26 problems

Conjecture on the $G(2)$-structure. The tensors $V(\mathfrak{g}_i)$ might be related to the $G(2)$-structure on a purely odd superspace.

Let $\mathfrak g$ be one of the five exceptional simple Lie algebras, let $\mathcal O_{\mathrm{min}}$ be its minimal nilpotent orbit, and let $\Lambda^x$ denote the operator associ…

Let $F_q^{\mathrm{E}_6}(L)$ denote the finitized fermionic polynomial. For $L\in\mathbb{Z}_{\geq0}$, define … … where the $q$-multinomial is understood to vanish unless its lower e…

Deligne's conjecture. There should exist a more general class of objects, depending on a parameter $t$, which specialises to the representation categories of the members of the exc…

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{E_7(a_5)}$ be the nilpotent orbit of Bala–Carter type $E_7(a_5)$ in $E_7$. Associated-variety…

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{F_4(a_3)}$ be the nilpotent orbit of Bala–Carter type $F_4(a_3)$ in $F_4$. Level $-6$ $F_4$ co…

Let $X_V$ denote the associated variety of a vertex algebra $V$, and let $\mathbb{O}_{E_6(a_1)}$ be the nilpotent orbit of Bala–Carter type $E_6(a_1)$ in $E_6$. Associated-variety…

Exceptional commutant conjecture. One has $B\cong A$ in all three cases $\mathfrak{g}=e_6,e_7,e_8$. The assertion is a case-by-case conjecture about the explicit branching rules of…

Let $\vartheta_{E_{7+1/2}}(\tau,\mathfrak z)$ be the weak Jacobi form of weight $4$ and index $24$ for $E_7$ defined from the formal denominator of the conjectural generalized affi…

Let $\chi_{10/13}$ and $\chi_{18/13}$ be the indicated affine characters of $(E_{7+1/2})_2$, and consider the $E_{7+1/2}$ representations appearing in their Fourier coefficients, w…

Let $w_1,\ldots,w_7$ be the fundamental weights, let $\lambda=\sum_{i=1}^7 n_iw_i$, let $\rho$ be the Weyl vector, and let $\Delta_+$ be the set of positive roots. Let…

Let $(E_{7+1/2})_k$ denote the affine vertex operator algebra associated with the intermediate Lie algebra $E_{7+1/2}$ at level $k$. Rationality conjecture. The affine VOA…

Let $N$ be a positive integer, and let $Obs_N(0)$ denote the local operators of the theory on a stack of $N$ fivebranes wrapping $C^3$ in $R\times C^5$. Let $\mathcal G_N$ be the l…

Let $E_6$, $E_7$, $E_8$, and $F_4$ denote the corresponding simple exceptional Lie algebras, and let the labels such as $2A_2$, $A_2+3A_1$, $E_6$, and $\widetilde A_2$ denote nilpo…

Let $p=2$, and let $\mathfrak{g}$ be of type $E_7$ with maximal subalgebra $M_7$ and $M_7$-invariant subspace $L'_{-1}$ as in Theorem none7p2. For the Weisfeiler filtration…

Let $p=2$, and let $\mathfrak{g}$ be of type $E_8$ with maximal subalgebra $M_8$ and $M_8$-invariant subspace $L'_{-1}$ as in Theorem none8p2. For the Weisfeiler filtration…

Let $G$ be an algebraic group of type $E_8$ with Lie algebra $\mathfrak{g}=\operatorname{Lie}(G)$ over a field of characteristic $5$. Let $M$ be a maximal subalgebra of Cartan or M…

Let $G$ be an algebraic group of type $E_8$ with Lie algebra $\mathfrak{g}=\operatorname{Lie}(G)$ over an algebraically closed field of characteristic $5$. Let $M$ be a maximal sub…

Let $G$ be one of the exceptional Lie groups, let $(V,\pi)$ be respectively the $7$-, $26$-, $27$-, $56$-, or $248$-dimensional representation for $G_2$, $F_4$, $E_6$, $E_7$, or…

Conjecture. These gradings exhaust the list of fine gradings, up to equivalence, on $\mathfrak{e}_6$, $\mathfrak{e}_7$, and $\mathfrak{e}_8$, with the exception of the root space d…

Exhaustiveness conjecture. The list exhausts the fine gradings, up to equivalence, on $\mathfrak{e}_6$, and it also exhausts them for $\mathfrak{e}_7$ and $\mathfrak{e}_8$.

Killing's conjecture. There exists a realization of $\mathfrak{g}_2$ as a Lie algebra of vector fields on a $5$-dimensional manifold.

Let $3^3{:}\mathrm{L}_3(3)$ act through the appropriate $52$-dimensional representation. Uniqueness conjecture. There is a unique algebra invariant under this representation. This…

Let $\mathfrak g$ be a simple complex Lie algebra of exceptional type, let $e\in\mathfrak g$ be nilpotent, and let $U(\mathfrak g,e)$ be the associated finite $W$-algebra. Premet's…

E8 algebraisation and flag conjecture. In the $E_8$ case, taking the FTI as coordinates should yield an algebraic form for the Hamiltonian, and the infinite flag of polynomial spac…