Given any action of a compact torus $T$ on a stably complex manifold, the tangent space to each fixed point carries the structure of a complex $T$-representation, called the isotropy representation. The isotropy representations at different fixed points are not independent, but deeply interrelated by the equivariant localization formula of Atiyah–Bott and Berline–Vergne.

Abstracting these relations to a system of conditions on an arbitary collection of representations, we ask the following question: if the conditions are satisfied, must the collection arise as the list of isotropy representations of some torus action on a manifold?

We show the answer is "yes" under additional conditions corresponding to (1) GKM actions with isolated fixed points and (2) semifree circle actions with isolated fixed points. Unpublished work of Alastair Darby is heavily implicated in the proof for the GKM case, which is joint with Elisheva Adina Gamse and Yael Karshon.