Homogeneous compact geometries

We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.Comment: To appear in: Transformation Group

Quantum Pieri rules for isotropic Grassmannians

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.Comment: 59 pages, LaTeX, 6 figure

Regular Polyhedra of Index Two, II

A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2 if it is combinatorially regular and its geometric symmetry group has index 2 in its combinatorial automorphism group; thus its automorphism group is flag-transitive but its symmetry group has two flag orbits. The present paper completes the classification of finite regular polyhedra of index 2 in 3-space. In particular, this paper enumerates the regular polyhedra of index 2 with vertices on one orbit under the symmetry group. There are ten such polyhedra.Comment: 33 pages; 5 figures; to appear in "Contributions to Algebra and Geometry

A new construction of homogeneous quaternionic manifolds and related geometric structures

Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map \Pi : \wedge^2 W \to V. If the skew symmetric vector valued bilinear form \Pi is nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q,g) is a homogeneous quaternionic pseudo-K\"ahler manifold. The construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any spin(V)-equivariant linear map \Pi : Sym^2 W \to V a homogeneous quaternionic supermanifold (M,Q) is constructed and, moreover, a homogeneous quaternionic pseudo-K\"ahler supermanifold (M,Q,g) if the symmetric vector valued bilinear form \Pi is nondegenerate.Comment: to appear in the Memoirs of the AMS, 81 pages, Latex source fil

Classification of maximal transitive prolongations of super-Poincar\'e algebras

Let $V$ be a complex vector space with a non-degenerate symmetric bilinear form and $\mathbb S$ an irreducible module over the Clifford algebra $Cl(V)$ determined by this form. A supertranslation algebra is a $\mathbb Z$-graded Lie superalgebra $\mathfrak m=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}$, where $\mathfrak{m}_{-2}=V$ and $\mathfrak{m}_{-1}=\mathbb S\oplus\cdots\oplus\mathbb{S}$ is the direct sum of an arbitrary number $N\geq 1$ of copies of $\mathbb S$, whose bracket $[\cdot,\cdot]|_{\mathfrak{m}_{-1}\otimes \mathfrak{m}_{-1}}:\mathfrak{m}_{-1}\otimes\mathfrak{m}_{-1}\rightarrow\mathfrak{m}_{-2}$ is symmetric, $\mathfrak{so}(V)$-equivariant and non-degenerate (that is the condition "$s\in\mathfrak{m}_{-1}, [s,\mathfrak{m}_{-1}]=0$" implies $s=0$). We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for $\dim V\geq3$ and classify them in terms of super-Poincar\'e algebras and appropriate $\mathbb Z$-gradings of simple Lie superalgebras.Comment: 32 pages, v2: general presentation improved, corrected several typos. Proofs and results unchanged. Final version to appear in Adv. Mat

Characterization of symmetric planes in dimension at most 4

AbstractA stable plane is a topological geometry with the properties that (i) any two points are joined by a unique line, and (ii) the operations of join and intersection are continuous and have open domains of definition. A stable plane is called symmetric if its point space is a (differentiable) symmetric space whose symmetries are automorphisms of the plane.Among locally compact stable planes of positive (topological) dimension ≦ 4, we determine those which admit a reflection at each point (i.e., an involutory automorphism fixing this point line-wise), and we list the possible groups containing reflections at all points. Together with an additional, purely geometric condition, this yields a characterization of symmetric planes and, indirectly, of the plane geometries defined by real and complex hermitian forms. No differentiability hypotheses and no algebraic axioms ruling the reflections are needed