Title. New symmetric tensor categories in positive characteristic
Abstract. We construct and study a nested sequence of finite symmetric tensor
categories \({\rm Vec}=\mathcal C_0\subset \mathcal C_1\subset\cdots\subset \mathcal C_n\subset\cdots\)
over a field of characteristic 2 such that \(\mathcal C_{2n}\) are
incompressible, i.e., do not admit tensor functors into
tensor categories of smaller Frobenius--Perron dimension.
This generalizes the category \(\mathcal C_1\) described by S. Venkatesh and the category \(\mathcal C_2\) defined by V. Ostrik.
The Grothendieck rings of the categories \(\mathcal C_{2n}\) and \(\mathcal C_{2n+1}\) are both isomorphic to the ring of real cyclotomic integers defined by a primitive \(2^{n+2}\)-th root of unity, \({\mathcal O}_n=\Bbb Z[2\cos(\pi/2^{n+1})]\). We expect that the category \(\mathcal C_{2n}\) is a reduction to characteristic 2 of the Verlinde category at the \(2^n\)-th root of unity, and that there exists similar non-semisimple reduction of the Verlinde category at the \(p^n\)-th root of unity to characteristic p when . This is joint work with Dave Benson.