New incompressible symmetric tensor categories in positive characteristic

We propose a method of constructing abelian envelopes symmetric rigid monoidal Karoubian categories over an algebraically closed field k. If char(k)=p>0, then we use this to construct the incompressible tensor Verpn, Verpn+ generalizing earlier constructions by Gelfand–Kazhdan and Georgiev–Mathieu for n=1, Benson–Etingof p=2. Namely, Verpn is envelope quotient category tilting modules SL2(k) nth Steinberg module, its subcategory generated PGL2(k)-modules. show that are reductions characteristic p Verlinde braided in 0, which explains notation. study structure these detail and, particular, they categorify real cyclotomic rings Z[2cos(2π∕pn)], embeds into Verpn+1. conjecture every moderate growth k admits fiber functor union Verp∞ nested sequence Verp⊂Verp2⊂⋯ . This would provide analogue Deligne’s theorem 0 generalization results Coulembier, Etingof, Ostrik, shows holds Frobenius exact (in semisimple) categories, moreover, lands Verp case fusion was shown Ostrik). Finally, classify object with invertible exterior square; class contains Verpn.