Discrete Morse theory for totally non-negative flag varieties

TL;DR

This paper proves that the totally non-negative parts of flag varieties have a cell decomposition with boundaries homotopic to spheres and closures contractible, using discrete Morse theory to confirm a conjecture about their topological structure.

Contribution

It applies discrete Morse theory to establish that the cell decomposition of totally non-negative flag varieties is homotopy-equivalent to a regular CW complex homeomorphic to a ball, confirming a conjecture.

Findings

Boundaries of cells are homotopic to spheres.

Closures of cells are contractible.

Top-dimensional cell's boundary is homotopic to a sphere.

Abstract

In a seminal 1994 paper, Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a "remarkable polyhedral subspace", and conjectured a decomposition into cells, which was subsequently proven by the first author. Subsequently the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's that (G/P)_{\geq 0} -- the closure of the top-dimensional cell -- is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere, is new for all G/P other than projective space.