20 problems

Let ${\mathfrak{g}}$ be a simple basic classical Lie superalgebra, and let $L(\lambda)$ be a finite-dimensional simple ${\mathfrak{g}}$-supermodule. Write…

Let $G$ be the reductive algebraic group, $G_1$ its first Frobenius kernel, $h$ the Coxeter number, and $W_p^+$ the set of minimum-length right coset representatives in…

Scheme-theoretic traditional Humphreys conjecture. For every $w\in W_{\mathrm{ext}}^\varnothing$, the coherent sheaf

Relative Humphreys conjecture.

Traditional Humphreys conjecture.

Assume that a local ring $R$ has minimal regular presentation $Q/I$, let $k$ be its residue field, and fix minimal generators $f_1,\ldots,f_n$ of $I$. Let … and let … where each…

Let ${\mathcal C}$ be a finite tensor category, and write … for its cohomology ring. Etingof–Ostrik finite generation conjecture. The cohomology ring…

Standard conjecture. The following assertions hold:

Assume $(A,\mathfrak m)$ is a complete local complete intersection of dimension $d$ and codimension $c$, with algebraically closed residue field $k=A/\mathfrak m$. Let $X$ be an al…

Let $G$ be a quasi-reductive supergroup, and let $M\in\operatorname{Rep}(G)$. Write $\mathfrak{g}_{\mathrm{neat}}$ for the neat locus in the odd part of the Lie superalgebra, and…

Suppose $k$ is algebraically closed. Let $G$ be a height-$1$ infinitesimal unipotent $k$-supergroup scheme, let $M$ be a finite-dimensional rational $G$-supermodule, and let…

Let $k$ be algebraically closed, and let $G$ be an infinitesimal unipotent $k$-supergroup scheme of height $\leq r$. Let $H(G,k)$ denote the cohomology algebra, let…

Let $G$, $G_1$, $W$, ${}^{\mathrm{f}}W^{\mathrm{f}}$, $\mathcal{C}_p$, $\iota_{\Bbbk}$, and the corresponding nilpotent orbit $\mathscr{O}^{\mathbb{C}}_{\mathbf{c}(w)}$ be as in th…

Let $G$ be a semisimple simply connected algebraic group over a field $k$ of characteristic $p>h$, let $G_1$ be its Frobenius kernel, and let $\mathbb{U}_{\zeta}(\mathfrak{g})$ be…

Let $G$ be a semisimple simply connected algebraic group over a field of characteristic $p$, with Frobenius kernel $G_1$, weight lattice $X(T)$, and nilpotent cone…

Let $G$ be a semisimple simply connected algebraic group, let $W_p^+$ be the dominant right-coset representatives, let $c_{[w]}$ be the weight cell associated with the right cell…

Let $G$ be semisimple and simply connected, with $p$ good, and let $\lambda\in X(T)$ be arbitrary. Suppose $w\in W$ is such that $w\cdot\lambda\in X(T)_+$ and that…

The atypicality conjecture. For every basic classical Lie superalgebra ${\mathfrak{g}}$ and every simple supermodule $L(\lambda)$ in $\mathcal{F}$,

Atypicality conjecture.

Support-variety wildness conjecture. If there exists a simple supermodule $S$ in $\mathcal{B}$ such that