Quaternionic Grassmannians Research Articles

Hitchin map on even very stable upward flows

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the [Formula: see text] case, we classify the type [Formula: see text] examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, [Formula: see text]-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.

Explicit harmonic morphisms and p-harmonic functions from the complex and quaternionic Grassmannians

We construct explicit complex-valued p-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the classical Laplace–Beltrami and the so-called conformality operator. A known duality principle implies that these p-harmonic functions and harmonic morphisms also induce such solutions on the Riemannian symmetric non-compact dual spaces.

Stability of Compact Symmetric Spaces

In this article, we study the stability problem for the Einstein–Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

Quaternionic Grassmannians and Borel classes in algebraic geometry

The quaternionic Grassmannian H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) is the affine open subscheme of the usual Grassmannian parametrizing those 2 r 2r -dimensional subspaces of a 2 n 2n -dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have HP n = H Gr ⁡ ( 1 , n + 1 ) \operatorname {HP}^n = \operatorname {H Gr}(1,n+1) . For a symplectically oriented cohomology theory A A , including oriented theories but also the Hermitian K \operatorname {K} -theory, Witt groups, and algebraic symplectic cobordism, we have A ( HP n ) = A ( pt ) [ p ] / ( p n + 1 ) A(\operatorname {HP}^n) = A(\operatorname {pt})[p]/(p^{n+1}) . Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes. The cell structure of the H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is HP n \operatorname {HP}^n where the cell of codimension 2 i 2i is a quasi-affine quotient of A 4 n − 2 i + 1 \mathbb {A}^{4n-2i+1} by a nonlinear action of G a \mathbb {G}_a .

Quaternionic projective bundle theorem and Gysin triangle in MW-motivic cohomology

In this paper, we show that the motive of the quaternionic Grassmannian $HP^n$ (as defined by I. Panin and C. Walter) splits in the category of effective MW-motives (as defined by B. Calm\`es, F. D\'eglise and J. Fasel). Moreover, we extend this result to an arbitrary symplectic bundle, obtaining the so-called quaternionic projective bundle theorem. Finally, we give the Gysin triangle in MW-motivic cohomology.

On the motivic commutative ring spectrum $\mathbf {BO}$

We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on SmOp/S the cohomology theory (X, U) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ −→ KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is SpecZ[ 1 2 ], this monoid structure and the induced ring structure on the cohomology theory BO are the unique structures compatible with the products KO [2m] 0 (X)× KO [2n] 0 (Y ) → KO [2m+2n] 0 (X × Y ). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO(T∧T ) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space 〈−1〉.

Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces

We find many examples of compact Riemannian manifolds (M, g) whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric g is replaced with $$g'$$ in a neighbourhood of g. Our examples (M, g) consist of certain minimal isoparametric hypersurfaces of spheres, their focal manifolds, the Lie groups $${\text {SU}}(n)$$ for $$n\le 17$$ and $${\text {Sp}}(n)$$ for all n, and all quaternionic Grassmannians.

Smooth maps into quaternionic Grassmannians inducing a prescribed 4-form

Let $$\gamma ^n_k\rightarrow G_k(\mathbb {H}^n)$$ be the Stiefel bundle of quaternionic k-frames in $$\mathbb {H}^n$$ over the the quaternionic Grassmannian $$G_k(\mathbb {H}^n)$$ . Let $$\sigma $$ denote the first symplectic Pontrjagin form associated with the universal connection on $$\gamma ^n_k$$ . We show that every 4-form $$\omega $$ on a smooth manifold M can be induced from $$\sigma $$ by a smooth map $$f:M\rightarrow G_k(\mathbb {H}^n)$$ (for sufficiently large k and n) provided there exists a continuous map $$f_0:M\rightarrow G_k(\mathbb {H}^n)$$ which pulls back the deRham cohomology class of $$\sigma $$ (referred as the symplectic Pontrjagin class of $$\gamma _k^n$$ ) onto that of $$\omega $$ .

On some Grassmannians carrying an even Clifford structure

We give an explicit description of the non-flat parallel even Clifford structures of rank 8, 6, 5 on some real, complex and quaternionic Grassmannians, and discuss the rôle of the octonions in them, in particular for some low dimensional examples.

Morse Theory and the Lusternik–Schnirelmann Category of Quaternionic Grassmannians

AbstractThe Lusternik–Schnirelmann category of the quaternionic Grassmannianis known to bek(n − k). In this paper we show that this result can be deduced from Morse theory.

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

In this paper, we prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say: 1. the classical simple compact Lie groups: special orthogonal groups SO(n), special unitary groups SU(n) and compact symplectic groups USp(n); 2. the real, complex and quaternionic Grassmannian varieties (including the real spheres, and the complex or quaternionic projective spaces when q = 1): SO(p + q)/(SO(p)×SO(q)), SU(p + q)/S(U(p)×U(q)) and USp(p + q)/(USp(p)×USp(q)); 3. the spaces of real, complex and quaternionic structures: SU(n)/SO(n), SO(2n)/ U(n), SU(2n)/USp(n) and USp(n)/UU(n). Denoting μt the law of the Brownian motion at time t, we give explicit lower bounds for dTV(μt,Haar) if \(t t_{\text{cut-of\/f}}\). This provides in particular an answer to some questions raised in recent papers by Chen and Saloff-Coste. Our proofs are inspired by those given by Rosenthal and Porod for products of random rotations in SO(n), and by Diaconis and Shahshahani for products of random transpositions in \(\mathfrak{S}_{n}\).

Radon transform on real, complex, and quaternionic Grassmannians

Let Gn,k(K) be the Grassmannian manifold of k-dimensional K-subspaces in Kn, where K=R,C,H is the field of real, complex, or quaternionic numbers. For 1≤k<k'≤n−1, we define the Radon transform (Rf)(η), η∈Gn,k'(K), for functions f(ξ) on Gn,k(K) as an integration over all ξ⊂η. When k+k'≤n, we give an inversion formula in terms of the Garding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive (k×k)-matrices over K. This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians

Harmonic morphisms from the Grassmannians and their non-compact duals

In this paper we give a unified framework for the construction of complex valued harmonic morphisms from the real, complex and quaternionic Grassmannians and their non-compact duals. This gives a positive answer to the corresponding open existence problem in the real and quaternionic cases.

The Geometry of Polygons in \open R5and Quaternions

We consider the moduli space \(\mathcal{M}\)r of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between \(\mathcal{M}\)r and a weighted quotient of n-fold products of the quaternionic projective line \(\mathbb{H}\mathbb{P}\)1 by the diagonal PSL(2, \(\mathbb{H}\))-action. We explore the relation between \(\mathcal{M}\)r and the fixed point set of an anti-symplectic involution on a GIT quotient Grℂ(2, 4)n/SL(4, ℂ). We generalize the Gel'fand—MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space \(\mathcal{M}\)r as a quotient of a subspace in the quaternionic Grassmannian Grℍ(2, n) by the action of the group Sp(1)n. We also give analogues of the Gel'fand—Tsetlin coordinates on the space of quaternionic Hermitean marices and briefly describe generalized action—angle coordinates on \(\mathcal{M}\)r.

On the signature of homogeneous spaces

We compute the signature of real and quaternionic Grassmannians, thereby completing the table of signatures of symmetric spaces given in a previous paper [4]. In addition, all homogeneous spaces of exceptional Lie groups with non-zero signature are listed.

Calibrated geometries in quaternionic Grassmannians

Soit H le corps des quaternions et H n l'ensemble de tous les vecteurs n-colonnes sur H. On considere H n comme un H espace a droite. On etudie le grassmannien quaternionique G p (H p+q ) qui est l'ensemble de tous les H-sousespaces a droite de H-dimension p dans H p+q

The index of complex and quaternionic Grassmannians via lefschetz formula

The index of complex and quaternionic Grassmannians via lefschetz formula

Morse theory on quaternionic Grassmannians

Morse theory on quaternionic Grassmannians