_i y_n)^2}{(1-x_i y_n)^2}, \end{aligned}$$ **4.**- The \(n=1\) case is small enough to be calculated explicitly: Substituting the Boltzmann weights into this expression, we obtain$$\begin{aligned} Z^{(1)}_\mathrm{HT} = \frac{(1-t) \sqrt{t}}{1-x_1 y_1} + \frac{1-tx_1 y_1}{1-x_1 y_1} \frac{(1-t)}{1-x_1 y_1} = \frac{(1-t)(1+\sqrt{t})(1-\sqrt{t} x_1 y_1)}{(1-x_1 y_1)^2}. \end{aligned}$$

**Theorem 4**

### 4.3 \(u\)-Deformed Cauchy identity at Schur and Hall–Littlewood level

**Corollary 1**

**Corollary 2**

These identities are recovered as the special cases \(q=t\) and \(q=0\) of Theorem 3. In view of the fact that the Schur polynomials on the left-hand side of (52) are determinants, it is possible to prove (52) by completely elementary means via the Cauchy–Binet identity.^{8} On the other hand, (53) remains a highly non-trivial identity, admitting no simple proof (that we know of) outside of the use of Macdonald difference operators, or a Lagrange interpolation style of proof similar to that of Sect. 2.3.

One can consider further specializations of (53), by setting the free parameter \(u\) to various values. Setting \(u=0\) produces the Cauchy identity for Hall–Littlewood polynomials, while setting \(u=t\) reproduces Eq. (1). Finally, in the case \(u=-\sqrt{t}\), we recover one half of the factors in Eq. (51) for \(Z_\mathrm{HT}\) (the remaining factors being those of the domain wall partition function).

## 5 \(u\)-Deformed Littlewood identity and doubly off-diagonally symmetric alternating sign matrices

This section proceeds largely in parallel with the previous one. The goal here is to study the \(q=0\) specialization of the conjecture (46) and its connection with the six-vertex model. The relevant domain in this case is the *doubly off-diagonally symmetric* lattice, whose configurations are in one-to-one correspondence with off-diagonally/off-anti-diagonally symmetric alternating sign matrices [8].

### 5.1 Corner vertices

*boundary vertices*, which consist of a single lattice line making a turn through a node. The type of boundary vertices of interest to us are the

*corner vertices*, shown in Fig. 5.

*corner reflection*equations shown in Fig. 6. The corner reflection equations, in conjunction with the regular Yang–Baxter equation, ensure that the \(Z_\mathrm{OO}\) partition function that we subsequently study is symmetric in its rapidities.

### 5.2 Partition function on doubly off-diagonally symmetric lattice

**Lemma 3**

**1.**Multiplying by \(\prod _{1\leqslant i<j \leqslant 2n} (1-x_i x_j)^2\), it is a polynomial in \(x_{2n}\) of degree \(4n-3\).

**2.**It is symmetric in \(\{x_1,\ldots ,x_{2n}\}\).

**3.**- It obeys the recursion relations$$\begin{aligned}&Z^{(2n)}_\mathrm{OO} \Big |_{x_{2n} = \bar{x}_{2n-1}/t} = - t^{4n-5/2} Z^{(2n-2)}_\mathrm{OO}, \end{aligned}$$(54)$$\begin{aligned}&\lim _{x_{2n} \rightarrow \bar{x}_{2n-1}} \Big ( (1-x_{2n-1} x_{2n})^2 Z^{(2n)}_\mathrm{OO} \Big ) \nonumber \\&\quad = (1-t)^2 \prod _{i=1}^{2n-2} \frac{(1-tx_i \bar{x}_{2n-1})^2}{(1-x_i \bar{x}_{2n-1})^2} \frac{(1-tx_i x_{2n-1})^2}{(1-x_i x_{2n-1})^2} Z^{(2n-2)}_\mathrm{OO}. \end{aligned}$$(55)
**4.**- When \(n=1\), it is given explicitly by$$\begin{aligned} Z^{(2)}_\mathrm{OO} = \frac{(1-t)(1+\sqrt{t})(1 - \sqrt{t} x_1 x_2)}{(1-x_1 x_2)^2}. \end{aligned}$$

*Proof*

**1.**Multiplying the partition function by \(\prod _{1\leqslant i<j \leqslant 2n} (1-x_i x_j)^2\) is equivalent to renormalizing every vertex by \((1-x_i x_j)\). This makes all Boltzmann weights degree-1 polynomials in \(x_i\), with the sole exception of the \(c_{+}\) vertex (which is a constant). Examining the left-most vertical line of Fig. 7, which gives rise to all \(x_{2n}\) dependence, we see that exactly one \(c_{+}\) vertex will occur on this line. Hence, \(Z^{(2n)}_\mathrm{OO}\) is a polynomial in \(x_{2n}\) of degree \(4n-3\).

**2.**Symmetry in the \(x\) variables can be deduced using both the Yang–Baxter equation and the two corner reflection equations in Fig. 6. These equations, in combination, allow for any two lattice lines bearing the labels \(x_i\) and \(x_j\) to be interchanged.

**3.**- Consider the top-most bulk vertex in Fig. 7. Setting \(x_{2n} = \bar{x}_{2n-1}/t\) rules out the possibility that this is an \(a_{+}\) vertex. It must therefore be a \(c_{+}\) vertex, and this causes a subset of the vertices to be in a frozen configuration, as shown on the left of Fig. 8. The total contribution from these frozen vertices is the weight \(-t (\sqrt{t})^{8n-7} = -t^{4n-5/2}\), while the surviving region is simply \(Z^{(2n-2)}_\mathrm{OO}\). Hence, we obtain Eq. (54). A similar argument applies to the bottom-most bulk vertex in Fig. 7. After multiplying by \((1-x_{2n-1} x_{2n})^2\) and sending \(x_{2n} \rightarrow \bar{x}_{2n-1}\), this cannot be a \(b_{+}\) vertex, meaning that it must be a \(c_{+}\) vertex. This causes some of the vertices to freeze, as shown on the right of Fig. 8, and they contribute a total weight ofwith the non-frozen part of the lattice representing \(Z^{(2n-2)}_\mathrm{OO}\). Hence, we recover Eq. (55).$$\begin{aligned} (1-t)^2 \prod _{i=1}^{2n-2} \frac{(1-tx_i \bar{x}_{2n-1})^2}{(1-x_i \bar{x}_{2n-1})^2} \frac{(1-tx_i x_{2n-1})^2}{(1-x_i x_{2n-1})^2}, \end{aligned}$$
**4.**- Calculating the \(n=1\) case explicitly, we find that Substituting the explicit expression for the Boltzmann weights, we obtain\(\square \)$$\begin{aligned} Z^{(2)}_\mathrm{OO} = \frac{(1-t)\sqrt{t}}{1-x_1 x_2} + \frac{1-tx_1 x_2}{1-x_1 x_2} \frac{(1-t)}{1-x_1 x_2} = \frac{(1-t)(1+\sqrt{t})(1 - \sqrt{t} x_1 x_2)}{(1-x_1 x_2)^2}. \end{aligned}$$

**Theorem 5**

### 5.3 \(u\)-Deformed Littlewood identity at Schur and Hall–Littlewood level

**Theorem 6**

*Proof*

**Theorem 7**

^{9}:

*Proof*

The idea of the proof is similar to that of Theorem 2. For the sake of brevity, we will simply point out the places where the proof deviates from the scheme exposed in Sect. 2.4.

**1.**\(\mathcal {G}_{2n}\) is symmetric in \(\{x_1,\ldots ,x_{2n}\}\).

**2.**\(\mathcal {G}_{2n} \times \prod _{1 \leqslant i<j \leqslant 2n} (1-x_i x_j) \) is a polynomial in \(x_{2n}\) of degree \(2n-1\).

**3.**\(\mathcal {G}_{2n}|_{x_{2n} = 1/(t x_{2n-1})} = -ut^{2n-2} \mathcal {G}_{2n-2}\).

**4.**\(\mathcal {G}_{2n}(0,\ldots ,0;u) = \prod _{i=1}^{n} (1-ut^{2i-2})\).

**5.**\(\mathcal {G}_{2} = (1-u+(u-t)x_1 x_2)/(1-x_1 x_2)\).

**4**, which at first glance seems to require taking a delicate limit. In fact, property

**4**can be quickly deduced by setting all \(x_i = 0\) in Eq. (59) \(\times \prod _{1 \leqslant i<j \leqslant 2n} (1-t x_i x_j)\). It is straightforward to show that these five properties uniquely determine \(\mathcal {G}_{2n}\). We remark that the additional property

**4**is needed here, because properties

**1**and

**3**determine \(\mathcal {G}_{2n}\) at \(2n-1\) values of \(x_{2n}\), which only specifies it up to a constant. The value of the constant is fixed by property

**4**.

**1**–

**5**. Properties

**1**and

**4**are trivial, while

**5**follows from

**2**, it suffices to show that \(\prod _{1 \leqslant i<j \leqslant 2n} (1- x_i x_j) \mathcal {G}_{2n}\) is a degree \(2n-1\) polynomial in \(x_{2n}\) at \(n+1\) different values of \(u\) (since \(\mathcal {G}_{2n}\) is a degree \(n\) polynomial in \(u\)). The \(n+1\) points that we choose are \(u=0\) (for which the claim is trivial, since in that case, we obtain the left-hand side of the Littlewood identity (29)) and \(u=t^{2k-2n}\) for \(1 \leqslant k \leqslant n\). For these latter values of \(u\), we find that

**3**, one repeats the procedure outlined in Sect. 2.4.3, but with obvious modifications to the formulae to cater for the more general coefficients appearing in the sum (60). The strategy is to expand the left-hand side of (60) (evaluated at \(x_{2n} = 1/(t x_{2n-1})\)) using two applications of the branching rule and to isolate the coefficients \(\mathcal {D}(\nu )\) of \(P_{\nu }(x_1,\ldots ,x_{2n-2};t)\) which arise from this expansion. In the case where \(\nu \) has only even columns, one finds that

**3**. \(\square \)

Theorems 6 and 7 are important results, since they serve as checks of Conjecture 1 at the particular values \(q=t\) and \(q=0\), respectively. Further specialization (of the parameter \(u\)) leads to various known results. For example, in the case of (60), setting \(u=0\) yields the Littlewood identity for Hall–Littlewood polynomials, whereas setting \(u=t\) gives rise to Eq. (2). In complete analogy with the previous section, when we set \(u=-\sqrt{t}\), we recover one half of the factors in Eq. (56) for \(Z_\mathrm{OO}\). The remaining factors in (56) are precisely those of the OSASM partition function, on the right-hand side of (2).

## 6 \(u\)-Deformed \(BC\)-type Cauchy identity and double U-turn alternating sign matrices

In this section, we conclude our study of the relationship between refined Cauchy/Littlewood identities and partition functions of the six-vertex model. We present one final example, conjecturing a \(u\)-deformed version of Eq. (3) and showing that its right-hand side contains half of the factors present in the partition function with U-turn boundaries on two sides of the lattice. The remaining factors are those of the UASM partition function, as given by the right-hand side of (3). The explicit formula for this partition function, as a product of two determinants, is again due to Kuperberg [8].

### 6.1 Redefinition of Boltzmann weights for bulk vertices

### 6.2 U-turn vertices, reflection, and fish equations

*U-turn vertices*, as shown in Fig. 9. The U-turn Boltzmann weights (denoted \(r_{\pm }\) and \(t_{\pm }\), since they are situated on the

*right*and

*top*edges of the partition function that we subsequently study) are given explicitly by

### 6.3 Partition function on double U-turn lattice

**Lemma 4**

**1.**Multiplying by \(\prod _{i,j=1}^{n} (1-x_i y_j)^2 (1-x_i \bar{y}_j)^2\), it is a polynomial in \(x_n\) of degree \(4n\).

**2.**It has zeros at \(x_n = \pm 1\).

**3.**It is symmetric in \(\{y_1,\ldots ,y_n\}\).

**4.**- It is quasi-symmetric under \(y_n \longleftrightarrow \bar{y}_n\):$$\begin{aligned}&\bar{y}_n (1-t y_n^2) Z_\mathrm{UU} (x_1,\ldots ,x_n; y_1,\ldots ,y_{n-1},y_n;t) \nonumber \\&\quad = y_n (1-t\bar{y}_n^2) Z_\mathrm{UU} (x_1,\ldots ,x_n; y_1,\ldots ,y_{n-1},\bar{y}_n;t). \end{aligned}$$(63)
**5.**- It obeys the recursion relations$$\begin{aligned}&\lim _{x_n \rightarrow \bar{y}_n} \Big ( (1-x_n y_n)^2 Z^{(n)}_\mathrm{UU} \Big ) = (1-t)^2 t^{n-1/2} (\bar{p} \bar{y}_n-p y_n) \frac{(1-t\bar{y}_n^2)}{(1-\bar{y}_n^2)} \nonumber \\&\qquad \times \, \prod _{i=1}^{n-1} \left[ \frac{(1-t x_i y_n)}{(1-x_i y_n)} \frac{(1-ty_i \bar{y}_n)}{(1-y_i \bar{y}_n)} \frac{(1-t\bar{y}_i \bar{y}_n)}{(1-\bar{y}_i \bar{y}_n)} \right] ^2 Z^{(n-1)}_\mathrm{UU}, \end{aligned}$$(64)$$\begin{aligned}&Z^{(n)}_\mathrm{UU} \Big |_{x_n = y_n/t} = t^{3n-1/2} (p y_n - \bar{p} \bar{y}_n) \frac{(1-y_n^2/t^2)}{(1-y_n^2/t)} \prod _{i=1}^{n-1} \frac{(1-tx_i y_n)^2}{(1-x_i y_n)^2} Z^{(n-1)}_\mathrm{UU}.\nonumber \\ \end{aligned}$$(65)
**6.**- When \(n=1\), it is given explicitly by$$\begin{aligned} Z^{(1)}_\mathrm{UU} = \frac{ \sqrt{t}(1\!-\!t)(1\!-\!x_1^2)(y_1-t\bar{y}_1) \left[ (pt+\bar{p}) (x_1 y_1 \!+\! x_1 \bar{y}_1) \!-\! (p\!+\!\bar{p}) (1+tx_1^2) \right] }{(1-x_1 y_1)^2 (1-x_1 \bar{y}_1)^2}. \end{aligned}$$

*Proof*

**1.**Multiplying the entire partition function by \(\prod _{i,j=1}^{n} (1-x_i y_j)^2 (1-x_i \bar{y}_j)^2\) is equivalent to renormalizing the individual Boltzmann weights, such that each is a degree-1 polynomial in \(x_i\) (except the \(c_{\pm }\) weights, which go as \(\sqrt{x_i}\)). We focus our attention on the bottom two rows of \(Z^{(n)}_\mathrm{UU}\), which is the sole place having \(x_n\) dependence. In every legal configuration, there must be an odd total number of \(c_{\pm }\) vertices in these final two rows. Combining this with the explicit parametrization of the right U-turn vertices ensures that \(Z^{(n)}_\mathrm{UU}\) is indeed a polynomial in \(x_n\). Furthermore, since there is minimally one \(c_{\pm }\) vertex in these two rows, the degree of the polynomial is \(4n\).

**2.**Starting from the U-turn vertex associated with the final two rows of \(Z^{(n)}_\mathrm{UU}\), we can immediately use the fish equation on the left of Fig. 11 to introduce an extra vertex, internal to the lattice. This also produces an overall multiplicative factor of \(\sqrt{t}(1-x_n^2)/(1-t^2 x_n^2)\). Using the Yang–Baxter equation repeatedly, it is possible to slide the new vertex horizontally through the lattice until it ultimately emerges from the left as a \(b_{+}\) vertex, with Boltzmann weight \(\sqrt{t}\). This process is shown in Fig. 13. The two zeros at \(x_n = \pm 1\) are due to the factor \((1-x_n^2)\) introduced at the start of this procedure.

**3.**Using both the Yang–Baxter and reflection equation, it is possible to interchange the order of any two double columns bearing the rapidities \(\{\bar{y}_i,y_i\}\) and \(\{\bar{y}_j,y_j\}\). This is a standard argument used in models with a double-row transfer matrix, see [8] and references therein for more details.

**4.**- One can attach a single \(a_{-}\) vertex at the base of the first two columns in \(Z^{(n)}_\mathrm{UU}\), which is equivalent to multiplying the partition function by \((1-t y_n^2)/(1-y_n^2)\). The inserted vertex can then be moved vertically through the lattice (using the Yang–Baxter equation) until it reaches the U-turn vertex at the top of the double column. Applying the fish equation on the right of Fig. 11, the internal crossing is removed and the order of the first two columns is interchanged, up to an overall factor of \(-(t-y_n^2)/(1-y_n^2)\). Hence, we find thatTrivial rearrangement of the prefactors gives the result (63).$$\begin{aligned}&\frac{(1-t y_n^2)}{(1-y_n^2)} Z_\mathrm{UU} (x_1,\ldots ,x_n; y_1,\ldots ,y_{n-1},y_n;t) \nonumber \\&\qquad = - \frac{(t-y_n^2)}{(1-y_n^2)} Z_\mathrm{UU} (x_1,\ldots ,x_n; y_1,\ldots ,y_{n-1},\bar{y}_n;t). \end{aligned}$$
**5.**The recursion relation (64) follows from the original lattice representation of \(Z^{(n)}_\mathrm{UU}\), in Fig. 12. Multiplying the partition function by \((1-x_n y_n)^2\) and taking \(x_n \rightarrow \bar{y}_n\) forces the bottom-left vertex of the lattice to be a \(c_{+}\) vertex. This restriction causes a larger subset of the vertices in \(Z^{(n)}_\mathrm{UU}\) to be in a frozen configuration, as shown on the left of Fig. 14. The second recursion relation (65) can be deduced from the lattice representation on the right-hand side of Fig. 13, obtained by a single application of the horizontal fish equation and repeated use of the Yang–Baxter equation. One starts by removing the frozen \(b_{+}\) vertex from the left side of the lattice, then setting \(x_n=y_n/t\) forces the bottom- left vertex to be of type \(c_{-}\). A subset of the vertices then freeze, as shown on the right of Fig. 14. In both cases, by reading off the Boltzmann weights for the frozen vertices, we deduce the prefactors in the recursion relations (64) and (65). One must also be mindful of the overall multiplicative factors which are introduced in the derivation of Fig. 13 and take these into account when arriving at Eq. (65). In either case, the surviving (unfrozen) region represents the partition function of one size smaller, \(Z^{(n-1)}_\mathrm{UU}\).

**6.**- The \(n=1\) case of the partition function can be computed as a sum of five terms: Using the expressions (61) and (62) for the Boltzmann weights, we obtain the explicit sumwhich can be simplified to$$\begin{aligned} Z^{(1)}_\mathrm{UU}= & {} - \frac{ \sqrt{t}(1-t)^3 x_1 (p - x_1) (1+\bar{p}\bar{y}_1) }{ (1-x_1 y_1) (1-x_1 \bar{y}_1)^2 } - \frac{ t^{3/2}(1-t) (p x_1 - 1) (1 + \bar{p} \bar{y}_1) }{ (1-x_1 \bar{y}_1)} \\&-\, \frac{ \sqrt{t} (1-t) (1-t x_1 y_1) (1-t x_1 \bar{y}_1) (p - x_1) (y_1 + \bar{p}) }{ (1-x_1 y_1)^2 (1-x_1 \bar{y}_1) } \\&+\, \frac{ \sqrt{t}(1-t)(1-tx_1 \bar{y}_1) (p - x_1) (t \bar{y}_1 + \bar{p}) }{(1-x_1 \bar{y}_1)^2 } \\&+\, \frac{ \sqrt{t} (1-t) (1-t x_1 \bar{y}_1) (p x_1 - 1) (t + \bar{p} y_1) }{ (1-x_1 y_1)(1-x_1 \bar{y}_1) }, \end{aligned}$$$$\begin{aligned} Z^{(1)}_\mathrm{UU} \!=\! \frac{ \sqrt{t}(1-t)(1-x_1^2)(y_1-t\bar{y}_1) \left[ (pt\!+\!\bar{p}) (x_1 y_1 \!+\! x_1 \bar{y}_1) \!-\! (p\!+\!\bar{p}) (1\!+\!tx_1^2) \right] }{(1\!-\!x_1 y_1)^2 (1\!-\!x_1 \bar{y}_1)^2}. \end{aligned}$$

**Theorem 8**

*Proof*

It is a simple matter to verify that (66) satisfies the six properties of Lemma 4. The fact that these properties uniquely determine \(Z_\mathrm{UU}\) is again a consequence of Lagrange interpolation. Indeed, the recursion relations (64) and (65) (together with the symmetry property **3** and quasi-symmetry property **4**) determine the polynomial \(Z_\mathrm{UU}\) at \(4n\) values of \(x_n\). Combined with the two known zeros at \(x_n = \pm 1\), these are sufficiently many points to fully determine \(Z_\mathrm{UU}\). \(\square \)

### 6.4 \(u\)-Deformed \(BC_n\) Cauchy identity at Schur and Hall–Littlewood level

In this subsection, we present a (conjectural) \(u\)-deformation of Eq. (3), involving lifted Koornwinder polynomials [11]. We build up to this via a simpler result at the level of symplectic Schur polynomials, which we are able to prove using the Cauchy–Binet identity. In that sense, the two results presented here are direct analogs of equations (52) and (53) in Sect. 4.3, and (59) and (60) in Sect. 5.3. In contrast with those other equations, we are currently unable to obtain (67) and (68) as the \(q \rightarrow 0\) case of some identity at Macdonald level.

**Theorem 9**

*Proof*

**Conjecture 2**

In analogy with previous sections, we wish to point out a further specialization of \(u\) which leads to a connection with the partition function (66). By choosing the boundary parameter in (66) to be \(\bar{p}=-\sqrt{t}\), and setting \(u=-t\) in (68), we obtain agreement between the determinants appearing in (66) and (68) up to an overall factor of \((\sqrt{t})^n\). Furthermore, by specializing \(t_0 = 1\), \(t_1 = -1\) and \(t_2 = \sqrt{t}\), \(t_3 = -\sqrt{t}\), we find that all of prefactors present in (68) are also present in (66). The leftover factors in (66) are those of the UASM partition function, given by the right-hand side of Eq. (3). Hence, this is yet another example of a Cauchy-type identity that is closely related to a partition function appearing in [8].

## Footnotes

- 1.
We use this as an umbrella term for any proof that involves:

**1**. Writing down a set of properties, one of which is a simple recursion relation, that uniquely determine an object, and**2.**Showing that a certain determinant or Pfaffian Ansatz satisfies these properties. - 2.
- 3.
We have departed slightly from the convention of [9] for the function \(\psi _{\lambda /\mu }(t)\), by incorporating the Kronecker delta into its definition (so that it is defined for all partitions \(\lambda \), \(\mu \)). This turns out to be convenient in many of the equations that follow, since it avoids keeping track of interlacing conditions when writing sums.

- 4.
- 5.
The superscript in \(w_{\lambda }^\mathrm{el}(N,t)\) and \(b^\mathrm{el}_{\lambda }(t)\) is for

*even legs*, since in the Macdonald case \(b^\mathrm{el}_{\lambda }(q,t)\) is defined as a product over all boxes in \(\lambda \) with even leg-length [9]. - 6.
- 7.
We are grateful to E. Rains for a comprehensive numerical test of the factorized dependence on \(q\) and \(u\) in (46), and to O. Warnaar for independently suggesting this conjecture to us while the manuscript was in preparation.

- 8.Dividing Eq. (52) by \((1-t)^n\), letting \(u = t^{-z}\) and taking the limit \(t \rightarrow 1\), the left-hand side of (52) becomesThis limiting form was already investigated in [1], although there the right-hand side was not expressed in determinant form. We thank F. Jouhet for showing us this reference.$$\begin{aligned} \sum _{\lambda } \prod _{i=1}^{n} (\lambda _i - i + n - z) s_{\lambda }(x_1,\ldots ,x_n) s_{\lambda }(y_1,\ldots ,y_n). \end{aligned}$$
- 9.

## Notes

### Acknowledgments

We express our sincere thanks to Ole Warnaar, for many valuable insights and suggestions which motivated this work, and in particular for suggesting the idea of \(u\)-deformations of the original identities (1)–(3), and to Eric Rains, for helping us in arriving at Conjecture 1. M.W. would like to thank Frédéric Jouhet for his interest in this work and for pointing out the reference [1], and Jean-Christophe Aval, Philippe Nadeau and Eric Ragoucy for invitations to present related work at LaBRI, ICJ and LAPTh, respectively. We finally wish to acknowledge the open-source package Sage whose built-in functions for Hall–Littlewood and Macdonald polynomials were indispensable in verifying some of the conjectures. This work was done under the support of the ERC Grant 278124, “Loop models, integrability and combinatorics”.

### References

- 1.Andrews, G.E., Goulden, I.P., Jackson, D.M.: Generalizations of Cauchy’s summation theorem for Schur functions. Trans. Am. Math. Soc.
**310**(2), 805–820 (1988)MathSciNetMATHGoogle Scholar - 2.Betea, D., Wheeler, M.: Refined Cauchy and littlewood identities, plane partitions, and symmetry classes of alternating sign matrices (2014). arXiv:1402.0229
- 3.Izergin, A.G.: Partition function of the six-vertex model in a finite volume. Dokl. Akad. Nauk SSSR
**297**(2), 331–333 (1987)MathSciNetGoogle Scholar - 4.Kirillov, A.N., Noumi, M.: \(q\)-difference raising operators for Macdonald polynomials and the integrality of transition coefficients. In: Algebraic Methods and \(q\)-Special Functions (Montréal, QC, 1996), CRM Proc. Lecture Notes, vol. 22. Am. Math. Soc, Providence, RI, pp. 227–243 (1999)Google Scholar
- 5.Korepin, V., Zinn-Justin, P.: Thermodynamic limit of the six-vertex model with domain wall boundary conditions. J. Phys. A
**33**(40), 7053–7066 (2000)MathSciNetCrossRefMATHGoogle Scholar - 6.Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys.
**86**(3), 391–418 (1982)MathSciNetCrossRefMATHGoogle Scholar - 7.Kuperberg, G.: Another proof of the alternating-sign matrix conjecture. Int. Math. Res. Not.
**3**, 139–150 (1996)MathSciNetCrossRefGoogle Scholar - 8.Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. Math.
**156**(3), 835–866 (2002)MathSciNetCrossRefGoogle Scholar - 9.Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. The Clarendon Press, Oxford University Press, New York (1995). With contributions by A Zelevinsky, Oxford Science Publications (1995)Google Scholar
- 10.Okounkov, A.: BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials. Transform. Groups
**3**(2), 181–207 (1998)MathSciNetCrossRefMATHGoogle Scholar - 11.Rains, E.M.: \(\text{ BC }_n\)-symmetric polynomials. Transform. Groups
**10**(1), 63–132 (2005)MathSciNetCrossRefMATHGoogle Scholar - 12.Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A
**21**(10), 2375–2389 (1988)MathSciNetCrossRefMATHGoogle Scholar - 13.Sundaram, S.: Tableaux in the representation theory of the classical Lie groups. In: Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl., vol. 19. Springer, New York, pp. 191–225 (1990)Google Scholar
- 14.Tsuchiya, O.: Determinant formula for the six-vertex model with reflecting end. J. Math. Phys.
**39**(11), 5946–5951 (1998)MathSciNetCrossRefGoogle Scholar - 15.Venkateswaran, V.: Symmetric and nonsymmetric Hall–Littlewood polynomials of type BC (2013). arXiv:1209.2933v2
- 16.Vuletić, M.: A generalization of MacMahon’s formula. Trans. Am. Math. Soc.
**361**(5), 2789–2804 (2009)CrossRefMATHGoogle Scholar - 17.Warnaar, S.O.: Bisymmetric functions, Macdonald polynomials and \(sl_{3}\) basic hypergeometric series. Compos. Math.
**144**(2), 271–303 (2008)MathSciNetMATHGoogle Scholar - 18.Warnaar, S.O.: Remarks on the paper “Skew Pieri rules for Hall–Littlewood functions” by Konvalinka and Lauve. J. Algebr. Comb.
**38**(3), 519–526 (2013)MathSciNetCrossRefMATHGoogle Scholar