On cohomology in symmetric tensor categories in prime characteristic

We describe graded commutative Gorenstein algebras ${mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)cong{mathcal E}_{n}(p)$, where $mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently constructed in cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,Coulembier}. We investigate the combinatorics of these algebras, and the relationship with Mincs partition function, as well as possible actions of the Steenrod operations on them. Evidence for the conjecture includes a large number of computations for small values of $n$. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in ${mathcal E}_n(p)$ with a homogeneous system of parameters in $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)$. These parameters have degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1le i le n$. This at least shows that $mathrm{Ext}^bullet_{mathsf{Ver}_{p^{n+1}}}(1,1)$ is a finitely generated graded commutative algebra with the same Krull dimension as ${mathcal E}_n(p)$. For $p=2$ we also show that $mathrm{Ext}^bullet_{mathsf{Ver}_{2^{n+1}}}(1,1)$ has the expected rank $2^{n(n-1)/2}$ as a module over the subalgebra of parameters.