276 problems

Let $cal C$ be a monoidal abelian length category, and let $cal A$ be a cluster algebra. A monoidal categorification of $cal A$ is an isomorphism of $bb Z$-algebras…

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…

Let $\bmod\text{-}\mathcal{H}_q^{l}$ be the subcategory of $\mathcal{H}_q$-modules on which $\gamma=q^l\operatorname{Id}$, and let…

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 an affine Lie algebra, let $k\in I\setminus\{0\}$, and let $W(l\varpi_k)$ be the Kirillov–Reshetikhin module parametrized by $k$ and a positive integer $l$. Wr…

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…

Fix a square root $\sqrt{q}$ of $q$. Let $\mathcal{O}^{\mathrm{sh}}_{\mathbb{Z}}$ be the abelian subcategory of shifted quantum affine algebra representations described above, and…

Chari–Pressley dimension conjecture.

Let $m,n$ be coprime integers. Let $\mathcal{F}_u$ be the Fock module, and let $s_\lambda^{m/n-\epsilon}$ and $s_\lambda^{m/n+\epsilon}$ be the stable-envelope bases associated wit…

Let $\triangle(n)$ be the cyclic quiver, let $\mathcal Z=\mathbb Z[\upsilon,\upsilon^{-1}]$, and let $\boldsymbol{\mathfrak D_{\!\vartriangle\!}}(n)$ be its double Ringel–Hall alge…

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 $m$ be a degree, let $n$ be a positive integer, and let $H_m$ be the type $A$ Hecke algebra specialized at an $n$th root of unity. Consider the degree $m$ part of the transitio…

Let $U_q(\widehat{sl}_{n+1})$ be the quantum affine algebra, and let $\widehat{H}_{\ell}(q^2)$ be the affine Hecke algebra. The Frobenius–Schur correspondence for…

The integral vertex-operator conjecture. For $\alpha\in\mathbb Z$, the following vertex operators exist:

Let $U_q[\widehat{gl(2|2)}]$ be the quantum affine superalgebra. For arbitrary $\beta$, define … and, for $\alpha\in\mathbb Z$, … Let $\overline V(\lambda^0_\beta)$,…

The Fock-space identification. There is an identification of highest weight $U_q[\widehat{gl(2|2)}]$-modules

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: