math.QA — Quantum Algebra

Deformations of Deligne's S_t

Contributed by Noah Snyder

Snyder and Spencer showed that every *ribbon* infinitesimal deformation of the trivalent category $\mathcal{S}_t$ underlying Deligne's $\mathrm{Rep}(S_t)$ comes from varying $t$. Here we remove the braided hypothesis: every pivotal deformation of $\mathcal{S}_t$, $t\neq0$, over a local Artinian $\mathbb{C}$-algebra comes from varying $t$. The pivotal hypothesis turns out to be automatic, so the same holds for monoidal deformations. Since $\mathcal{S}_t$ is not generated as a pivotal category by its trivalent vertex, the crossing must be added as a generator, and the Reidemeister moves only survive in a deformed form, with lower order terms in the $2$- through $6$-box spaces. An open condition on rotational eigenvalues lets us lift the crossing in a rotation invariant way. A handful of overlap computations, in the spirit of Liu's classification of Yang--Baxter relation planar algebras, then kill every lower order term up to a change of the chosen crossing.