Lie Groups Seminar
bis
Building 2, 142, Suffolk
Speaker: Steven Karp (University of Notre Dame) Title: Totally positive maximal tori, opposed Bruhat intervals, and amplituhedra Abstract: Motivated by positivity properties of canonical bases, Lusztig introduced total positivity for algebraic groups G in the 1990's. This generalizes the classical theory of totally positive matrices studied since the 1930's. This talk focuses on total positivity for maximal tori of G, a notion introduced by Lusztig (2024). We show that every totally positive maximal torus can be obtained from a totally positive element of G, verifying a conjecture of Lusztig. We also define a notion of total nonnegativity for maximal tori and show that it leads to an interesting new relation (which we call "opposition") between Bruhat intervals of the Weyl group. We explain how all this is connected to spaces called amplituhedra which encode scattering amplitudes in high-energy physics, leading to a Lie-theoretic generalization of amplituhedra. This is joint work with Grant Barkley, and with Barkley and Kaitao Xie.