  This paper computes the generic fusion rules of the Grothendieck ring of
Rep(PSL(2,q)), q prime-power, by applying the Schur orthogonality relations on
the generic character table. It then proves that this family of fusion rings
interpolates to all integers q>1, providing (when q is not prime-power) the
first example of infinite family of non group-like simple integral fusion
rings. Furthermore, they pass all the known criteria of (unitary)
categorification. This provides infinitely many serious candidates for solving
the famous open problem of whether there exists an integral fusion category
which is not weakly group-theoretical. A braiding criterion is finally
discussed.
