Related papers: Lagrangian Pairs and Lagrangian Orthogonal Matroid…
Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie…
We show how the exceptional isogenies of classical groups to orthogonal groups of quadratic spaces of dimensions up to 8 over fields of characteristic different from 2 may be obtained by explicit algebraic constructions using the…
An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer…
We investigate forms on supermanifolds defined as Lagrangians of ``copaths'' (that is, systems of equations, which may or may not specify submanifolds). For this, we consider direct products $M^{n|m}\times\Bbb R^{r|s}$ and study…
The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth…
In this short note we provide the examples of pairs of closed, connected Legendrian non-isotopic Legendrian submanifolds $(\Lambda_{-}, \Lambda_{+})$ of the $(4n+1)$-dimensional contact vector space, $n>1$, such that there exist Lagrangian…
We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons…
We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the \emph{algebra of symmetries} $\mathscr{S}(\Box^r)$ of powers of the Laplacian $\Box$ on…
We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…
Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group…
Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework,…
We introduce the concept of gauged Lagrangian $1$-forms, extending the notion of Lagrangian $1$-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian $1$-form on the cotangent bundle of…
We construct pseudotoric structures (\`{a} la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important…
New symmetry transformations for the n-dimensional Toda lattice are presented. Their existence allows for the construction of several first order Lagrangian structures associated to them. The multi-Hamiltonian structures are derived from…
The graph of a real symplectic linear transformation is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating…
A distinctive duality present in 3d $\mathcal{N}=4$ theories is the 3d mirror symmetry. Under this duality, the Coulomb (Higgs) branch of one theory corresponds to the Higgs (Coulomb) branch of its mirror dual. This paper is divided into…
This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is…
Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent…
A well-known conjecture states that the Whitney numbers of the second kind of a geometric lattice (simple matroid) are logarithmically concave. We show this conjecture to be equivalent to proving an upper bound on the number of new copoints…
Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally…