Narrow Results

This article is a survey on the combinatorics of semimagic/pandiagonal magic squares and explains their connection to polytopes. On the side of combinatorics, we count magic squares with fixed strict upperbounds or fixed sum. As for polytopes, we show the relationship between magic squares and various polytopes such as the Birkhoff polytopes.

We study the fan structure of Dressians Dr(d, n) and local Dressians Dr($\mathcal{M}$) for a given matroid $\mathcal{M}$. In particular we show that the fan structure on Dr($\mathcal{M}$) given by the three term Plücker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope P($\mathcal{M}$). As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse.

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e.g., in the study of cluster algebras, scattering amplitudes, and solitons. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian.