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Let $A$ be a quasi-hereditary algebra. We prove that in many cases, a tiltingmodule is rigid (i.e. has identical radical and socle series) if it does nothave certain subquotients whose composition factors extend more than one layerin the radical series or the socle series. We apply this theorem to give newresults about the radical series of some tilting modules for $SL_4(K)$, where$K$ is a field of positive characteristic.

Radically filtered quasi-hereditary algebras and rigidity of tilting modules