24 problems

Let $\mathfrak{g}$ be of type $C_n$, let $\lambda/\mu$ be a skew diagram, and let $\chi_{\lambda/\mu,a}$ be the associated Jacobi–Trudi determinant for $a\in\mathbb{C}$. Let…

Transfer-matrix q-character conjecture. The identification

Let $M=M_P$ be a standard module, let $Q/R$ be an $l$-weight of $M$, and let $M(Q/R)$ be the corresponding $l$-weight space. Define a filtration on $M(Q/R)$ by … where … Let…

Let $(i,a)\in I\times_2\mathbb Z$ and $w\in W$, and let $Q_{w\varpi_i,a}$ be the corresponding formal solution of the extended $QQ$-system. Let $\chi_{\ell}$ denote the $\ell$-char…

Let $(i,a)\in I\times_2\mathbb Z$, let $w\in W$, and let $Q_{w\varpi_i^{\vee},a}$ be the corresponding $Q$-variable. Let $\chi_{\ell}$ denote the $\ell$-character and let…

Folding conjecture. The $q$-characters of the simple modules $L(\boldsymbol{\psi})$ of $U_q(L{\mathfrak{g}})$ and $L(\rho(\boldsymbol{\psi}))$ of $U_q(L{\mathfrak{g}}^\sigma)$ sati…

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…

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 $\Psi$ be a monomial in $Y_{i,a}$ for $1\leq i\leq M$ and $a\in\mathbb{C}^{\times}$, so that $L(\Psi)$ is a polynomial module. Assume that every $\ell$-weight of $L(\Psi)$ othe…

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

Positivity conjecture. Every simple module is quantizable. The supplied status evidence says that this conjecture is proved for reachable simple modules in affine types $C_n^{(1)}$…

Let $bpsi$ be the highest $l$-weight of an irreducible $l$-highest-weight module $L(bpsi)$ over the quantum affine Borel algebra $\mathcal{U}_qmathfrak{b}$. Let…

Let w\frac{in W, let $bpsi=T_w(bpsi_{i,a}^{-1})$, and write the projected limit $bpsipi_{q,\infty}^w$ as $c\times a$, with $c$ its constant part and $a$ its non-constant part. Let…

Let $W$ be the Weyl group, let w\frac{in W, fix i\frac{in I and a\frac{inmathbb{C}^*, and let $m$ be a monomial in the variables $Y_{j,b}$. Let $V_\infty^w(m)$ be the correspo…

Fix i\frac{in I}, a\frac{inmathbb{C}^*, a Weyl group element w\frac{in W, and a monomial $m$ independent of $k$. Let $M_k$ be the Kirillov–Reshetikhin monomial specified in t…

Let $I$ be the set of simple-root indices, let $W$ be the Weyl group, and let $a\in\mathbb{C}^*$. For $i\in I$ and $w\in W$, let $Q_{w(\varpi_i),a}$ be the corresponding $Q$-variab…

Let $\xi$ be a height function and let $\ell\geq 2$. Let $\mathscr{C}^{\leq \xi}_\ell$ be the corresponding module subcategory, and let $L(m)\in\mathscr{C}^{\leq \xi}_\ell$ be a re…

Let $\mathcal{U}_q(\mathcal{L}\mathfrak{g})$ and $\mathcal{U}_q(\mathcal{L}\mathfrak{g}^{\sigma})$ be the non-twisted and twisted quantum affine algebras, respectively. Let…

Let $\mathcal{Y}$ be the monomial ring, let $m$ be a dominant monomial in $\mathcal{Y}$, let $L_t(\underline{m})$ denote the $(q,t)$-character of the simple module $L(m)$, and let…

Frenkel–Reshetikhin conjecture. For every eigenvalue $\lambda_j$ of $\mathcal{T}_V(z)$ on $W$, the preceding substitution in $\chi_q(V)$ yields $\lambda_j$; here each $Q_{i,j}(z)$…

Truncated $q$-character relation. In $\operatorname{Frac}(K_0(\mathcal{O}^+))$,

Let $m$ be a dominant monomial in the variables $Y_{i,r}\in\mathbf{Y}^-$. Write … with $z_{i,s}=1$ for $s>0$, and define … The integer vector $g(m)\in\mathbb{Z}^{(W^-)}$ is the $g$…

Let $\mathcal{A}$ be the cluster algebra identified with the ring of truncated $q$-characters of the category $\mathcal{C}^-$. A simple $U_q(\widehat{\mathfrak{g}})$-module $S$ is…

Frenkel–Mukhin conjecture. For any dominant monomial $m_+$, the algorithm never fails and stops after finitely many steps. Moreover,