19 problems

Let $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$ be a highest-weight label in the decomposition denoted by $(\lambda(1),\emptyset,\lambda(3),\lambda(4))$, where … and … Here…

Let $\widetilde{\mathfrak{g}}$ be a twisted loop algebra with $\epsilon(\mathfrak{h})=0$. Let $\widehat{\rho}$ be the Weyl vector determined by the simple roots of…

Let $k$ and $\ell$ be nonnegative integers, let $\lambda$ be a partition, and let $m_\lambda(k,\ell)$ denote the multiplicity associated with the hook Schur decomposition for the s…

Homogeneous realization conjecture. Every $\mathbb{Z}_{2}$-grading of $E$ is $\mathbb{Z}_{2}$-graded isomorphic to some homogeneous $\mathbb{Z}_{2}$-grading of $E$.

Finite-dimensional representability conjecture. There exists a finite-dimensional $W$-(super)algebra $B$ such that

Let $\widetilde{\mathcal{O}}^{\mathrm{SL}(r|s)}$ denote the relevant superalgebra, and let $Y_{\underline{i}|\underline{\hat{\mu}}}$,…

Let ${\rm R}$ be an integral domain in which $2$ is invertible. Let $\mathfrak{H}_{\Delta}(n)^{g}$ be the cyclotomic Sergeev algebra, let…

Let $\mathbb{K}$ be the ground field, let $q$ be the parameter, let $\underline{Q}=(Q_1,Q_2,\ldots,Q_m)\in(\mathbb{K}^*)^m$, and let $f=f^{(\bullet)}_{\underline{Q}}$ with…

Cocharacter support conjecture. The multiplicity $m_{\langle\lambda\rangle}$ is nonzero if and only if one of the following holds: (i) $\lambda(3)=\lambda(4)=\emptyset$ and…

Cocharacter multiplicity conjecture. The multiplicity $m_{\langle\lambda\rangle}$ is nonzero if and only if either $w_1\leq 2$, or $w_1\geq 3$ and…

Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair, let $\mathfrak{a}$ be a chosen Cartan subspace, and let $\overline{\Delta}$ be its reduced root system with restricted W…

Let $n\geq 4$ and $r\geq 0$, set $\mathfrak{g}=\mathfrak{sl}(n+r+2|r)$, and let $\Pi$ be a rank-one Heisenberg vertex operator algebra. Define … and … Let $h^\vee$ be the dual Coxe…

Equivalence conjecture. The functor $\Phi:\cal P^{n-m}\to W^\chi\text{-mod}$ defines an equivalence of categories.

Let $R_{{\mathfrak{b}}}(\nu)$ be the odd KLR algebra associated with a dimension vector $\nu$, and let $Z(R_{{\mathfrak{b}}}(\nu))$ denote its supercenter. Let…

Center conjecture. The image of the canonical map

Graded symmetry conjecture. The algebra $B_\xi$ is a graded symmetric algebra.

Finite-dimensional realization conjecture. There exists a finite-dimensional associative superalgebra $C=C_{\bar{0}} \oplus C_{\bar{1}}$ over $F$ with superinvolution that satisfie…

Null-vector characterization conjecture. Each ideal of $H_{1,\eta}(A_{n-1})$ is the set of null-vectors of the degenerate bilinear form $B_{sp}$ for some $\varkappa$-trace $sp$ on…

Simple multiplicity conjecture. The fusion rules of $\mathfrak W_{n,\ell}$ are