12 problems

Let ${\mathfrak{g}}$ be of finite type and let ${\boldsymbol{\psi}}$ be a reachable highest $\ell$-weight. Write ${\boldsymbol{\psi}}$ in terms of vectors ${\mathbf{l}}=(l_{i,s})$…

Toledano Laredo's monodromy conjecture. The monodromy of the trigonometric Casimir connection $\widehat{\nabla}_C$ should be described by the action of the affine braid group $B_G$…

Let $\mathfrak g$ be a general simple Lie algebra, let $\mathcal M_+$ be the set of dominant monomials, and let $\chi_{q,t}(L(m))$ and $\chi_q(L(m))$ denote the $(q,t)$-character a…

Let $V_{2j_1}(u)\tensor V_{2j_2}(u)$ be the tensor-product vector space, and let $(\mathcal{A}_2,\mathcal{A}_{12};\mathcal{B}_{12},\mathcal{B}_1)$ be the indicated ordered 4-tuple…

Let $\mathfrak g$ be simply laced, let $\sigma$ be the folding data, and let $\phi^\sigma_{\mathfrak g}$ and $\Phi^\sigma_{\mathfrak g}$ be the monomial and Grothendieck-ring foldi…

Consider the folding homomorphism from the Grothendieck ring of finite-dimensional representations of an untwisted quantum loop algebra to that of the corresponding twisted quantum…

Let $Q$ be an orientation of a type ADE Dynkin diagram, and let a total convex order on $\Delta^+$ refine the Auslander–Reiten partial order. Let $f_\alpha(x)$ be the fused current…

Let $\Delta^+$ be the positive roots of a finite-type ADE Lie algebra with a fixed total convex order, and let…

Let $U_q(L\mathfrak{g})$ be the quantum loop group of a finite-type ADE Lie algebra, with negative half $U_q(L\mathfrak{n}^-)$ and a completion $\widehat{U}_q(L\mathfrak{n}^-)$, an…

Let $V_1,V_2$ be finite-dimensional representations of the Yangian, and let $J^{\uparrow/\downarrow}_{V_1,V_2}(s)$ be the meromorphic tensor structure relating the standard tensor…

Holomorphicity conjecture. $J_{V_1,V_2}(s)$ is holomorphic in $s$ and can therefore be evaluated at $s=0$, yielding a tensor structure on $\mathsf{\Gamma}_{}$ with respect to the s…

Let $Y_\hbar(\mathfrak{g})$ be the Yangian of the Lie algebra $\mathfrak{g}$ and $U_q(L\mathfrak{g})$ its quantum loop algebra. The trigonometric Casimir connection yields represen…