The quest for of a stringy analog of the general relativistic equivalence principle has been hampered, in part, by a lack of understanding of the off-shell features of supersymmetric gauge theories and supergravity theories in cases having 16 and 32 supercharges. In this talk we present some mathematical progress towards a goal of codifying supersymmetry representation theory as a linear algebra problem, analogous to the system of roots and weights in Lie algebra representation theory. We present a comprehensive supersymmetry representation theory for supersymmetric quantum mechanics having arbitrary number of supercharges, and explain how this supplies a useful skeletal basis for the analogous problem in quantum field theories.