Discrete Morse theory for totally non-negative flag varieties

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 theo...

Shelling Totally Nonnegative Flag Varieties

In this paper we study the partially ordered set Q of cells in Rietsch’s [20] cell decomposition of the totally nonnegative part of an arbitrary flag variety P ≥0 . Our goal is to understand the geometry of P ≥0 : Lusztig [13] has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P ≥0 is homeomorphic to a ball. The order comp...

Discrete Morse Theory for Filtrations

Acknowledgements Enumerating all the ways in which I am grateful to Konstantin would essentially double the length of this document, so I'll save all that stuff for my autobiography. But I will note here that he was simultaneously patient, engaged, proactive, and – best of all – ruthlessly determined to refine and sculpt all our vague big-picture ideas into digestible and implementable concrete...

Equivariant discrete Morse theory

In this paper, we study Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the C2 × Sn−2-homotopy type of the complex of non-connected graphs on n nodes.

Persistence in discrete Morse theory