Algebra Seminar: Yacobi -- Perversity of categorical braid group actions

Oded Yacobi (University of Sydney) 

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

Title: Perversity of categorical braid group actions 

Abstract: Let g be a semisimple Lie algebra with simple roots I, and let C be a category
endowed with a categorical 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(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(C).  (This
generalises a theorem of Chuang-Rouquier who proved it for g = 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 C.  This agrees with
the cactus group action arising from the g-crystal (due to
Halacheva-Kamnitzer-Rybnikov-Weekes).  

Note: This is the same talk I recently gave at the meeting in Mooloolaba.