116 problems

Let $\mathscr{C}_{\ell}$ be the Hernandez–Leclerc monoidal subcategory and let $K_{0}(\mathscr{C}_{\ell})$ be its Grothendieck ring, identified with a cluster algebra. A simple mod…

Kirillov–Reshetikhin crystal-base conjecture. For some $\zeta^{(i)}_s\in\mathbb{C}(q)^\times$, the module $W^{(i)}_s(\zeta^{(i)}_s)$ has a crystal base $\mathcal{B}^{i,s}$ that is…

Let $B^{2,s}$ be a crystal of type $D_n^{(1)}$, and let $B(k\varpi_2)$ denote the classical crystal of highest weight $k\varpi_2$. Let $D_{B^{2,s}}$ be an energy function on…

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…

Hernandez–Leclerc's conjecture. The cluster monomials of the cluster algebra are in bijection with the isomorphism classes of real simple objects in $\mathscr{C}^-$.

Let $\mathcal{A}_{\Gamma_\infty'}$ be the cluster algebra associated with the quiver $\Gamma_\infty'$, and let $\mathcal{O}^{sh}$ be the shifted category appearing in the isomorphi…

Let $\mathcal{C}_{\epsilon^\mathbb{Z}}$ be the monoidal category associated to $\mathfrak{g}=\mathfrak{sl}_k$ at a parameter $\epsilon\in\mathbb{C}^*$ such that $\epsilon^2$ is an…

Let $\Lambda$ be the coweight lattice, let $\Lambda^+\subset\Lambda$ be the dominant coweights, and let $\Psi_D$ be the homomorphisms defined in for $\Lambda$-valued divisors $D$ o…

Let $I_0$ index the fundamental representations, let $\bar k=\bigcup_{n>0}\mathbb C((q^{1/n}))$, and let $\mathfrak m=\bigcup_{n>0}q^{1/n}\mathbb C[[q^{1/n}]]$. For every…

Let $I_0$ index the fundamental representations, let $R^{{\operatorname{nor}}}_{ij}(x,y)$ be the normalized $R$-matrix for $i,j\in I_0$, and let $\gamma$ be the parameter defined i…

Let $I_0$ index the fundamental representations, let $i_1,\ldots,i_l\in I_0$, and let $a_1,\ldots,a_l\in\bar k^\times$, where $\bar k=\bigcup_{n>0}\mathbb C((q^{1/n}))$. Write…

Let $H_{+}$ be the subalgebra generated by $1,B_1,B_2,\dots$, and let $U_0$ denote the operator acting on the level-1 module under consideration. At the specialization $p=q^{2n}$,…

Let $U_q(\widehat{\mathfrak{sl}_2})$ denote the quantum affine algebra. Let the twisted algebra be generated by $a_{1m}$ for $m\in 2\mathbb Z+1$, by $\gamma$, and by $x_{1n}^{\pm}$…

Level-restricted X=M conjecture. One has

Inhomogeneous X=M conjecture. There exists $b_0\in B_0$ with $\varphi(b_0)=k\Lambda_0$ such that

X=M conjecture. One has

Conjectural Kirillov–Reshetikhin module family. For every $(r,s)$, there exists an irreducible finite-dimensional $U'_q(X_n^{(1)})$-module $W^{(r)}_s$ such that:

Universal formula conjecture. The formula

Let ${\mathcal A}_q$ be the unital associative algebra generated by $x,y$ subject to the two cubic $q$-Serre relations … … For each $i\in\mathbb Z_4$, let the homomorphism from…

Let … where $\mathcal{F}_{M;m,\ell,n}$ are the Fock-space components in the decomposition of the $U_{x,p}(\widehat{\mathfrak{sl}}_2)$-module, and let $a$ satisfy…

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 $\overline{K}^{\diamondsuit}_{L,\lambda}(t)$ and $\overline{M}^{\diamondsuit}_{L,\lambda}(t)$ be the $K$- and fermionic formulas associated with an allowed diagram shape…

Let $\diamondsuit$ be one of the four allowed diagram shapes and let $\{\mathfrak{g}_n\}$ be a nonexceptional family of affine algebras associated with $\diamondsuit$. Let…

Let $B_n$ be the compatible bases of the level-one representation obtained by applying the canonical basis of the completed quantum loop-algebra positive part to the vectors $v_n$,…

Let $L_{\Lambda_0}$ be the level-one representation, let $W_0$ be its principal subspace, let $B_0$ be the canonical basis induced on $W_0$, and let $B^+$ be the Leclerc–Thibon can…