Iva Halacheva: Weaving group moves, crystals and cacti.
Let g denote a semi-simple Lie algebra. Lusztig introduced the effect of the braid group on any integrable representation of the quantum group of g. This action is clearly implemented in the work of Chuang-Rouquier, as shown by Cautis-Kamnitzer, where each woven group generator is upgraded to a functor complex called the Rickard complex. I will describe the corresponding effect of the cactus group on the g crystals from the characterization. In joint work with Licata, Losev, and Yacobi, we showed that when considering the positive lift of some parabolic longest Weyl elements in g, this effect can be clearly recovered from the Rickard complex.