Analytic proof of the even-index Yang–Baxter conjectures for descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras

Finch, Dancer, Isaac and Links (Nucl. Phys. B 847 (2011) 387) constructed spectral-parameter-dependent solutions of the Yang–Baxter equation, viewed as descendants of the zero-field six-vertex model, from the Drinfeld doubles $D(D_n)$ of dihedral group algebras. For odd $n$ the construction was later tied to a limit of the $N$-state Fateev–Zamolodchikov model, but for even index the supporting evidence remains purely computational: the candidate $R$-matrix satisfies the Yang–Baxter equation only as verified symbolically for even $m\le 16$, the two-parameter extension $R(z,\mu)=R^{+}(z)+\mu R^{-}(z)$ satisfies its mixed-parameter Yang–Baxter relation only for even $n\le 12$, and the solutions provably fail regularity and provably fail spectral equivalence with the specific two-rapidity Fateev–Zamolodchikov limit analyzed in the source, with no structural explanation. This problem asks for a rigorous, uniform-in-index resolution of these conjectures. The even-index conjectures are open.

Background

An $R$-matrix is a matrix $R(z)$, depending on a complex spectral parameter $z$, that solves the Yang–Baxter equation (YBE) $R_{12}(x)\,R_{13}(xy)\,R_{23}(y) = R_{23}(y)\,R_{13}(xy)\,R_{12}(x),$ where $R_{12}$, $R_{13}$, $R_{23}$ act on pairs of factors of a triple tensor product of a finite-dimensional vector space. Solutions generate commuting transfer matrices and thereby exactly solvable two-dimensional statistical-mechanical models and integrable one-dimensional quantum chains. The prototypical solution is the six-vertex model R-matrix $r(z)$; a solution is called regular if $R(1)=P$, the permutation matrix, the normalization needed to read off a local Hamiltonian by the quantum inverse scattering method.

Most known spectral-parameter solutions come from quasi-triangular Hopf algebras — quantum groups such as $U_q(\mathrm{sl}(2))$ — whose universal R-matrix evaluated on representations can be given a spectral parameter (Baxterisation). For quantum algebras the tensor product graph method determines the entries of the spectral-parameter R-matrix from the decomposition of tensor products of representations into irreducibles, assigning to each irreducible component a scalar function of the spectral parameter fixed by crossing, unitarity and limiting conditions.

A second, much smaller class of quasi-triangular Hopf algebras consists of the Drinfeld doubles $D(G)$ of finite group algebras. Given a finite group $G$, its Drinfeld double is a finite-dimensional quasi-triangular Hopf algebra built from the group algebra and its dual; its canonical element satisfies the constant (parameter-free) YBE, so every representation of $D(G)$ yields a constant matrix solution. These algebras describe the braiding of non-Abelian anyons in discrete gauge theories. The paper at issue studies the dihedral groups $D_n$ (symmetries of a regular $n$-gon, generated by a rotation $\sigma$ and a reflection $\tau$ with $\sigma^n=\tau^2=e$): the irreducible representations of $D(D_n)$ have dimensions 1, 2 and $n$ for $n$ odd, and 1, 2 and $n/2$ for $n$ even.

A solution $R(z)\in\mathrm{End}(W\otimes W)$ is called a descendant of a smaller R-matrix $r(z)\in\mathrm{End}(V\otimes V)$, $\dim V<\dim W$, if there is an invertible L-operator $L(z)\in\mathrm{End}(V\otimes W)$ with $r_{12}(xy^{-1})L_{13}(x)L_{23}(y)=L_{23}(y)L_{13}(x)r_{12}(xy^{-1})$ and $L_{12}(x)L_{13}(y)R_{23}(x^{-1}y)=R_{23}(x^{-1}y)L_{13}(y)L_{12}(x)$. The classical instance is Bazhanov and Stroganov's discovery that the chiral Potts model is a descendant of the six-vertex model; the N-state Fateev–Zamolodchikov model (Fateev and Zamolodchikov, 1982), an N-state solvable model whose edge weights $W$, $\bar W$ satisfy the star-triangle relation, is likewise a descendant of the zero-field six-vertex model, sitting at the intersection of the chiral Potts and Kashiwara–Miwa models.

Finch, Dancer, Isaac and Links (2011), building on the earlier $n=3$ analysis of Dancer, Isaac and Links (2006), constructed L-operators from the higher-dimensional irreducible representations of $D(D_n)$ and adapted the tensor product graph method to the finite-group-double setting, producing explicit candidate descendants of the zero-field six-vertex model. For odd $n$ the resulting $R(z)$ is regular, unitary ($R^{-1}(z)=R(z)$ up to the stated conventions) and was verified by direct symbolic computation to satisfy the YBE for all odd $n\le 17$; for even index ($D(D_{2m})$ with $m$ even) the candidate $R(z)$, built from scalar functions $g(a,j;z)=\frac{1}{m}\sum_{b=0}^{m-1}w^{2bj}\prod_{k=1}^{a}\frac{1+z\,w^{2(a-b+1-2k)}}{z+w^{2(a-b+1-2k)}}$ with $w$ a primitive $2m$-th root of unity and matrix units $e_{i,j}$ taken modulo $m$, satisfies the YBE for all even $m\le 16$ but fails regularity ($\lim_{z\to 1}R(z)\ne P$). For even $n>2$ a two-parameter family $R(z,\mu)=R^{+}(z)+\mu R^{-}(z)$ exists, with $R^{\pm}$ distinguished by which conjugacy-class functions are set to one; it satisfies a mixed-parameter (braid-form) YBE for $n\le 12$ and in the limit $x=y=0$. Finally, while for odd $n$ the descendant was shown (computationally for $n\le 11$, and at the point $z=0$ for all odd $n$) to be gauge-equivalent to a limit of the N-state Fateev–Zamolodchikov R-matrix, for even index the eigenvalue multiplicities of $R^{+}(z)$ and $R^{-}(z)$ differ, and neither eigenvalue spectrum matches that of the specific two-rapidity Fateev–Zamolodchikov limit $R(z)$ (Eq. (26) of arXiv:1003.0501) analyzed in the source, so the even-index descendants are not basis-transform-equivalent to that limit — with the structural reason left unexplained.

Problem Statement

The common objective is to place the even-index members of the dihedral-double descendant construction on the same rigorous footing as their odd-index counterparts. Two linked conjectures from the source realize this objective.

(1) Even-index Yang–Baxter conjecture. Fix an even integer $m\ge 2$ and a primitive $2m$-th root of unity $w$, let $e_{i,j}$ denote matrix units on $\mathbb{C}^{m}$ with indices taken modulo $m$, and define $g(a,j;z)=\frac{1}{m}\sum_{b=0}^{m-1}w^{2bj}\prod_{k=1}^{a}\frac{1+z\,w^{2(a-b+1-2k)}}{z+w^{2(a-b+1-2k)}},\qquad R(z)=\sum_{i,j,a=0}^{m-1}g(a,j;z)\,e_{i+j,\,i+a}\otimes e_{i+a+j,\,i}$ (Eq. (20) of arXiv:1003.0501; Eqs. (28)–(30) of the published version). Prove, or refute by an explicit even $m$ and explicit parameter values, that for every even $m\ge 2$ this $R(z)$ satisfies the difference-property Yang–Baxter equation $R_{12}(x)R_{13}(xy)R_{23}(y)=R_{23}(y)R_{13}(xy)R_{12}(x)$ for all $x,y\in\mathbb{C}$ — equivalently, that the functions $g$ satisfy the finite identity system of Proposition 4.1 of the source paper. The conjecture is currently supported only by symbolic verification for even $m\le 16$.

(2) Two-parameter family. Fix an even integer $n>2$ and a primitive $n$-th root of unity $w$, and define $R^{+}(z)$ and $R^{-}(z)$ by $R^{\pm}(z)=\sum_{i,j,a=0}^{n-1}\Big[\frac{1}{n}\sum_{b=0}^{n-1}(\pm 1)^{b-a}\,w^{bj}\prod_{p=1}^{a}\frac{z+w^{(2p-1+b-a)}}{1+z\,w^{(2p-1+b-a)}}\Big]e_{i+j,\,i+a}\otimes e_{i+a+j,\,i},$ and $R(z,\mu)=R^{+}(z)+\mu R^{-}(z)$, invertible for $\mu\ne\pm 1$. Prove, or refute by an explicit even $n$ and explicit parameter values, that for every even $n>2$ the check-form $\check R(z,\mu)=P\,R(z,\mu)$ satisfies the mixed-parameter Yang–Baxter relation $\check R_{12}(x,\lambda)\,\check R_{23}(xy,\mu)\,\check R_{12}(y,\nu)=\check R_{23}(y,\nu)\,\check R_{12}(xy,\mu)\,\check R_{23}(x,\lambda)$ (Eq. (25) of the arXiv version; Eq. (36) of the published version), known so far only for $n\le 12$ and in the limit $x=y=0$.

These two demands are not independent: identifying the index $n$ of (2) with the index $m$ of (1) and taking the root of (2) to be the square of the primitive $2m$-th root of (1), the matrix $R^{+}(z)$ coincides with the descendant $R(z)$ of (1), so the mixed-parameter relation (2) at $(\lambda,\mu,\nu)=(0,0,0)$ is the difference-property Yang–Baxter equation of (1) and a resolution of (2) settles (1) for every even $m>2$, leaving demand (1) independently necessary only for the base case $m=2$. A resolution covering only (1) addresses that special case and does not by itself settle (2).

References

1. P. E. Finch, K. A. Dancer, P. S. Isaac, J. Links, Solutions of the Yang–Baxter equation: Descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras, Nuclear Physics B 847 (2011) 387–412. DOI: 10.1016/j.nuclphysb.2011.01.034; arXiv:1003.0501, https://arxiv.org/abs/1003.0501 (source paper).
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