The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called shuffles, and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps; first, we reinterpret the shuffle equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite-dimensional problem, which we solve. For the 2nd step, we introduce a finite-dimensional analogue of the affine Grassmannian of $SL_n$, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

Remarks on Plücker embedding and totally antisymmetric gauge fields

We make a number of comments about the way the Plücker embedding, which can be derived via the Grassmann-Plücker relations, can be associated to totally antisymmetric gauge fields. As a first step we discuss the case of the electromagnetic field strength, showing that the Plücker map implies both the true degrees of freedom of the electromagnetic field and the 1-brane (string) structure. The procedure is generalized in order to prove that the true degrees of freedom of a totally antisymmetric field and the p-brane structure are, in part, consequence of the Plücker coordinates.

Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.

Semi-infinite Plücker Relations and Weyl Modules

The goal of this paper is two-fold. First, we write down the semi-infinite Plücker relations, describing the Drinfeld–Plücker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, that is, the quotient by the ideal generated by the semi-infinite Plücker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula.

Plücker varieties and higher secants of Sato’s Grassmannian

Every Grassmannian, in its Plücker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we callPlücker varieties. A Plücker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural numberk, thek-th secant variety ofanyPlücker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Plücker variety in the dual of a highly symmetric space known as theinfinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxiliary result that for every natural numberpthe space ofp-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Plücker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties ofp-tuples of matrices.

Integrable Systems in Four Dimensions Associated with Six-Folds in Gr(4, 6)

Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a six-fold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). These naturally fall into two subclasses. • Systems of Monge–Ampère type. The corresponding six-folds X are codimension 2 linear sections of the Plücker embedding Gr(4, 6)$ \hookrightarrow \mathbb{P}^{14}$. • General linearly degenerate systems. The corresponding six-folds X are the images of quadratic maps $\mathbb{P}^{6}\dashrightarrow \ $Gr(4, 6) given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.

Invariant theory for linear differential systems modeled after the grassmannian Gr(n, 2n)

We find invariants for the differential systems of rank 2n in n2 variables with n unknowns under the linear changes of the unknowns with variable coefficients. We look for a set of coefficients that determines the other coefficients, and give transformation rules under the linear changes above and coordinate changes. These can be considered as a generalization of the Schwarzian derivative, which is the invariant for second order ordinary differential equations under the change of the unknown by multiplying a non-zero function. Special treatment is done when n = 2: the conformal structure obtained through the Plücker embedding is studied, and a relation with line congruences is discussed.