28 problems

Convergence conjecture. This rewriting system is convergent, meaning that it is terminating and confluent.

Fan's conjecture. The algebra $TL(H_n)$ is generically semisimple.

2BTL representation-identification conjecture. For $n>0$, the irreducible representation $W^{(N,n)}_{\epsilon_1,\epsilon_2}$ is equivalent to $V^{(N,n)}_{\epsilon_1,\epsilon_2}$, a…

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…

The one-boundary Temperley–Lieb algebra has generators whose representations yield the Hamiltonians $H^{nd}$, $H^d$, and $H^{lp}$; the representation giving $H^d$ is explicitly con…

Let $\Lambda_n$ index the cell modules $\Delta(\lambda)$ and simple modules $L(\mu)$ of the orientifold quiver Temperley--Lieb algebra, and let $\langle k\rangle$ denote grading sh…

Let $\psi_0$ be the surjective map constructed from the algebra of diagrammatic generators to the relevant algebra of endomorphisms, with proposed free generators…

Let $\kappa$ be the coefficient field, let $\operatorname{Ch}^-(\mathsf{BN}_n)$ be the homotopy category of bounded-above complexes of $n$-strand Bar–Natan 1-morphisms, and let…

Let $w_{0},w_{1}$ be link states, let $\mathcal{S}(w_{0})$ and $\mathcal{S}(w_{1})$ be the associated spin-chain representations, and define … by…

Temperley–Lieb–Virasoro braided equivalence conjecture. There is an equivalence of braided tensor categories

Affine Temperley–Lieb quotient conjecture. There is an isomorphism of algebras

Eigenvalue inclusion conjecture. With inhomogeneous spectral parameters, every eigenvalue of $T^{(1)}$ is also an eigenvalue of $T^{(0)}$.

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 $si$ 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…

The two-boundary Temperley–Lieb $O(n)$ loop model has a ground-state vector $si$ whose components are Laurent polynomials with nonnegative integer coefficients in the parameters ap…

For $n\geq 3$, let $\mathcal{B}_n^{\pm}$ be the specialized algebra, let $\mathcal{I}_n^{\pm}$ be its defining ideal, and let…

Let ${\rm PTL}_n$ be the Partition Temperley–Lieb algebra, let $\mathsf{P}_n$ denote the set-partition monoid on $\{1,\ldots,n\}$, and let $F$ be a reduced word in the generators u…

Irreducibility conjecture. These radicals are irreducible whenever they are non-trivial.

Generalized boundary symmetry conjecture. For all $n$, one has

Boundary Temperley–Lieb representation conjecture. The map $\Theta_{i_1\dots i_m}$ is a representation of the boundary Temperley–Lieb algebra.

Junction symmetry conjecture. For every $x\in U_{q_{i_1\dots i_m}}(sl_2)$, one has

Quantum-group symmetry conjecture. The family of representations ${\mathrm f}_{i_1\dots i_m}$ should give non-trivial exact symmetries of the junction representation.

Let $W_c(\widetilde{C}_n)$ denote the set of fully commutative elements in the affine Coxeter group of type $\widetilde{C}_n$. Assume that $n$ is even, and consider the elements…

Gram-matrix parity conjecture. One has

Barad's factorization conjecture. The determinant satisfies