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.