\title{New incompressible symmetric tensor categories in positive characteristic}



\author{Dave Benson} 

\address{Institute of Mathematics,
                              University of Aberdeen,
                              Fraser Noble Building,
                              King's College,
                              Aberdeen AB24 3UE,
                              Scotland, UK.}


\author{Pavel Etingof}

\address{Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA}

\author{Victor Ostrik}

\address{Department of Mathematics,
University of Oregon, Eugene, OR 97403, USA}

\address{Laboratory of Algebraic Geometry,
National Research University Higher School of Economics, Moscow, Russia}

\begin{abstract} We propose a method of constructing abelian envelopes 
of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\k$. 
If ${\rm char}(\k)=p>0$, we use this method to construct generalizations 
$\Ver_{p^n}$, $\Ver_{p^n}^+$ of the incompressible abelian symmetric
tensor categories defined in \cite{BE} for $p=2$ and in \cite{GK,GM} for $n=1$. 
Namely, $\Ver_{p^n}$ is the abelian envelope of the quotient of the category of 
tilting modules for $SL_2(\k)$ by the $n$-th Steinberg module, and $\Ver_{p^n}^+$ 
is its subcategory generated by $PGL_2(\k)$-modules. We show that $\Ver_{p^n}$ are reductions 
to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. 
We study the structure of these categories in detail, and in particular show that 
they categorify the real cyclotomic rings $\ZZ[2\cos(2\pi/p^n)]$, and that $\Ver_{p^n}$ embeds 
into $\Ver_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth
over $\k$ admits a fiber functor to the union $\Ver_{p^\infty}$ of
the nested sequence $\Ver_{p}\subset \Ver_{p^2}\subset\cdots$.
This would provide an analog of Deligne's theorem in characteristic zero and a generalization of 
the result of \cite{O}, which shows that this conjecture holds for fusion categories, and 
moreover the fiber functor lands in $\Ver_p$.
\end{abstract}