Unitary Grassmannians

Unitary Grassmannians of Division Algebras

We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [17], where a similar issue for orthogonal and symplectic involuti...

A survey of symmetric functions, Grassmannians, and representations of the unitary group

Incompressibility of Products by Grassmannians of Isotropic Subspaces

We prove that the product of an arbitrary projective homogeneous variety Y by an orthogonal, symplectic, or unitary Grassmannian X is 2-incompressible if and only if the varieties XF (Y ) and YF (X) are so. Some new properties of incompressible Grassmannians are established on the way.

Filtrations, Factorizations and Explicit Formulae for Harmonic Maps

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for harmonic maps of finite uniton number from a Riemann surface to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formulae for harmonic maps into the special orthogonal and symplectic groups, real, complex and quaternionic Grassmannians, and the...

A Closed Form for Unitons

Unitons, i.e. harmonic spheres in a unitary group, correspond to ‘uniton bundles’, i.e. holomorphic bundles over the compactified tangent space to the complex line with certain triviality and other properties. In this paper, we use a monad representation similar to Donaldson’s representation of instanton bundles to obtain a simple formula for the unitons. Using the monads, we show that real tri...