14 problems

Higher Verlinde fibre conjecture. All symmetric tensor categories of moderate growth fibre over $\mathsf{Ver}_{p^\infty}$. If so, the study of such categories would reduce, via the…

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $\operatorname{Ver}_{p^n}$ denote the higher Verlinde categories, including…

Asymptotic classification conjecture. All finitely-generated semisimple symmetric tensor categories of moderate growth can be constructed from the categories $\mathsf{Ver}_p(G)$, f…

Let $p$ be a prime, and let $\operatorname{Ver}_{p^\infty}$ denote the corresponding infinite Verlinde tensor category. Let (GR) and (MN1-2) denote the properties defined in the so…

Let $p$ be the characteristic, let $n\geq 2$, and let $L_i$ be a simple object of the Verlinde category $\mathsf{Ver}_{p^n}$ that is not in the image of the embedding … Write…

Verlinde-fiber conjecture. Every STC fibers over $\operatorname{Ver}_{p^\infty}$.

Steinberg tensor product conjecture. Any irreducible representation of $G$ is uniquely a tensor product of a one-irreducible representation and one inflated from $G_0$.

Classification conjecture. Every incompressible category of moderate growth over $\mathbf{k}$ is a tensor subcategory of $\mathsf{Ver}_{p^infty}$.

Higher Verlinde target conjecture. Every tensor category of moderate growth admits a tensor functor to $\mathsf{Ver}_{p^infty}$.

Let $p>0$, let ${\rm Ver}_p^+$ be the relevant Verlinde category, and let $\mathfrak{g}$ be a simple Lie algebra in ${\rm Ver}_p^+$ containing a simple Lie algebra $\mathfrak{s}$ s…

Let $p\geq5$, let ${\rm Ver}_p$ be the Verlinde category, and let $G$ be a simple linearly reductive height-$1$ group scheme in ${\rm Ver}_p$, with Lie algebra…

Let $p\geq5$, let ${\rm Ver}_p$ be the Verlinde category, and let $G$ be a linearly reductive height-$1$ subgroup scheme of $GL(Y)$ for an object $Y\in {\rm Ver}_p$. Let…

Let $\mathcal{C}$ be an abelian symmetric tensor category of moderate growth over a field $\mathbf{k}$ of characteristic $p$. The Frobenius functor is the functor whose exactness d…

Assume that $p>0$. A pre-Tannakian category is a $\boldsymbol{k}$-linear, rigid, symmetric monoidal abelian category with finite-dimensional Hom spaces and finite length objects. I…