46 problems

Frenkel–Reshetikhin conjecture. The eigenvalues of the transfer matrices can be obtained by computing the q-character of an associated representation and performing a suitable subs…

Nakai–Nakanishi conjecture. The Jacobi–Trudi type determinant gives the $q$-character of an irreducible representation of the corresponding quantum affine algebra.

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

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…

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 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 $\mathfrak{g}$ 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…

Let $\mathcal{C}_q$ be the category of finite-dimensional representations of a quantum loop algebra $\mathcal{U}_q(\widehat{\mathfrak{g}})$, with $q$ not a root of unity. For an ob…

Let $C$ be a Cartan matrix such that, for $i\neq j$, $C_{i,j}C_{j,i}\leq 3$. The classical algorithm and deformed algorithm are the algorithms associated with $C$; the former stops…

Let $C$ be a finite Cartan matrix. The deformed algorithm is the algorithm whose output is denoted by $\hat{F}_t$; it stops when, equivalently,…

Let $\mathfrak g$ be a Lie algebra, let $\pi_i$ be irreducible finite-dimensional $\mathfrak g$-modules, and let $Z=(z_1,\ldots,z_n)\in\Sigma\subset\mathbb C^n$ have pairwise disti…

Transfer-matrix q-character conjecture. The identification

The hook-shaped q-character conjecture. For every $k\geq N+1$ and $0\leq i\leq N-1$, $\sigma_iH^{(i)}_k(u+\frac{i}{2})$ is precisely the irreducible $q$-character having the displa…

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 $(i,a)\in I\times_2\mathbb Z$, 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…

The paper considers a refinement of the $q$-character by the $q,t$-character, whose dominant part lies in $\mathbb{Z}[\mathscr{D}_N,q,q^{-1}]$, together with the transition coeffic…

Let $\phi_n$ be the homomorphism folding $q$-characters of integrable $\mathcal{U}_q({\widehat{\mathfrak{sl}_{\infty}}})$-modules to virtual $q$-characters of integrable…

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 $\Psi$ 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…

For each $i\in I$ and $a\in\mathbb{C}^*$, let $\chi_{i,a}\in\mathbb{Z}[Z_{j,aq^r}^{\pm1}]_{j\in I,r\in\mathbb{Z}}$ be the mixed-specialized interpolating $(q,t)$-character construc…

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

Fix an integer $R$, let ${\boldsymbol{\psi}}$ be the regular polynomial $\ell$-weight specified in the source, and let ${\boldsymbol{\tau}}$ be a polynomial $\ell$-weight satisfyin…

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…