Posets, Polytopes and Positroids

This dissertation explores questions about posets and polytopes through the lenses of positroids and geometry. The introduction and study of positroids, a special class of matroids, was pioneered by Postnikov in his study of the totally nonnegative Grassmannian and has subsequently been applied to various fields such as cluster algebras, physics, and free probability. Postnikov showed that positroids, the matroids realized by full rank $k\times n$ real matrices whose maximal minors are nonnegative, are in bijection with several combinatorial objects: Grassmann necklaces, decorated permutations, Le-diagrams and plabic graphs. In the first chapter, following work of Skandera and Reed, we define the unit interval positroid arising from a unit interval order poset via its associated antiadjacency matrix. We give a simple description of the decorated permutation representation of a unit interval positroid, and show it can be recovered from the Dyck path drawn on the associated antiadjacency matrix. We also describe the unit interval positroid cells in the totally nonnegative Grassmannian and their adjacencies. Finally, we provide a new description of the $f$-vector of posets.The second chapter concerns the $f$-vector of a $d$-dimensional polytope $P$, which stores the number of faces of each dimension. When $P$ is a simplicial polytope the Dehn--Sommerville relations condense the $f$-vector into the $g$-vector, which has length $\lceil{\frac{d+1}{2}}\rceil$. Thus, to determine the $f$-vector of $P$, we only need to know approximately half of its entries. This raises the question: Which $\left(\lceil{\frac{d+1}{2}}\rceil\right)$-subsets of the $f$-vector of a general simplicial polytope are sufficient to determine the whole $f$-vector? We prove that the answer is given by the bases of the Catalan matroid.In the final chapter, we explore the combinatorial structure of Knuth equivalence graphs $G_{\lambda}$. The vertices of $G_{\lambda}$ are the permutations whose insertion tableau is a fixed tableau of shape $\lambda$, and the edges are given by local Knuth moves on the permutations. The graph $G_{\lambda}$ is the $1$-skeleton of a cubical complex $C_{\lambda}$, and one can ask whether it is CAT(0); this is a desirable metric property that allows us to describe the combinatorial structure of $G_{\lambda}$ very explicitly. We prove that $C_{\lambda}$ is CAT(0) if and only if $\lambda$ is a hook