  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 give new
results about the radical series of some tilting modules for $SL_4(K)$, where
$K$ is a field of positive characteristic.
