Discrete Morse theory for totally non-negative flag varieties

Abstract

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part ( G / P ) ⩾ 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 Rietsch (1998) [33]. In Williams (2007) [40] 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 (1998) [28], that ( G / P ) ⩾ 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.