Semisimple Super Tannakian Categories with a small Tensor Generator

We consider semisimple super Tannakian categories generated by an object whose symmetric or alternating tensor square is simple up to trivial summands. Using representation theory, we provide a criterion to identify the corresponding Tannaka super groups that applies in many situations. As an example we discuss the tensor category generated by the convolution powers of an algebraic curve inside its Jacobian variety.