New incompressible symmetric tensor categories in positive characteristic

Abstract

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If $\mathrm{char}(\mathbf{k})=p>0$, then we use this method to construct the incompressible abelian symmetric tensor categories ${\mathrm{Ver}}_{p^n}$, ${\mathrm{Ver}}_{p^n}^{+}$ generalizing earlier constructions by Gelfand–Kazhdan and Georgiev–Mathieu for $n=1$, and by Benson–Etingof for $p=2$. Namely, ${\mathrm{Ver}}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for ${\mathit{SL}}_2(\mathbf{k})$ by the nth Steinberg module, and ${\mathrm{Ver}}_{p^n}^{+}$ is its subcategory generated by ${\mathit{PGL}}_2(\mathbf{k})$-modules. We show that ${\mathrm{Ver}}_{p^n}$ are reductions to characteristic p of Verlinde braided tensor categories in characteristic 0, which explains the notation. We study the structure of these categories in detail and, in particular, show that they categorify the real cyclotomic rings $\mathbb{Z}[2cos(2\mathrm{\pi }∕p^n)]$, and that ${\mathrm{Ver}}_{p^n}$ embeds into ${\mathrm{Ver}}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over k admits a fiber functor to the union ${\mathrm{Ver}}_{p^{\mathrm{\infty }}}$ of the nested sequence ${\mathrm{Ver}}_p\subset {\mathrm{Ver}}_{p^2}\subset \cdots$ . This would provide an analogue of Deligne’s theorem in characteristic 0 and a generalization of the results of Coulembier, Etingof, and Ostrik, which shows that this conjecture holds for Frobenius exact (in particular, semisimple) categories, and, moreover, the fiber functor lands in ${\mathrm{Ver}}_p$ (in the case of fusion categories, this was shown earlier by Ostrik). Finally, we classify symmetric tensor categories generated by an object with invertible exterior square; this class contains the categories ${\mathrm{Ver}}_{p^n}$.