Orthogonality of the dual symmetric functions of the half-space six-vertex model (Conjecture 5.5 of Garbali–de Gier–Mead–Wheeler)

The stochastic six-vertex model on a half-quadrant lattice with generic integrable boundary weights gives rise, via commuting double-row (reflection) transfer matrices, to two families of multivariate rational symmetric functions $G$ and $F$ indexed by finite particle configurations on the half-line. Garbali, de Gier, Mead and Wheeler proved a skew Cauchy identity between these families, a Pfaffian formula generalizing Kuperberg's evaluation of off-diagonally symmetric alternating-sign-matrix partition functions, and a multiple contour-integral formula for $G$. From these ingredients they deduced heuristically, and stated as Conjecture 5.5, that in the degeneration where the boundary parameter $c$ tends to infinity the dual family $\lim_{c\to\infty}F_\kappa$ is orthogonal with respect to an explicit contour-integral scalar product: pairing $\lim_{c\to\infty}F_\kappa$ against a prescribed kernel yields the Kronecker delta $\delta_{\kappa,\nu}$. The conjecture is an identity of rational functions in the quantization parameter $q$, the boundary parameter $a$ and the vertical rapidities $Y$, and its heuristic derivation rests on an unproved linear-independence (completeness) property of the family $\lim_{c\to\infty}G_\kappa$. The homogeneous specialization arising in the half-line open ASEP was proved in December 2025 by de Gier, Mead, Remenik and Wheeler; the full inhomogeneous six-vertex statement remains open and is the problem recorded here.

Background

The six-vertex model is an exactly solved lattice model of up-right directed paths in which each vertex conserves the number of paths passing through it (two entering and two leaving edges, giving six admissible local configurations). In a stochastic parametrization of the weights the model becomes a discrete-time Markov chain of interacting particles on the line, and its partition functions are multivariate rational symmetric functions of the horizontal inhomogeneity parameters (spectral rapidities) $x_1,\dots,x_L$. This interplay is a backbone of integrable probability: in full space, the higher-spin (vertical lines carry several paths) and coloured (multi-species) generalizations studied by Borodin–Petrov and Borodin–Wheeler produce families of symmetric rational functions carrying Cauchy identities, integral (sum-over-residues) formulas and, crucially, orthogonality with respect to an explicit contour-integral scalar product — a structure underlying the spectral/Plancherel theories used to extract asymptotics in the KPZ (Kardar–Parisi–Zhang) universality class, the generic scaling regime of one-dimensional stochastic growth.

Half-space versions, with a single reflecting boundary, are much less developed: the boundary breaks the full-space machinery and, when its weights are generic, allows paths to be created or destroyed at the reflecting line. Garbali, de Gier, Mead and Wheeler (arXiv 2023; published in Ann. Henri Poincaré 2025) constructed the first such theory with fully generic integrable boundary weights. Their lattice is the half-quadrant: horizontal lines carry rapidities $x_i$ or $z_j$, inhomogeneous vertical lines carry rapidities $y_1,y_2,\dots$, and the boundary carries weights depending on two parameters $a,c$ obtained by solving the reflection equation — the boundary analogue of the Yang–Baxter equation (the local weight-consistency relation of the six-vertex model); the reflection equation guarantees commutativity of the double-row (boundary) transfer matrices. Acting with products of these commuting double-row operators between particle configurations (finite subsets $\mu,\nu$ of the positive integers, viewed as particle positions on the half-line), they defined two families of multivariate rational functions, symmetric in their respective alphabets: $G_{\nu/\mu}(x_1,\dots,x_L|Y)$ and its dual $F_{\mu/\nu}(z_1,\dots,z_M|Y)$, both depending on the vertical alphabet $Y=(y_1,y_2,\dots)$. They proved a skew Cauchy identity summing $\sum_\kappa G_{\kappa/\mu}(x)F_{\kappa/\nu}(z)$ to a closed expression; showed that $G_{\emptyset}(x_1,\dots,x_L)$ coincides with the half-quadrant 'triangular' partition function, for which they obtained a Pfaffian formula (a Pfaffian is the square root of the determinant of a skew-symmetric matrix) generalizing Kuperberg's Pfaffian evaluation of the partition function of off-diagonally symmetric alternating-sign matrices (OSASMs — square matrices with entries $0,\pm1$ whose nonzero entries alternate in sign along every row and column, enumerated by six-vertex partition functions); and derived a multiple contour-integral formula for $G_\nu$.

The orthogonality conjecture arises in the degeneration $c\to\infty$, in which the boundary vertex that removes paths from the system vanishes (in the associated particle dynamics the exit rate at the origin tends to zero). In this limit the triangular partition function factorizes completely, the integrand of the integral formula for $G_\nu$ factorizes, and the Cauchy kernel becomes visible inside it. Expanding that kernel through the Cauchy identity and formally matching coefficients of $G_\kappa$ on both sides yields Conjecture 5.5 of that paper: the dual family $\lim_{c\to\infty}F_\kappa$ is orthogonal with respect to an explicit integral scalar product, the pairing with the kernel displayed in the problem statement returning $\delta_{\kappa,\nu}$. The coefficient-matching step presupposes that the degenerate family $\lim_{c\to\infty}G_\kappa(x_1,\dots,x_L)$ is linearly independent; the authors prove neither this nor the conjecture, note that the conjecture would follow from such a linear-independence statement, and record that it has been tested extensively for index sets of size at most three. The same degeneration connects the conjecture to probability: a further scaling limit of the $c\to\infty$ formulas produces transition probabilities of the asymmetric simple exclusion process (ASEP — the standard interacting-particle model of the KPZ class) on the half-line with generic injection rate $\alpha$ at the origin, and the orthogonality is exactly the initial condition of that transition kernel. In December 2025, de Gier, Mead, Remenik and Wheeler proved the homogeneous specialization of the orthogonality at the ASEP level (vertical rapidities $y_j\equiv1$, with $a=\alpha/(\alpha+q-1)$), as the initial-condition lemma in their derivation of half-space ASEP transition probabilities from general initial data. The full inhomogeneous six-vertex statement — an identity of rational functions in $q$, $a$ and $Y$ — remains open, and establishing it is the problem stated here.

Problem Statement

Prove or disprove the following conjecture in its full generality. Let $q$, $a$ be complex parameters and $Y=(y_1,y_2,\dots)$ an alphabet of vertical rapidities. For every finite subset $\nu=\{\nu_1>\cdots>\nu_n\geq 1\}$ of the positive integers and every finite subset $\kappa$ with $|\kappa|\leq n$,

$$\oint_{\mathcal{C}}\frac{dw_1}{2\pi i}\cdots\oint_{\mathcal{C}}\frac{dw_n}{2\pi i}\ \prod_{1\leq i<j\leq n}\left[\frac{w_j-w_i}{qw_j-w_i}\cdot\frac{1-qw_iw_j}{1-w_iw_j}\right]\ \prod_{i=1}^{n}\left[\frac{w_i-a}{w_i(1-aw_i)}\cdot\frac{1-qw_i^2}{1-w_i^2}\cdot\frac{y_{\nu_i}}{1-qw_iy_{\nu_i}}\prod_{j=1}^{\nu_i-1}\frac{1-w_iy_j}{1-qw_iy_j}\right]\ \lim_{c\to\infty}F_{\kappa}(w_1,\dots,w_n|Y)=\delta_{\kappa,\nu},$$

where $F_{\kappa}=F_{\kappa/\emptyset}$ denotes the dual family of half-space six-vertex symmetric functions of Garbali–de Gier–Mead–Wheeler (a multivariate rational function of $w_1,\dots,w_n$, of $q$, of the boundary parameters and of $Y$, defined as a six-vertex partition function in Section 3 of their paper) and $\mathcal{C}$ is a small, positively oriented circle surrounding the points $y_j^{-1}$, $j\geq 1$, and no other singularities of the integrand, such that $q\cdot\mathcal{C}$ is disjoint from the interior of $\mathcal{C}$ (for fixed $\nu$ and $\kappa$ the integrand depends on only finitely many entries of $Y$, so only finitely many of the points $y_j^{-1}$ are enclosed). The identity is to be established as an identity of rational functions in $q$, $a$ and the finitely many entries of $Y$ on which both sides depend; equivalently, for all parameter values at which the contour conditions above can be realized. A proof along the route envisaged in the source — establishing the linear independence of the family $\lim_{c\to\infty}G_{\kappa}(x_1,\dots,x_L)$ and rigorously justifying the contour deformations, the uniform convergence and the interchange of summation with integration used in the heuristic derivation — is admissible but not required: any complete and rigorous argument establishing the displayed identity at this level of generality resolves the problem.

References

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