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Finite $$F$$-representation type is an important notion in characteristic-$$p$$commutative algebra, but explicit examples of varieties with or without thisproperty are few. We prove that a large class of homogeneous coordinate ringsin positive characteristic will fail to have finite $$F$$-representation type. Todo so, we prove a connection between differential operators on the homogeneouscoordinate ring of $$X$$ and the existence of global sections of a twist of$$(\mathrm{Sym}^m \Omega_X)^\vee$$. By results of Takagi and Takahashi, thisallows us to rule out FFRT for coordinate rings of varieties with$$(\mathrm{Sym}^m \Omega_X)^\vee$$ not ``positive''. By using results positivityand semistability conditions for the (co)tangent sheaves, we show that severalclasses of varieties fail to have finite $$F$$-representation type, includingabelian varieties, most Calabi--Yau varieties, and complete intersections ofgeneral type. Our work also provides examples of the structure of the ring ofdifferential operators for non-$$F$$-pure varieties, which to this point havelargely been unexplored.