Orthogonal Grassmannians Research Articles

The Cramer varieties [formula omitted

In this paper we study quasi-homogeneous affine algebraic varieties, that is, varieties obtained as closures of orbits of suitable group representations. We also discuss one interesting case that has links with the Orthogonal Grassmannian OGr(5,10). The main aim is to write the tangent bundle and the canonical class of quasi-homogeneous affine algebraic varieties in terms of group representations.

ABJM amplitudes and the positive orthogonal Grassmannian

A remarkable connection between perturbative scattering amplitudes of four-dimensional planar SYM, and the stratification of the positive grassmannian, was revealed in the seminal work of Arkani-Hamed et. al. Similar extension for three-dimensional ABJM theory was proposed. Here we establish a direct connection between planar scattering amplitudes of ABJM theory, and singularities there of, to the stratification of the positive orthogonal grassmannian. In particular, scattering processes are constructed through on-shell diagrams, which are simply iterative gluing of the fundamental four-point amplitude. Each diagram is then equivalent to the merging of fundamental OG_2 orthogonal grassmannian to form a larger OG_k, where 2k is the number of external particles. The invariant information that is encoded in each diagram is precisely this stratification. This information can be easily read off via permutation paths of the on-shell diagram, which also can be used to derive a canonical representation of OG_k that manifests the vanishing of consecutive minors as the singularity of all on-shell diagrams. Quite remarkably, for the BCFW recursion representation of the tree-level amplitudes, the on-shell diagram manifests the presence of all physical factorization poles, as well as the cancellation of the spurious poles. After analytically continuing the orthogonal grassmannian to split signature, we reveal that each on-shell diagram in fact resides in the positive cell of the orthogonal grassmannian, where all minors are positive. In this language, the amplitudes of ABJM theory is simply an integral of a product of dlog forms, over the positive orthogonal grassmannian.

A Giambelli formula for even orthogonal Grassmannians

Abstract Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the classical and quantum cohomology rings of X as a polynomial in certain special Schubert classes. Our analysis reveals a surprising relation between the Schubert calculus on even and odd orthogonal Grassmannians. We also study eta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X.

Codes and caps from orthogonal Grassmannians

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding εkgr of an orthogonal Grassmannian Δk. In particular, we determine some of the parameters of the codes arising from the projective system determined by εkgr(Δk). We also study special sets of points of Δk which are met by any line of Δk in at most 2 points and we show that their image under the Grassmann embedding εkgr is a projective cap.

Some results on caps and codes related to orthogonal Grassmannians — a preview

Abstract In this note we offer a short summary of some recent results, to be contained in a forthcoming paper [Cardinali, I., and L. Giuzzi, Caps and codes from orthogonal Grassmannians, submitted], on projective caps and linear error correcting codes arising from the Grassmann embedding e k g r of an orthogonal Grassmannian Δ k . More precisely, we consider the codes arising from the projective system determined by e k g r ( Δ k ) and determine some of their parameters. We also investigate special sets of points of Δ k which are met by any line of Δ k in at most 2 points proving that their image under the Grassmann embedding is a projective cap.

Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian

A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).

Grassmann and Weyl embeddings of orthogonal grassmannians

Given a non-singular quadratic form q of maximal Witt index on \(V := V(2n+1,\mathbb{F})\), let Δ be the building of type B n formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δ k be the k-grassmannian of Δ. Let e k be the embedding of Δ k into PG(⋀ k V) mapping every point 〈v 1,v 2,…,v k 〉 of Δ k to the point 〈v 1∧v 2∧⋯∧v k 〉 of PG(⋀ k V). It is known that if \(\mathrm{char}(\mathbb{F})\neq2\) then \(\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}\). In this paper we give a new very easy proof of this fact. We also prove that if \(\mathrm{char}(\mathbb{F}) = 2\) then \(\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}\). As a consequence, when 1 2 or a number field, n>k and k=2 or 3, then e k is universal.

Embeddings of orthogonal grassmannians

In this paper I survey a number of recent results on projective and Veronesean embeddings of orthogonal Grassmannians and propose a few conjectures and problems.

Complex Static Skew-Symmetric Output Feedback Control

We study the problem of feedback control for skew-symmetric and skew-Hamiltonian transfer functions using skew-symmetric controllers. This extends work of Helmke et al., who studied static symmetric feedback control of symmetric and Hamiltonian linear systems. We identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a Grassmannian, give necessary and sufficient conditions for pole placement by static skew-symmetric complex feedback, and use Schubert calculus for the orthogonal Grassmannian to count the number of complex feedback laws when there are finitely many of them. Finally, we also construct a real skew-symmetric linear system with only real feedback for any set of real poles.

Sufficiently generic orthogonal grassmannians

We prove the following conjecture due to Bryant Mathews (2008). Let Qi be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1⩽i⩽m:=[(dimq)/2]). Assume that for a given i, the form q has the following property (possessed by the generic quadratic form): the degree of each closed point on Qi is divisible by 2i and the Witt index of q over the function field of Qi is equal to i. Then the variety Qi is 2-incompressible.Assuming that the form q is sufficiently close to the generic one in a different sense, we prove a stronger property of Qi saying that its Chow motive with coefficients in F2 (the finite field of 2 elements) is indecomposable. This result contrasts with recent results of Zhykhovich (2010) [21] on decomposability of the motives of incompressible twisted grassmannians.The above two main results of the paper were known for the quadric Q1 and the maximal grassmannian Qm due to the works of A. Vishik.The proofs are based on the theory of upper motives. The results allow one to compute the canonical 2-dimension of any projective homogeneous variety (i.e., orthogonal flag variety) associated to the generic quadratic form.This paper is an extended version of Karpenko (2011) [10] including the results of Karpenko (2010) [5].

K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux

Abstract. We present a proof of a Littlewood–Richardson rule for the K-theory of odd orthogonal Grassmannians OG(n,2n+1), as conjectured by Thomas–Yong (2009). Specifically, we prove that rectification using the jeu de taquin for increasing shifted tableaux introduced there, is well-defined and gives rise to an associative product. Recently, Buch–Ravikumar (2012) proved a Pieri rule for OG(n,2n+1) that confirms a special case of the conjecture. Together, these results imply the aforementioned conjecture.

Unitary grassmannians

We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.

Pieri rules for the K-theory of cominuscule Grassmannians

We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent work of Thomas and Yong and of E. Clifford has shown that the full Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K-theoretic Pieri coefficients both as integers determined by positive recursive identities and as the number of certain tableaux. The proof is based on a computation of the sheaf Euler characteristic of triple intersections of Schubert varieties, where at least one Schubert variety is special.

Quantum Giambelli formulas for isotropic Grassmannians

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors’ cohomological Giambelli formulas.

Incompressibility of orthogonal grassmannians

We prove the following conjecture due to Bryant Mathews (2008). Let Q be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1 ⩽ i ⩽ ( dim q ) / 2 ). If the degree of each closed point on Q is divisible by 2 i and the Witt index of q over the function field of Q is equal to i, then the variety Q is 2-incompressible.

Incompressibility of orthogonal Grassmannians of rank 2

For a nondegenerate quadratic form φ on a vector space V of dimension 2 n + 1 , let X d be the variety of d-dimensional totally isotropic subspaces of V. We give a sufficient condition for X 2 to be 2-incompressible, generalizing in a natural way the known sufficient conditions for X 1 and X n . Key ingredients in the proof include the Chernousov–Merkurjev method of motivic decomposition as well as Pragacz and Ratajskiʼs characterization of the Chow ring of ( X 2 ) E , where E is a field extension splitting φ.

VAFA-INTRILIGATOR TYPE FORMULAS AND QUANTUM EULER CLASSES FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANNS

We use the result of Peterson to give very simple and explicit formulas of genus zero Gromov–Witten invariants for Lagrangian and orthogonal Grassmannians in terms of [Formula: see text]- and [Formula: see text]-polynomials of Pragacz–Ratajski which represent their cohomology classes. We also find an explicit expression for the quantum Euler classes and give a direct proof for the semi-simplicity of the quantum cohomology rings of these varieties.

The Wronski Map and Shifted Tableau Theory

The Mukhin–Tarasov–Varchenko theorem, conjectured by B. and M. Shapiro, has a number of interesting consequences. Among them is a well-behaved correspondence between certain points on a Grassmannian—those sent by the Wronski map to polynomials with only real roots—and (dual equivalence classes of) Young tableaux. In this paper, we restrict this correspondence to the orthogonal Grassmannian OG(n,2n+1)⊂Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and only if the corresponding tableau has a certain type of symmetry. From this, we recover much of the theory of shifted tableaux for Schubert calculus on OG(n,2n+1), including a new, geometric proof of the Littlewood–Richardson rule for OG(n,2n+1).

Reality and transversality for Schubert calculus in $\OGn$

We prove an analogue of the Mukhin-Tarasov-Varchenko theorem (formerly the Shapiro-Shapiro conjecture) for the maximal type B_n orthogonal Grassmannian OG(n,2n+1).

Zero-cycles on projective varieties and the norm principle

AbstractUsing the Gille–Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer’s theorem), for twisted forms of maximal orthogonal Grassmannians (a theorem of Bayer-Fluckiger and Lenstra), and for E6- (a theorem of Rost) and E7-varieties.