Discrete Morse Theory for Totally Non-negative Flag Varieties

In a seminal 1994 paper [20], 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 [27]. In this article we use discrete Morse theory to show that the cell decomposition of (G/P )≥0 is polyhedral in the following sense: closures of cells are collapsible and hence contractible. This answers a question posed by Lusztig in 1996 [23], and generalizes a later result of Lusztig’s [21], that (G/P )≥0 – the closure of the top-dimensional cell – is contractible. Furthermore, we show that the boundary of each cell – hence in particular the boundary of (G/P )≥0 – is homotopy equivalent to a sphere.