The Izergin–Korepin nineteen-vertex domain wall partition function beyond the root of unity: closed forms, reflection boundaries, eigenvector components, and structural characterization

The Izergin–Korepin (IK) nineteen-vertex model is the integrable lattice model associated with the twisted quantum affine algebra $U_q(A_2^{(2)})$, the next case after the six-vertex model in the hierarchy of exactly solvable vertex models. For the six-vertex model, the domain wall partition function (DWPF) is known in a single compact determinant form (the Izergin–Korepin determinant), and this determinant underpins correlation functions, Bethe-ansatz eigenvector components, alternating-sign-matrix enumeration, a proof that the DWPF is a Kadomtsev–Petviashvili (KP) tau-function, natural Hall–Littlewood expansions, and determinantal/Pfaffian formulas for reflecting boundaries. For the IK model, Garbali (J. Stat. Mech. 2016) derived the recurrence relations for the DWPF at generic $q$ and solved them at the cubic root of unity $q=e^{i\pi/3}$ ($q^3=-1$) as a Jacobi–Trudi-like determinant, explicitly leaving open: (1) whether the generic-$q$ partition function admits a single compact determinant or comparable closed form; (2) extensions to domain-wall-type partition functions with reflection matrices on one or two sides of the domain; (3) whether the root-of-unity determinant is a KP tau-function; (4) whether the generic-$q$ partition function expands in symmetric-polynomial families such as Hall–Littlewood polynomials; and (5) construction of the full highest-spin transfer-matrix eigenvector, of which only the domain-wall components are known. The source leaves these five directions open.

Background

An integrable vertex model is a two-dimensional statistical lattice model whose local configuration weights are built from a solution $R$ of the Yang–Baxter equation — the cubic consistency condition that guarantees commuting transfer matrices and exact solvability. Each edge of the lattice carries a discrete state, and the weight of a configuration is the product of vertex weights, one per lattice site, each depending on the pair of spectral parameters (rapidities) attached to the horizontal and vertical lines crossing at that site. The archetype is the six-vertex model, in which each edge carries one of two arrow states and local arrow conservation leaves exactly six allowed vertices. Under domain wall boundary conditions (DWBC) — all arrows on the left and right boundaries of the $N\times N$ square point inward and all arrows on the top and bottom point outward — the partition function $Z_N(\zeta_1,\dots,\zeta_N|z_1,\dots,z_N)$ is a symmetric function of the horizontal rapidities $\zeta_i$ and, separately, of the vertical rapidities $z_j$.

The six-vertex DWPF is exactly solvable in a strong sense. Korepin (1982) showed that $Z_N$ is characterized by simple recurrence relations in the system size (obtained by specializing one rapidity so that a boundary row and column freeze to a unique configuration), and Izergin solved these recurrences as a single $N\times N$ determinant (the Izergin–Korepin determinant; see Korepin–Bogoliubov–Izergin 1993). That determinant is the workhorse of the six-vertex exact theory: correlation functions of the spin-1/2 XXZ chain, norms of Bethe states, and the enumeration of alternating-sign matrices — square matrices with entries $0,\pm 1$ whose nonzero entries alternate in sign along every row and column, in bijection with six-vertex domain-wall configurations — all flow from it, and in the algebraic Bethe ansatz (ABA) — the method that constructs transfer-matrix eigenvectors by repeated application of creation operators to a reference state — its coefficients are the components of the highest-spin eigenvector of the periodic transfer matrix. The same object carries structural results of independent interest: it is a KP tau-function, meaning a function of infinitely many time variables satisfying the bilinear Hirota equations of the Kadomtsev–Petviashvili (KP) integrable hierarchy, equivalently an expectation value of charged free fermions (Foda–Wheeler–Zuparic 2009, via rewriting the Lascoux determinant — an alternative determinant form of the six-vertex DWPF due to A. Lascoux — as a Cauchy–Binet product of rectangular matrices); it expands naturally in Hall–Littlewood polynomials, a one-parameter family of symmetric polynomials interpolating between monomial and Schur polynomials (Warnaar 2008); and its relatives with one reflecting boundary — a reflection or $K$-matrix, a boundary weight matrix solving the reflection equations that implements an integrable mirror — are again determinants and Pfaffians (a Pfaffian being the square root of the determinant of an antisymmetric matrix; Tsuchiya 1998; Kuperberg 2002).

The Izergin–Korepin (IK) nineteen-vertex model is the vertex model built from the trigonometric $R$-matrix of the twisted quantum affine algebra $U_q(A_2^{(2)})$ — a $q$-deformation of the twisted affine Kac–Moody algebra $A_2^{(2)}$ whose fundamental representation yields the three-state $R$-matrix of the model. Its edges carry three states, so local conservation leaves nineteen allowed vertices whose weights contain square-root-type factors; the model underlies the integrable spin-1 chain and, at special values of $q$, relates to the dilute Temperley–Lieb loop model, a loop-gas representation of the same configurations in which every closed curve receives weight $n$ (here $n=1$, relevant to critical percolation). Here $q$ is the deformation parameter, and a root of unity is a value $q=e^{i\pi k/\ell}$ at which the representation theory degenerates and weights simplify.

For the IK model the same program stalled. Garbali (2016) derived, for generic $q$, the two specialization recursions for the normalized DWPF $\tilde Z_N$ (at $\zeta_j=z_i$ and $\zeta_j=-q^{-3}z_i$ a row and a column freeze) and combined them by Lagrange interpolation into a representation of $\tilde Z_N$ as a weighted sum of copies of $\tilde Z_{N-1}$ (eqs. (35)–(36) of that paper); no single-determinant form emerged. At the cubic root of unity $q=e^{i\pi/3}$ (so $q^3=-1$ and $q+q^{-1}=1$) the vertex weights acquire a common factor; after extracting the product $\prod_{i,j}(z_i-\zeta_j)$ the recurrence collapses to a single one with proportionality factor $P_N$, whose generating function defines a family $\Delta_{i,N}$ of symmetric polynomials in the $2N$ rapidities, and the solution is the $(N-1)\times(N-1)$ Jacobi–Trudi-like determinant (so named after the classical Jacobi–Trudi formula expressing a Schur function as a determinant of elementary symmetric polynomials)

$$\bar Z_N=\det_{1\le i,j\le N-1}\Delta_{3j-i,N}(\zeta_1,\dots,\zeta_N,z_1,\dots,z_N),$$

with $\Delta_{i,N}=\sum_l(q^{-i+2l+1}+q^{i-2l-1})E_l(z)E_{i-l}(\zeta)$, where $E_k$ denotes the elementary symmetric polynomials. This determinant is close in form to the Lascoux determinant for the six-vertex DWPF — the very form through which the six-vertex KP property is proved — and the paper's conclusion asks for the remaining steps toward six-vertex parity. For comparison, the other classical nineteen-vertex model, the Fateev–Zamolodchikov model (the nineteen-vertex model associated with the spin-1 XXZ chain), does admit a generic-$q$ determinant (Caradoc–Foda–Kitanine 2006, by identification with a spin-1 six-vertex model), so the difficulty is specific to the twisted $A_2^{(2)}$ weights.

Problem Statement

The problem is to complete the exact theory of the domain wall partition function (DWPF) of the Izergin–Korepin nineteen-vertex model ($U_q(A_2^{(2)})$) — which Garbali (2016) solved only at the cubic root of unity — to the level of structural completeness that the six-vertex DWPF enjoys. The source poses the generic-$q$ closed-form question as an open problem in its main text and four further interrelated directions in its Conclusion; together they frame the program of bringing the IK domain-wall theory to six-vertex parity.

(1) Generic-$q$ closed form. Determine whether the normalized IK DWPF $\tilde Z_N$ at generic $q$ can be written as a single compact determinant or another closed form of comparable compactness — i.e., whether the Lagrange-type representation (eqs. (35)–(36) of the source), which expresses $\tilde Z_N$ through a weighted sum of copies of $\tilde Z_{N-1}$, can be resummed.

(2) Reflection boundaries. For the IK model with reflection ($K$-)matrices on one or two sides of the $N\times N$ domain, investigate the corresponding domain-wall-type partition functions — their recurrence relations at generic $q$, and whether determinantal or Pfaffian closed forms exist at $q=e^{i\pi/3}$ — in analogy with the six-vertex results of Tsuchiya and Kuperberg.

(3) KP hierarchy. Determine whether the root-of-unity determinant $\bar Z_N=\det_{1\le i,j\le N-1}\Delta_{3j-i,N}$ (eq. (41) of the source) is a KP tau-function, as the six-vertex Izergin–Korepin determinant is: either prove the identification with a stated hierarchy (for example via a charged-free-fermion expectation-value representation or a Cauchy–Binet structure), or refute it by exhibiting an explicit instance for which the required bilinear identities fail.

(4) Symmetric-function expansion. Determine whether the generic-$q$ DWPF $Z_N$ expands in Hall–Littlewood polynomials, or an appropriate analog adapted to the $A_2^{(2)}$ weight structure, as the six-vertex $Z^{6v}$ does in Hall–Littlewood polynomials — with explicit coefficients and in a family fixed independently of the model; the trivial expansion of $Z_N$ in monomial symmetric functions, or in a basis built from $Z_N$ itself, does not count.

(5) Highest-spin eigenvector. For the IK transfer matrix at $q^3=-1$, construct the full highest-spin eigenvector of the algebraic Bethe ansatz — whose known components at $q^3=-1$ are precisely the domain-wall partition functions — and determine whether it admits a closed form at $q=e^{i\pi/3}$ (a recurrence relation is a natural route, but any sound method is admissible). The source expects the non-domain-wall components not to have nice closed forms, since they are not separately symmetric in the $\zeta$'s or $z$'s.

The common objective is six-vertex parity for the IK domain-wall theory. Directions (1), (3) and (4) concern the same object $Z_N$ and its root-of-unity specialization — a generic-$q$ closed form is the hub from which the other two most naturally follow — while (2) and (5) extend the same root-of-unity technology to the neighboring domain-wall-type objects. The five directions are independently attackable, and each is a self-contained open problem: a complete solution to any one of them resolves the task, with the remaining directions left open. The source's proposed routes (derive recurrences first, then specialize $q=e^{i\pi/3}$) are suggestions, not requirements; any sound method is admissible.

References

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