Séminaire Combinatoire et Théorie des Nombres ICJ

The Nonnegative Flag Variety and its Tropicalization

by Jonathan Boretsky (Harvard)

Salle Fokko du Cloux (ICJ)

Salle Fokko du Cloux (ICJ)


The flag variety of rank r=(r_1,...,r_k) has points corresponding to collections of subspaces (V_1,..., V_k) with V_i of dimension r_i such that V_i is contained in V_{i+1}. It can be embedded into a multi-projective space, where it is cut out by the incidence Plücker relations. We explore two natural extensions of this variety: First, we study the nonnegative flag variety, which corresponds to a subset of the flag variety consisting of flags that can be represented by totally positive matrices. Second, we study the tropicalization of the flag variety and, more specifically, its nonnegative part. In both cases, we provide equivalent descriptions of these spaces for flag varieties of rank r=(a,a+1,...,b), where r consists of consecutive integers. We also explore descriptions of the nonnegative tropical flag variety in terms of polytopal subdivisions. The supports of points in the nonnegative flag Dressian give a natural notion of a flag positroid. Once again restricting to the consecutive rank case, we give a combinatorial description of flag positroids. This talk is based on joint work with Chris Eur and Lauren Williams.