Related papers: Grassmannians over rings and subpolygons
Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients of the Grassmannian of r-planes coincide with the Littlewood-Richardson tensor product structure constants for…
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…
Berezin integration of functions of anticommuting Grassmann variables is usually seen as a formal operation, sometimes even defined via differentiation. Using the formalism of geometric algebra and geometric calculus in which the Grassmann…
The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic…
In the spirit of the action of the symmetric group on the ring of polynomials in $n$ variables, we consider the actions of the seven frieze groups on rings of formal infinite linear combinations of monomials of restricted degree. For each…
We make progress towards characterizing the algebraic matroid of the determinantal variety defined by the minors of fixed size of a matrix of variables. Our main result is a novel family of base sets of the matroid, which characterizes the…
A new generalization of Grassmannians in supergeometry, called $\nu-$Grassmannians, are constructed by gluing $\nu-$domains. By a $\nu-$domain, we mean a superdomain with an odd involution say $\nu$ on its structure sheaf, as morphism of…
In this talk I discuss properties of the two Symanzik polynomials which characterise the integrand of an arbitrary multi-loop integral in its Feynman parametric form. Based on the construction from spanning forests and Laplacian matrices,…
Frieze patterns, as introduced by Coxeter in the 1970's, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in…
We study the integrability to second order of the infinitesimal Einstein deformations of the symmetric metric $g$ on the complex Grassmannian of $k$-planes inside $\mathbb{C}^n$. By showing the nonvanishing of Koiso's obstruction…
Let $F$ be a finite field with characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis of the $\mathbb{Z}_{2}$-graded polynomial…
A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology…
We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher)…
We show that the higher Grothendieck-Witt groups, a.k.a. algebraic hermitian K-groups, are represented by an infinite orthogonal Grassmannian in the A1-homotopy category of smooth schemes over a regular base for which 2 is a unit in the…
We establish versions of Matsuki duality for loop groups. The main result is a bijection between symmetric loop group orbits and real polynomial loop group orbits on the affine Grassmannians or affine flag varieties. Along the way we obtain…
Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the…
The goal of this paper is to clarify the connection between certain structures from the theory of totally nonnegative Grassmannians, quiver Grassmannians for cyclic quivers and the theory of local models of Shimura varieties. More…
We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations,…
Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$…
Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the…