Higher Verlinde Category

A higher Verlinde category is the finite symmetric tensor category in characteristic obtained as the abelian envelope of the quotient of the tilting-module category of by the tensor ideal generated by the Steinberg module of highest weight . Here is algebraically closed and . At level one it is the semisimplification of the representation category of the cyclic group of order , obtained by removing morphisms whose trace pairings all vanish. Its simple objects are represented by the Jordan blocks of sizes 1 through . Thus is equivalent to finite-dimensional vector spaces, while is equivalent to finite-dimensional super vector spaces. Higher levels are not semisimple. They are proposed as targets for fiber functors from finite symmetric tensor categories (Benson et al. 2023).