In this paper, we develop the theory of flag manifold over a semifield for any Kac-Moody root datum. We show that admits natural action monoid associated with datum and cellular decomposition. This extends previous work Lusztig, Postnikov, Rietsch others on totally nonnegative manifolds (of finite type) Speyer, Williams tropical type). As by-product, prove conjecture Lusztig duality type.

This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $CP^{n-1}$ and Grassmannian models. They naturally arise in description continuum limits spin chains, their phase structure sensitive values topological angles, determined by representations spins chain. Gapless phases can certain cases be explained presence discr...

We prove several relations on the f -vectors and Betti numbers of flag complexes. For every complex Δ, we show that there exists a balanced with same -vector as whose top-dimensional number is at least thereby extending theorem Frohmader by additionally taking homology into consideration. obtain upper bounds Δ in terms its face numbers. also give quantitative refinement Meshulam establishing lo...

The application of flags to network coding has been introduced recently by Liebhold, Nebe, and Vazquez-Castro. It is a variant random linear explicit routing solutions for given networks. Here we study lower upper bounds the maximum possible cardinality corresponding flag code with parameters.