232 problems

Kac–Wakimoto's conjecture. These are all irreducible highest weight $\mathfrak{g}$-modules $L(\Lambda)$ with this property.

Frenkel–Kac–Wakimoto conjecture. Drinfeld–Sokolov reduction sends every admissible $\widehat{\mathfrak{g}}_\kappa$-module to either zero or a cohomological shift of a minimal serie…

Let $\mathfrak{g}$ be an affine Lie algebra with Dynkin index set $I$, and let $i\in I\setminus\{0\}$. A positive geometric crystal for $\mathfrak{g}$ is a quadruple … where $X$ is…

Let $B^{r,s}$ be a Kirillov–Reshetikhin crystal associated with a Kirillov–Reshetikhin module $W^{r,s}$, where $r$ is a node of the affine Dynkin diagram other than $0$ and…

Let $(A_1,D_3)$ be the Argyres-Douglas theory, and let $\mathfrak{su}(2)_{-4/3}$ denote the affine Lie algebra at level $-4/3$. Chiral-algebra conjecture. The chiral algebra corres…

CMPP's conjecture. The generating function for $\Lambda$-admissible colored partitions is

Let $ell$ be an admissible level for the simple Lie algebra $\bf g$. Vertex tensor and ribbon category conjecture. 1. $L_{ell}(\bf g)\operatorname{-wtmod}$ is a vertex tensor categ…

Let $P$ be the weight lattice, $P^+$ the dominant integral weights, $P_{\ell}\subset P^+$ the weights of level $ell$, and let $W_{ell}$ be the affine Weyl group acting on the weigh…

Let $\ell$ be a level, let $\Gamma_q$ be the equivariance group, and let $L_{\widehat{\mathcal{A}}}(\ell,0)$ be the corresponding quasi vertex operator algebra. Consider an…

Let $\operatorname{Sb}[\mathbb{C}^{16}]$ denote the vertex operator algebra of sixteen symplectic bosons, and let $\widehat{SU}(2)_{-4}$ act through its affine current algebra. Sym…

Let $\mathfrak g=\mathrm D_{n+1}^{(2)}$, let $m$ be a nonnegative integer, and set $\Lambda=2m\Lambda_0$. Define … Let $V(\Lambda)$ be the integrable highest-weight representation…

Let $B$ be a perfect crystal, let $\{w^{(k)}\}$ be a sequence of Weyl group elements, and let $\lambda=l\Lambda_0$ be the level-$l$ dominant integral weight for which the sequence…

Let $\mathfrak{g}=C^{(1)}_n$, let $B$ be the perfect crystal considered for this case, and set $\lambda=l\Lambda_i$ with $i\neq 0,n$. For the associated sequence $\{w^{(k)}\}$ and…

Let ${\cal L}(\Lambda_n)$ be the level $1$ integrable module of $A^{(2)}_{2n}$, and let $Y(B_n)$ be the Yangian of type $B_n$. Suppose … where $BS$ is the set of border strips with…

Let $\kappa(\vec{h})$ be the border strip associated with the spectral datum $\vec{h}$, let $s^{\rm L}_{\kappa(\vec{h})}$ be its L-admissible tableau character, and let $Y(B_n)$ be…

Let ${\cal L}(\Lambda_k)$ be the level $1$ integrable module of $\widehat{sl}_n$, and let the character decomposition in equation (26a) be the decomposition obtained from the irred…

Let $U_q(\widehat{sl}_n)$ be the quantized affine algebra, and let the level $1$ integrable modules of the affine Lie algebra $\widehat{sl}_n$ carry their Yangian $Y(sl_n)$-module…

Let $k$ be a rational level, and let the formulas referred to as … provide all solutions to the system $(,)$ and conjectures that, when $k$ is rational, they actually give horizont…

Let $n$ be a positive integer, let $\widehat{\mathfrak{sl}}_n$ be the affine Lie algebra of type $A^{(1)}_{n-1}$, and let $K_m$ be the complexified Grothendieck group of finitely g…

Let $n,l,r$ be as in the source, let ${\cal H}(l\Lambda_r,\mu)$ be the set of paths defined there, let $E(p)$ be their energy, and let $\operatorname{af}(\operatorname{wt}p)$ denot…

Let $(P^+)_l$ be the level-$l$ dominant weights, let $\overline{P}^+$ be the dominant weights of the classical subalgebra, and let $Q$ be the root lattice. For a partition…

Let $n$ and $l$ be positive integers, and let $X^{(l)}{ u}(rho)$, $X^{(l)'}{ u}(rho)$, $F^{(l)}{ u}(q)$, and $F^{(l)'}{ u}(q)$ denote the restricted one-dimensional sums and fermio…

Let $\mathfrak{g}$ be a simple complex Lie algebra, let $\hat{\mathfrak{g}}$ be its affine Lie algebra, and let $\bar{\hat{\mathfrak{g}}}$ denote the affine Lie algebra at the dual…

Let $\mathfrak{g}$ be a simple complex Lie algebra, let $k$ be an integral central charge different from the critical level, and let $\bar{k}=-2h^{\vee}-k$. Let $U\mathcal{A}$ be t…

Let $B^{r,s}$ be a crystal of type $D_n^{(1)}$. Let $B(\Lambda)$ denote the classical $D_n$-crystal of highest weight $\Lambda$, and let the complement of a partition $\Lambda$ in…