Totally nonnegative maximal tori and opposed Bruhat intervals

TL;DR

This paper verifies Lusztig's conjecture on the surjectivity of a map to totally positive maximal tori, explores opposition of Borel subgroups via Bruhat intervals, and links these concepts to the amplituhedron in physics.

Contribution

It confirms Lusztig's conjecture, introduces a new opposition relation between Bruhat intervals, characterizes opposition for SL_n, and connects the space of totally positive tori to the amplituhedron.

Findings

Lusztig's conjecture on the surjectivity of the map to $ ext{Totally positive maximal tori}$ is verified.

A new combinatorial relation called 'opposition' between Bruhat intervals is introduced and characterized for SL_n.

The space $ ext{T}_{>0}$ is shown to be a 'universal flag amplituhedron', linking algebraic groups to physics.

Abstract

Lusztig (2024) recently introduced the space $\mathcal{T}_{>0}$ 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 $\mathcal{T}_{>0}$ and conjectured that it is surjective. We verify this conjecture. We also examine the closure of $\mathcal{T}_{>0}$, 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 $G = \text{SL}_n$ (and $W$ is the symmetric group). Along the way, we disprove another conjecture of Lusztig (2021) on totally nonnegative Borel subgroups. Finally, we connect $\mathcal{T}_{>0}$ to the amplituhedron introduced by Arkani-Hamed and Trnka (2014) in theoretical physics, by showing that $\mathcal{T}_{>0}$ can be regarded as a 'universal flag amplituhedron'. This gives further motivation for studying $\mathcal{T}_{>0}$ and its closure.