University of Sydney Algebra Seminar

Oded Yacobi (University of Sydney)

Friday 13 March, 12-1pm, Place: Carslaw 275

Perversity of categorical braid group actions

Let \(\mathfrak{g}\) be a semisimple Lie algebra with simple roots \(I\), and let \(\mathcal{C}\) be a category endowed with a categorical \(\mathfrak{g}\)-action. Recall that Chuang-Rouquier construct, for every \(i \in\) I, the Rickard complex acting as an autoequivalence of the derived category \(D^b(\mathcal{C})\), and Cautis-Kamnitzer show these define an action of the braid group B_g. As part of an ongoing project with Halacheva, Licata, and Losev we show that the positive lift to B_g of the longest Weyl group element acts as a perverse auto-equivalence of \(D^b(\mathcal{C})\). (This generalises a theorem of Chuang-Rouquier who proved it for \(\mathfrak{g}= \mathfrak{sl}(2)\).) This implies, for instance, that for a minimal categorification this functor is t-exact (up to a shift). Perversity also allows us to "crystallise" the braid group action, to obtain a cactus group action on the set of irreducible objects in \(\mathcal{C}\). This agrees with the cactus group action arising from the \(\mathfrak{g}\)-crystal (due to Halacheva-Kamnitzer-Rybnikov-Weekes). Note: This is the same talk I recently gave at the meeting in Mooloolaba.