Radically filtered quasi-hereditary algebras and rigidity of tilting modules

Let A be a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to show that the restricted tilting modules for SL 4(K) are rigid, where K is an algebraically closed field of characteristic p ≥ 5.