155 problems

Fully Packed Loops (FPL) are obtained by coloring edges of a square grid so that exactly two edges incident to each inner vertex are colored, while every other boundary edge is col…

Let $n\geqslant 1$ be an integer, and consider the ground-state eigenvector of the periodic XXZ spin chain with anisotropy parameter $\Delta=-\frac12$ and $N=2n+1$ sites. Normalize…

Let $a$ be a spin taking values modulo $N$, let $\omega$ be an $N$th root of unity, and define the order parameter by … Here $k$ is the universal chiral-Potts-model constant, and…

Refined Razumov–Stroganov conjecture. Every entry $\boldsymbol{\Psi}_n^{(1)}(\pi\mid t)$ is a polynomial in $t$ of degree at most $n-1$, and

Yang–Baxter conjecture. The operators $X_i(u)$ obey the Yang–Baxter equation.

Determinant conjecture. Similarly to the sine-Gordon model, the one-point functions in the fermionic basis satisfy

Let $u$ be sufficiently small, with $|u|\ll 1$, let $s$ be a positive integer, and let $a\in\{1,2,\dots,s\}$. For each positive integer $m$, let $T^{(a)}_{m}(v)$ denote the transfe…

Let $Z_N^{\rm P}(q,x)$ be the partition function of the $sl_n$ Polychronakos model, and define the vertex-model partition function by … Here ${\cal S}_N$ is the finite set of verte…

Let $V={\bf C}^n$, let $V^{\otimes N}$ be the spin space of the $sl_n$ Polychronakos model, and let $Y(sl_n)$ act on this space commuting with the Hamiltonian. Let the spaces $W_d$…

Let $U_q(\widehat{sl}_n)$ be the quantized affine algebra, and let the level $1$ integrable modules of the affine Lie algebra $\widehat{sl}_n$ carry their Yangian $Y(sl_n)$-module…

ABF Boltzmann-weight conjecture. The connection matrices for four point functions having arbitrary numbers of screening operators should be expressible in terms of the Boltzmann we…

The regime-III ABF models have vacuum sectors, and the work constructs a weight-preserving bijection there between paths, strings (or rigged configurations), and standard tableaux.…

Consider the higher-spin eight-vertex model with spin parameter $\ell$, and let the fused transfer matrix corresponding to the spin chain with local interaction determine the physi…

Let $(\nu, \{\lambda_1, \dots, \lambda_M\})$ be a solution of the Bethe equations, and let $\nu”$ and $\nu'$ be the parities of the Bethe vector…

2BTL representation-theoretic conjecture. The cases in which Bethe Ansatz equations can be constructed for the six-vertex model with integrable boundary terms are related to proper…

Let $\Lambda_{a\pm2s,a}(u)$ be the tail operator, let $g_{P+h}$ be the gauge factor, and let $\widehat{F}^{+}$ and $F(v_j)$ denote the half-current and its integral-kernel realizat…

Let $\widehat{\Phi}_{\varepsilon}(u)$ be the gauge-transformed vertex operators, with $\varepsilon\in\{-k,-k+2,\ldots,k\}$, and let $W^{(k,k)}$ denote the fused face weights. Commu…

Brauer-loop groundstate conjecture. (i) For each system size, the overall normalisation can be chosen so that the smallest element of $\psi_0$ is $1$ and all other elements are int…

Stationary-state enumeration conjecture. For any of these four triples, $a_F$ equals $M_F(G)$, the number of fully packed loop diagrams on $G$ with matching $F\in{\mathcal F}$. Thi…

Perron eigenvector conjecture. The vector $\Psi$ is the unique eigenvector of $H$ corresponding to its largest eigenvalue $2n$.

For an odd lattice size $2n+1$, let $\Phi_{2n+1}^{(s)}$ be the maximally nested component with red arcs joining $(i,2n+1-i)$, green arcs joining $(i+1,2n+2-i)$, a red line emanatin…

Consider the polynomial solution of the level $1$ qKZ equation with open, or reflecting, boundaries, and let CSTCPPs denote cyclically symmetric transpose-complementary plane parti…

Alcaraz–Stroganov conjecture. The free-fermion part of the spectrum of the $SU_q(k+1)$ Perk–Schultz model at $q=-\exp(\pi i/(k+1))$ is a consequence of nice properties of the inhom…

Let $\Psi^{(L)}(\tau)$ be the homogeneous open boundary quantum Knizhnik–Zamolodchikov solution, with components indexed by link patterns, and let the admissible endpoints of link…

At $q=e^{2i\pi/3}$ and in the homogeneous limit, let … for $C_r$ is the sum of entries for $C_{r-1}$. The assertion is presented after empirical sum and pairing formulas, and the s…