We introduce techniques for uniformly studying the gamma filtration of projective
homogeneous varieties. These techniques are utilized in some cases of inner-twisted
flag varieties (of type A) to show that functorality known for the Chow rings of these
varieties also extends to the associated graded rings for the gamma filtrations of the
same varieties. As an application, we show that the associated graded groups for
the gamma filtration of these varieties are torsion free in low homological
degrees.
