12 problems

Let $\pi_a$ range over link patterns on $2n$ external links, let $A_n(\pi_a)$ be the corresponding FPL numbers, and let … be the eigenvector of the periodic Temperley–Lieb Hamilton…

Parabolic Temperley–Lieb relations conjecture. These functors should satisfy the natural isomorphisms corresponding to the defining relations of the Temperley–Lieb category, with…

Let $\mathcal O^{k,n-k}$ be the relevant parabolic category, and let $\mathcal U_i$ for $1\leq i\leq n-1$ be the projective functors defined by $\mathcal U_i=T_i\circ T^i$. Product…

For integers $n$ and $k$, let $D^b(\mathcal O_{k,n-k})$ be the bounded derived category of the relevant parabolic category, and define functors … … by…

Let $\Psi_n$ be the polynomial solution of the level $1$ $U_q(\widehat{\mathfrak{sl}_2})$ $q$KZ equation in the link-pattern basis, let $\pi_0$ denote the reference link pattern, a…

Even two-boundary FPL conjecture. The coefficients of the ground state enumerate fully packed loop configurations with the appropriate link patterns on the $(L+1)\times L$ patch an…

VHASMs ground-state sum conjecture. For $a=1$, the sum of the components of $\psi_0(1)$ for a system of size $L$ equals

Doubly refined two-boundary conjecture. The ground state $psi_0(a,b)$ is related to weighted fully packed loop diagrams on an $(L+1)\times L$ grid with special boundary conditions.

Periodic FPL conjecture. The state $\ket0$ is obtained by counting fully packed loop configurations on an $L\times L/2$ rectangular grid. The conjecture was checked in the source t…

Let $F^{n}_{A_{2n-1}}$ be the relevant Fock space, and let $v_c$ denote the graded vector associated with a crossingless matching $c$. Temperley–Lieb representation conjecture. The…

The two-step Temperley–Lieb algebraic Floquet protocol has evolution operators generated by two time steps $T_1$ and $T_2$, with $\beta$ denoting the Temperley–Lieb parameter. Inte…

Tutte's conjecture. There is such a relation, similar to Tutte's identity at $\phi+1=B_5$, for each Beraha number $B_{n+1}$.