We are taught that the dimension of a vector space must be a whole number. In representation theory, however, it is sometimes possible to “interpolate” dimension and study symmetries in arbitrary complex dimensions.

This talk explores the bridge between the classical representation theory of the general linear groups GL(n) and the world of Deligne’s interpolation categories, where the parameter n is allowed to be any complex number.

We begin with Schur–Weyl duality, the classical correspondence linking representations of matrix groups and symmetric groups through the combinatorics of Young diagrams. We then introduce Pierre Deligne’s remarkable interpolation categories, which extend many features of the representation theory of GL(n) beyond integer dimensions.

When the parameter is not an integer, these categories behave in a remarkably simple way. At integer values, however, a richer structure emerges, closely connected to the representation theory of supergroups and supersymmetry. In fact, Deligne categories at integral dimensions can be viewed as limits of categories of representations of supergroups.