Six-vertex domain-wall partition function at roots of unity beyond the third: functional-equation evaluation

The partition function of the inhomogeneous six-vertex model on an $n\times n$ square with domain-wall boundary conditions is given by the Izergin–Korepin determinant. Yu. G. Stroganov (Theoretical and Mathematical Physics 146 (2006) 53–62) showed that when the crossing parameter is a root of unity, $\eta=2\pi/N$, this partition function satisfies a linear functional equation; at $\eta=2\pi/3$ it collapses to a three-term relation that determines the partition function by interpolation up to a constant, implies the striking symmetry of the partition function in the union of all $2n$ spectral parameters, and yields refined and doubly-refined enumeration formulae for alternating sign matrices. The paper derives its basic functional equation for every integer $N\ge3$ and remarks that it 'appears to be useful for the study of the partition function $Z$', but limits its analysis to the case $N=3$ ('In this paper we limit ourselves to the case $N=3$'); within that case it left the normalization constants, the closed form of the adjacent-boundary doubly-refined numbers $C(n;r,\tilde r)$, and the treatment of coincident spectral parameters undetermined. The problem posed is this root-of-unity extension, retaining the generality in $N$ of the source's basic functional equation.

Background

The six-vertex model (also called square ice) places arrows on the edges of a square lattice, allowing only the six vertex configurations in which two arrows point into and two out of each site (the ice rule); it is the archetype of an exactly solved two-dimensional lattice model. On an $n\times n$ square with domain-wall boundary conditions (DWBC) — all boundary arrows pointing inward on the horizontal sides and outward on the vertical ones — and with spectral parameters $x_1,\dots,x_n$ and $y_1,\dots,y_n$ attached to the horizontal and vertical lines, the partition function $Z_n(\{x\},\{y\})$ is given by the Izergin–Korepin determinant: a determinant of the matrix $M_{ij}=1/(\sin(x_i-y_j+\eta/2)\,\sin(x_i-y_j-\eta/2))$ times explicit trigonometric prefactors, found by Izergin on the basis of Korepin's work and recorded in the standard monograph of Korepin, Bogoliubov and Izergin. In the trigonometric weights used by Stroganov, $a(u)=\sin(u+\eta/2)/\sin\eta$, $b(u)=\sin(u-\eta/2)/\sin\eta$, $c=1$, the anisotropy is $\Delta=\cos\eta$, where $\eta$ is the crossing parameter. Inhomogeneous DWBC partition functions are also weighted enumerators of alternating sign matrices (ASMs): square matrices with entries in $\{-1,0,1\}$ whose nonzero entries alternate in sign along every row and column and sum to one in each. The famous ASM count of Mills, Robbins and Rumsey, and its refinements recording the column of the 1 in the first row, arise as homogeneous specializations of $Z_n$ at the so-called combinatorial point (the specialization where all three vertex weights are equal).

Stroganov's paper (the source of this problem) observes that when $\eta=2\pi/N$ is a root of unity, $Z_n$ satisfies a linear functional equation: summing the Izergin–Korepin determinants with the spectral parameter $x_1$ shifted by $k\eta$, $k=0,\dots,N-1$, with explicit trigonometric coefficients, gives identically zero. This basic equation is derived for every integer $N\ge3$, but it becomes tractable at $N=3$, where the identity $\sin u\,\sin(u+2\pi/3)\sin(u+4\pi/3)=-\tfrac14\sin 3u$ collapses it to the three-term relation $f(u)+f(u+2\pi/3)+f(u+4\pi/3)=0$ for $f(u)=Z_n(u,u_1,\dots,u_{2n-1})\prod_{i=1}^{2n-1}\sin(u-u_i)$, the $u_i$ being the remaining spectral parameters. Since $f$ is a trigonometric polynomial of degree at most $3n-2$ which vanishes at each $u_i$ (its product factor does) and satisfies $f(u+\pi)=(-1)^n f(u)$, the three-term relation and the vanishing conditions determine $f$, hence $Z_n$, up to a multiplicative constant, by interpolation — under the assumption that the $u_i$ are distinct. Two far-reaching consequences follow at $\eta=2\pi/3$. First, $Z_n$ is symmetric not only within the sets $\{x\}$ and $\{y\}$ separately (true for all $\eta$) but in the union $\{x\}\cup\{y\}$ of all $2n$ spectral parameters. Second, homogeneous specializations reproduce the refined ASM enumeration and give formulae for doubly-refined enumerations: $B(n;r,\tilde r)$, the number of $n\times n$ ASMs whose top-row and bottom-row 1s sit in columns $r$ and $\tilde r$ (opposite boundaries), and $C(n;r,\tilde r)$, the number whose top-row 1 sits in column $r$ and whose left-column 1 sits in row $\tilde r$ (adjacent boundaries). Stroganov proved the linear relation $B(n;r,\tilde r)=C(n;r,\tilde r+1)+C(n;r+1,\tilde r)-C(n;r+1,\tilde r+1)$ but did not determine $C$ in closed form.

The modern understanding of the third-root points is complete. Independently of Stroganov, Okada showed that $Z_n$ there equals an explicit prefactor times the Schur function (the irreducible character of the general linear group, a symmetric polynomial) of the double-staircase partition $(n-1,n-1,n-2,n-2,\dots,1,1)$ evaluated at all $2n$ spectral parameters; this single identity fixes every normalization constant, makes the total symmetry manifest, and underlies the subsequent closed theory of doubly-, triply- and quadruply-refined ASM enumerations built by Karklinsky–Romik, Fischer, Ayyer–Romik and Behrend. Aval later gave an elementary proof of the total symmetry that avoids determinants and extends it to half-turn-symmetric ice models (models invariant under a $180^\circ$ rotation of the lattice). The two third-root points $\eta=2\pi/3$ and $\eta=\pi/3$ are related by the standard parameter transformation $\eta\to\pi-\eta$ (with $x\to\pi/2-x$, $y\to-y$), so both are settled.

What has no known analogue is every other root of unity: $\eta=2\pi/N$ with $N\ge4$, a list that includes the free-fermion point $N=4$ ($\eta=\pi/2$, $\Delta=0$) and, apart from $N=6$, is untouched by the functional-equation and interpolation machinery. Galleas derived general-$\eta$ functional relations for the DWBC partition function by Yang–Baxter-algebra methods (functional relations for the algebra generated by the model's row monodromy matrix — a different type of equation, not the root-of-unity interpolation), and Garbali carried out the analogous root-of-unity program for the nineteen-vertex Izergin–Korepin model (the integrable spin-1 generalization of the six-vertex model); neither addresses the six-vertex case at $N\ge4$. The enhanced total symmetry of the $N=3$ theory does not persist at these points: it already fails at $N=4$ for $n=2$, where the Izergin–Korepin determinant (equivalently, the $2\times2$ domain-wall state sum) takes different values under the swap of one horizontal with one vertical spectral parameter; consequently no representation of $Z_n$ as a single symmetric function of all $2n$ parameters, comparable to the double-staircase Schur formula, can exist there. The free-fermion point $N=4$, where the Izergin–Korepin matrix becomes a Cauchy-type matrix and an elementary treatment is therefore expected, is the natural entry point; for $N\ge5$ no tool of comparable power is currently available.

Problem Statement

Carry out, for general roots of unity of the crossing parameter, the program that the source paper develops only at $N=3$: extend its $N=3$ functional-equation analysis beyond the third root of unity, retaining its basic functional equation, which is valid for every integer $N\ge3$. For the inhomogeneous six-vertex model on the $n\times n$ square with domain-wall boundary conditions and weights $a(u)=\sin(u+\eta/2)/\sin\eta$, $b(u)=\sin(u-\eta/2)/\sin\eta$, $c=1$, take the crossing parameter $\eta=2\pi/N$ with integer $N\ge4$ (this includes the free-fermion point $N=4$, $\eta=\pi/2$, $\Delta=0$; the case $N=6$, $\eta=\pi/3$, is excluded as settled, being related to the $N=3$ case by the standard transformation $\eta\to\pi-\eta$ (which maps $x\to\pi/2-x$, $y\to-y$ and leaves the state sum unchanged)):

(1) Start from the basic root-of-unity equation, valid for every $N\ge3$:

$$\sum_{k=0}^{N-1}Z_n(x_1+k\eta,x_2,\dots,x_n;y_1,\dots,y_n)\prod_{i=2}^{n}\sin(x_1-x_i+k\eta)\prod_{j=1}^{n}\frac{1}{\sin(x_1-y_j+(k+\tfrac12)\eta)\,\sin(x_1-y_j+(k-\tfrac12)\eta)}=0,$$

and reduce it — as the identity $\sin u\,\sin(u+2\pi/3)\sin(u+4\pi/3)=-\tfrac14\sin 3u$ does at $N=3$ — to a tractable closed functional equation for a modified partition function analogous to $f(u)=Z_n(u,u_1,\dots,u_{2n-1})\prod_{i=1}^{2n-1}\sin(u-u_i)$, and solve it by interpolation or an equivalent closed form (determinant, symmetric-function representation, or other exact formula) valid for distinct spectral parameters, with the overall multiplicative constant fixed as an explicit function of $n$ and $N$.

The objective is the extension of the root-of-unity functional-equation evaluation of the Izergin–Korepin determinant beyond the third root of unity. If the closed form obtained in part (1) admits specialization or enumeration consequences at those roots of unity — analogues of the refined and doubly-refined boundary-statistics distributions of the $N=3$ theory — they may be reported, but they are not a required part of the target. The other issues left undetermined in the source (the $N=3$ normalization constants and the closed-form determination of $C(n;r,\tilde r)$ at $N=3$) are settled by later work and are not part of the target; a solution may use them freely as the $N=3$ reference point.

References

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