Doctoral Defense (Note time)

Speaker:  Williem Rizer

Title:  Combinatorial models for nonnegativity in flag varieties

Abstract:  

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh and Carolina Benedetti has further developed the combinatorics and geometry of these structures. 

In this talk, we will extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. We introduce flag positroid pipe dreams, a diagrammatic model analogous to Le diagrams, together with associated directed graphs and networks that parameterize Richardson cells indexed by flag positroids. Using this framework, we give a constructive proof of a conjecture from Benedetti-Chavez-Tamayo in the case of nonnegatively representable quotients, giving a complete characterization of all positroids of which a fixed positroid is such a quotient in terms of decorated permutations and diagrammatic data. 

Our approach also highlights the connection between flag positroids and intervals in the Bruhat order. Building on work of onathan Boretsky, Christopher Eur and Williams, we show that flag positroid pipe dreams are in bijection with Bruhat intervals, where the number of ones in the diagram encodes the interval length and hence the dimension of the corresponding Richardson cell. Together, these results provide a unified combinatorial perspective on nonnegativity in flag varieties.