Totally nonnegative maximal tori and opposed Bruhat intervals
Abstract
Lusztig (2024) recently introduced the space T>0 of totally positive maximal tori of an algebraic group G. Each such torus is the intersection of a totally positive Borel subgroup and a totally negative Borel subgroup. Lusztig defined a map from the totally positive part of G to T>0 and conjectured that it is surjective. We verify this conjecture. We also examine the closure of T>0, by studying when a totally nonnegative Borel subgroup is opposed to a totally nonpositive Borel subgroup. Our main result reduces this problem to a new combinatorial relation between pairs of Bruhat intervals of the Weyl group W, which we call 'opposition'. We provide a characterization of opposition when G = SLn (and W is the symmetric group). Along the way, we disprove another conjecture of Lusztig (2021) on totally nonnegative Borel subgroups. Finally, we connect T>0 to the amplituhedron introduced by Arkani-Hamed and Trnka (2014) in theoretical physics, by showing that T>0 can be regarded as a 'universal flag amplituhedron'. This gives further motivation for studying T>0 and its closure.