Full Flag Variety Research Articles

Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space

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 ⊂ GL m + n ( C ) . These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M m , n ( C ) , isomorphic to intersections of dual Schubert cells in the full flag variety of GL m + n ( C ) . Three different presentations of the T-orbits of symplectic leaves in M m , n ( C ) are obtained: (a) as pullbacks of Bruhat cells in GL m + n ( C ) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GL m ( C ) and GL n ( C ) . In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group S m + n , such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed row-echelon form. Finally, decompositions of generalized double Bruhat cells in M m , n ( C ) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.

Hilbert–Kunz functions and multiplicities for full flag varieties and elliptic curves

We compute the Hilbert–Kunz functions and multiplicities for certain projective embeddings of flag varieties G/ B and elliptic curves, over algebraically closed fields of positive characteristics. The group theoretic nature of both these classes of examples is used, albeit in different ways, to explicitly describe the cokernels in each degree of the Frobenius twisted multiplication maps for the corresponding graded rings. This detailed information also enables us to extend our results to arbitrary products of such varieties.

Braid cobordisms, triangulated categories, and flag varieties

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.