Related papers: Total positivity for the Lagrangian Grassmannian

As a generalization of Postnikov's construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson…

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all maximal by inclusion weakly separated collections of minors have the…

Baker and Bowler showed that the Grassmannian can be defined over a tract, a field-like structure generalizing both partial fields and hyperfields. This notion unifies theories of matroids over partial fields, valuated matroids, and…

Positroids are matroids realizable by real matrices with all nonnegative maximal minors. They partition the ordered matroids into equivalence classes, called positroid envelope classes, by their Grassmann necklaces. We give an explicit…

We provide a short proof of a classical result of Kasteleyn, and prove several variants thereof. One of these results has become key in the parametrization of positroid varieties, and thus deserves the short direct proof which we provide.

Let $\sigma$ be a simple involution of an algebraic semisimple group $G$ and let $H$ be the subgroup of $G$ of points fixed by $\sigma$. If the restricted root system is of type $A$, $C$ or $BC$ and $G$ is simply connected or if the…

The study of the flag variety $\mathrm{Fl}_n$ and its subvarieties, including Schubert and Richardson varieties, plays a fundamental role in algebraic geometry and algebraic combinatorics. In this paper, we introduce and develop the theory…

The vector space of symmetric matrices of size $n$ has a natural map to a projective space of dimension $2^n-1$ given by the principal minors. This map extends to the Lagrangian Grassmannian ${\rm LG}(n,2n)$ and over the complex numbers the…

We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally…

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite…

The open projected Richardson varieties form a stratification for the partial flag variety $G/P$. We compare the Segre--MacPherson classes of open projected Richardson varieties with those of the corresponding affine Schubert cells by…

Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n)…

Alex Postnikov has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr_{kn}+, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded…

Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial.…

We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian $G=G(2,N)$ parametrizing lines in projective space, for the natural arithmetic Lefschetz operator defined via the Pl\"ucker embedding of $G$ in…

Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and…

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov;…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…

A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lnenicka and Mat\'us are equivalent to compatibility with…

The Grassmannian, which is the manifold of all $k$-dimensional subspaces in the Euclidean space $\mathbb{R}^n$, was decomposed through three equivalent methods connecting combinatorial geometries, Schubert cells and convex polyhedra by…