Deformation cohomology of Schur–Weyl categories

The deformation cohomology of a tensor category controls deformations of the constraint of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur–Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra. The results make precise an intriguing connection between the combinatorics of partitions and invariants of the exterior of the general linear algebra observed by Kostant.