33 problems

At the Razumov–Stroganov point $q=e^{2i\pi/3}$, take the homogeneous limit $w_i=1$ for all $i$ in the level-one $D_r$ $q$KZ solution, viewed as the Perron–Frobenius eigenvector…

At the Razumov–Stroganov point $q=e^{2i\pi/3}$ and in the homogeneous limit $w_i=1$, let $\Psi$ be normalized so that its smallest entry is $\Psi_{\pi_0}=1$. Let $v$ be the left ei…

Extra symmetry conjecture. As operators on these BRST cohomology groups,

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…

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

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…

At $q=e^{2i\pi/3}$, let … is … For odd $r=2n-1$, the preceding discussion presents the corresponding product as computed, while the even case follows by a specialization from the o…

At $q=e^{2i\pi/3}$ and in the homogeneous limit, let … , with arches connecting consecutive points, is $A_V(r)$. The sum-of-entries formula is established in the supplied discussio…

Let … , corresponding to the link pattern with arches connecting consecutive points, is $A(n-1)$. This refines the known result that the sum of entries is $A(n)$; the largest-entry…

Snake in the Snail conjecture. For odd $m$, the extended T-systems appear in successive lines of the snail operator, and the component corresponding to

Analytic-properties conjecture. The function $D_{1,\dots,m}$ is meromorphic in $\lambda_1,\dots,\lambda_m$, with at most simple poles when

Let $L=2n+1$ with $n\geqslant 0$, let $\eta=pi/3$, and let $V$ be the two-dimensional spin space. For inhomogeneity parameters $u1,\dots,u{2n+1}$, consider the eigenvalue problem ……

Let $\mathcal{C}_{N+1}$ be the set of signed permutation matrices of size $2N\times2N$ invariant under diagonal and anti-diagonal reflections and avoiding $(-2,-1)$. Let $B(c)$ be…

Let $\mathcal{S}$ be the set of symmetric binary matrices with no row sum greater than one. For $s\in\mathcal{S}$, let $A(s)$ be the set of upper-triangular positions containing a…

At $q=1$, write the Type BII sum as … Type BII coefficient-polynomial conjecture. For the relevant coefficient index $i$, … where … with $p_{i,k}\in\mathbb{Z}$. The source reports…

At $q=1$, write the Type BII sum as … with $S_{N,j}\in\mathbb{N}$. Type BII first-coefficient conjecture. … The paper reports verification through $N=20$, but no general proof or r…

Let $\mathcal{C}_n$ be the set of signed permutation matrices of size $2(n-1)\times2(n-1)$ invariant under diagonal and anti-diagonal reflections and avoiding the pattern $(-2,-1)$…

Let $\mathcal{A}_N$ be the set of symmetric binary $N\times N$ matrices with no row sum greater than one. For $a=(a_{i,j})$, define … Let $S_{N,i}$ be the coefficient of $Q^i$ in t…

At $q=1$, write the Type A and Type BIII sums as … with nonnegative integer coefficients $S_{N,i}$ and $S'_{N,i}$. Coefficient-polynomial conjecture. The coefficients satisfy … and…

Let $si$ be the ground-state vector, and let $S^{\mathrm{X}}_{N}(q,Q)$ denote the sum of all its components on the Kazhdan–Lusztig basis of boundary type $\mathrm{X}$, where…

Throughout this subsection, let $N=2n$, and let $P_\lambda(x_1,\dots,x_N;q,t)$ denote Macdonald polynomials, with $b^{\rm el}_\lambda(q,t)$ the even-columns Littlewood coefficient.…

Let $T^{(2)}(z)$ be the transfer matrix of the nineteen-vertex model with twist angle $\phi=\pi$, and let … be its eigenvalue for inhomogeneity parameters $w_1,\dots,w_N$. Non-dege…

Let $N$ be the strip width, $d$ the number of defects, and $A^M_{m,n}$ the sets of admissible two-column configurations specified by the preceding selection rules, with choices of…

Let $Y_L$ be the quantity defined in the paper from the finite-size expectation-value calculation, and let $q$ be the model parameter. Let $w$ be its spectral variable. Invariance…

Let $F_L$ be the expectation value defined from the two-boundary Temperley–Lieb ground-state eigenvectors, and let $Z_L$ be the corresponding normalisation. Let $P$ denote a polyno…