52 problems

Minimal-odd-root Dynkin diagram conjecture. There always exists exactly one Dynkin diagram with the minimal number of odd roots satisfying the definition of a hyperbolic Kac–Moody…

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 the superalgebra be defined by the relations in Eqs. (5.2.5), (5.2.6), and (5.2.7), with $\tilde{f}^I_J$ and $\gamma^I_J$ treated as independent generators, and let…

Let $n\in\mathbb{N}$, let $\mathbb{K}$ be an algebraically closed field, and let $\mathcal{H}^{f}_{\mathbb{K}}$ be the cyclotomic Hecke-Clifford superalgebra associated with…

Let $P=P_0\oplus P_1$ be a Poisson superalgebra, and let $\mathrm{Kan}(P)=P\oplus\bar P$ be its Kantor double, a Jordan superalgebra. The functor $\mathrm{Kan}$ sends Poisson $P$-s…

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 a shifted affine super Yangian be a modification of the affine super Yangian adapted to the non-rectangular setting, and let a type $A$ $W$-superalgebra be the corresponding…

Let $A=U\big(\widehat{\mathfrak{gl}}(m_1+m_2|n_1+n_2)\big)$ and $B=U\big(\widehat{\mathfrak{gl}}(m_1|n_1)\big)$. Define the centralizer … Let … be the homomorphism from…

Character decomposition conjecture.

Let $\mathscr O^{(p|q)}_N(\epsilon_1,\epsilon_2)$ be the algebra of observables of the $U(N)$ Chern–Simons matrix model with internal symmetry $U(p|q)$, and let $\mathcal H_\infty$…

Let $U_q(\widehat{\mathfrak g})$ be an untwisted quantum affine algebra of type $C_n^{(1)}$, and consider its $q$-oscillator representations together with finite-dimensional repres…

Let $(\mathfrak{g},\mathfrak{k})$ be a special supersymmetric pair, let $G/K$ be the corresponding supersymmetric space, and let…

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 $R^\Lambda_\theta$ be a cyclotomic quiver Hecke superalgebra, where $\theta$ is a positive-root combination and $\Lambda$ is a dominant weight. Symmetry conjecture. Every cyclo…

Let $\rho$ be a $\bar p$-core and $d\in\mathbb{Z}_{\geq0}$, and let $\mathcal{R}^{\Lambda_0}_{\operatorname{cont}(\rho)+d\delta}$ be the associated cyclotomic quiver Hecke superalg…

Let $\rho$ be a $\bar p$-core and $d\in\mathbb{Z}_{\geq0}$. Let $\mathcal{R}^{\Lambda_0}_{\operatorname{cont}(\rho)+d\delta}$ be the associated cyclotomic quiver Hecke superalgebra…

Let $\rho$ and $\rho'$ be $\bar p$-cores, let $d,d'\in\mathbb{Z}_{\geq0}$, let $\operatorname{cont}(\rho)$ denote the residue content, and let $\delta$ be the null-root parameter u…

Let $(A,L)$ be a Lie-Rinehart superalgebra, where $A$ is a supercommutative associative superalgebra and $L$ is a Lie superalgebra. A compatible action and anchor pair is called tr…