Single-sum basic-hypergeometric form for the general higher-spin six-vertex K-matrix elements

The most general non-diagonal solution of the reflection equation for the higher-spin (fused) six-vertex model was obtained by Mangazeev and Lu through a generating function that is a terminating balanced ${}_4\phi_3$ basic hypergeometric series; expanding it yields the boundary K-matrix elements as an explicit double sum, while the bulk fused R-matrix has long been known in single-sum basic hypergeometric form. The source paper poses the question of whether that double sum can be reduced to a single sum — a standard basic hypergeometric function — for generic boundary parameters and arbitrary spin. Such a reduction would supply compact closed-form boundary weights for open higher-spin quantum chains and half-space stochastic vertex models.

Background

Integrable lattice models in two dimensions are built from an R-matrix, a solution of the Yang–Baxter equation that supplies the elementary Boltzmann weights; for the six-vertex model it acts on a tensor product of two two-dimensional spaces. Replacing the fundamental representation by the $(J+1)$-dimensional symmetric tensor representation (spin $J/2$) of the quantum affine algebra $U_q(\widehat{sl_2})$ — a procedure called fusion — produces the higher-spin six-vertex model. A stochastic gauge of the fused R-matrix, with nonnegative weights normalized so that the weights of each vertex configuration sum to one, defines the higher-spin stochastic six-vertex model of integrable probability, and its bulk matrix elements are known in a compact single-sum basic hypergeometric form. Throughout, a q-Pochhammer symbol is $(a;q)_n=\prod_{m=0}^{n-1}(1-aq^m)$, with the multi-parameter convention $(a_1,\dots,a_r;q)_n=\prod_{i=1}^r(a_i;q)_n$, and a basic hypergeometric series ${}_{r+1}\phi_r$ is a series whose summand is composed of such factors so that the ratio of consecutive terms is a rational function of the summation index; a 'terminating balanced' series is one that stops at a finite order because one upper parameter is $q^{-N}$, with the standard balancing constraint on the parameters.

Open (half-infinite) chains are treated with Sklyanin's formalism: one introduces a boundary K-matrix solving the reflection equation $R_{12}(x/y)K_1(x)R_{21}(xy)K_2(y)=K_2(y)R_{12}(xy)K_1(x)R_{21}(x/y)$, which guarantees that a transfer matrix built from R- and K-matrices commutes across different spectral parameters and so yields commuting conserved charges for the open chain. For spin $1/2$ the general non-diagonal K-matrix is a $2\times2$ matrix depending on four arbitrary complex boundary parameters ($t_+,t_-,\mu,\nu$ in the source's conventions). For arbitrary spin $J/2$ the most general non-diagonal solution was derived by Mangazeev and Lu (Nuclear Physics B 945 (2019) 114665). They introduced a generating function $F(u,v)=\sum_{j,l\ge0}N_{j,l}u^jv^l$ for the matrix elements, derived a pair of coupled q-difference equations for it from the reflection equation, and solved them: $F(u,v)$ equals a prefactor times a terminating balanced ${}_4\phi_3$ series. Expanding this solution back in powers of $u$ and $v$ gives every element $N_{j,l}$ — and hence the K-matrix $K_J$ through a known prefactor — only as a double sum. In the upper- and lower-triangular boundary specializations the elements collapse to simple products of q-Pochhammer symbols, but the general case retains the double summation.

This asymmetry with the bulk — single-sum hypergeometric R-matrix elements versus a double-sum K-matrix — is more than aesthetic. Users of the formula in half-space stochastic models have found it 'quite unwieldy' and unsuitable for asymptotic analysis (He 2022), and later constructions of integrable boundaries either restrict to the triangular regime or deliberately avoid the explicit K-matrix altogether. This motivated the authors' closing question of whether the double sum can be rewritten as a single sum, i.e. some basic hypergeometric function.

Problem Statement

Let $q,y,t,\nu$ be generic complex parameters and $J$ a nonnegative integer, and let $0\le j,l\le J$. Mangazeev and Lu express the elements of the general non-diagonal spin-$J/2$ boundary K-matrix of the higher-spin (stochastic) six-vertex model as the double sum

$$N_{j,l}=\sum_{k=0}^{J}\sum_{s=0}^{\min(j,l)} \frac{(-1)^s q^{(k-2s)(k+1)}}{t^{j+l-2(k+s)}} \frac{(q^{-2J},y^{-4};q^2)_k}{\left(q^2,\,-\frac{q^{-J}}{\nu y^2},\,\frac{q^{2-J}\nu t^2}{y^2};q^2\right)_k} \frac{(q^{-2(J-k)};q^2)_s}{(q^2;q^2)_s} \prod_{m\in\{j,l\}}\frac{(q^{-2k};q^2)_{m-s}}{(q^2;q^2)_{m-s}},$$

where $(a;q^2)_n=\prod_{m=0}^{n-1}(1-aq^{2m})$ and $(a_1,\dots,a_r;q^2)_n=\prod_i(a_i;q^2)_n$; the K-matrix itself is recovered through the fixed prefactor $K_J(y)_j^{\,l}=(-1)^lq^{2j}t^{j-l}\frac{(q^2;q^2)_l}{(q^{-2J};q^2)_l}N_{j,l}$ at the stochastic normalization, which corresponds to setting the source's auxiliary parameter $\mu$ to $1$ (a choice the authors show can be made without loss of generality and under which the K-matrix is stochastic). This is Eq. (7.24) of Mangazeev and Lu (arXiv:1903.00274).

The problem: determine whether this double-sum formula for the matrix elements can be rewritten in the form of a single sum, i.e. some basic hypergeometric function — meaning a single terminating basic hypergeometric series in the natural parameters $q,y,t,\nu$ and the indices $j,l,J$, valid for generic parameters and all $0\le j,l\le J$ and $J\ge0$ — or establish that no such reduction to a standard basic hypergeometric function exists. The upper- and lower-triangular parameter specializations, where the expression already collapses to simple products of q-Pochhammer symbols, are settled special cases; the question concerns the general non-triangular case.

References

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