15 problems

Let $\mathfrak{g}$ be of type $B_n^{(1)}$, let $1\leq l<n-1$, and let $m\in 2\mathbb{Z}_{\geq 1}+1$. Let $\mathtt{V}(\boxed{i^r})_a$ denote the corresponding Kirillov–Reshetikhin m…

Let $r>1$ and let $t$ be the parameter in the expansion … where $H(x,y,u,v)$ is the leading-order polynomial and $\theta$ is its leading exponent. Leading-exponent conjecture. For…

Let $\widehat{I}$ be the index set used for the quantum $T$-systems, let $m_{k,p}^{(i)}$ be the dominant monomial associated with $(i,p)\in\widehat{I}$ and $k\in\mathbb{Z}_{\ge 0}$…

Let $\mathfrak{g}$ be a simple Lie algebra, let $\mathcal{K}_t(\mathscr{C}_{\mathfrak{g}})$ be its quantum Grothendieck ring, and let $F_t(m)$ denote the elements used to formulate…

Let $G$ be an affine-affine $ADE$ bigraph in the classification, and let $A(t)$ be a function of the system parameters at time $t$. It is a time-dependent conserved quantity if the…

Let $G$ be an affine-affine $ADE$ bigraph in the classification. A backward Devron singularity is an initial state with $T_v(-1)=0$ for every vertex of color $\epsilon_v=1$, and a…

Growth-rate conjecture. If $Q$ admits a strictly subadditive labeling, then $f$ is bounded and periodic; otherwise, if $Q$ admits a subadditive labeling, then $f$ grows exponential…

Let $Q$ be a bipartite recurrent quiver, and suppose that a subadditive labeling is a labeling of its vertices satisfying the subadditivity inequalities used for recurrent quivers.…

Let $U_q(\mathfrak{g})$ be the finite quantum group associated with the type $C_3$ algebra, and let $\operatorname{Res}(V)$ denote restriction of a $U_q(\hat{\mathfrak{g}})$-module…

Extended T-systems are families of relations among modules of quantum affine algebras, and an extended T-system involves minimal affinizations of such algebras. Extended T-system e…

Positivity conjecture. The solution to the $A_1$ $T$-system is a positive Laurent polynomial in the initial data at the empty circles.

Let $T_j(x)$ be the commuting transfer matrices in the fusion hierarchy of the quantum transfer matrix, with spectral parameter $x$ and fusion index $j$. Zero-distribution conjectu…

Let $T^{(a)}_m(u)$ be the transfer matrix constructed from the restricted RSOS Boltzmann weights for ${\mathfrak g}$ at level $\ell$, with $a\in I$ and $m$ in the allowed level ran…

Let $U_q(X_n^{(1)})$ be a quantum affine algebra, and let $q$-characters of Kirillov–Reshetikhin modules be the associated representation-theoretic objects. A q-character T-system…

Let $X_r$ be a simple type with dual Coxeter number $h^{\vee}$, let $I$ be its index set, let $t_a$ be the symmetrizing integers, and let $\omega$ be the involution of $I$ specifie…