On the cohomology of twistor flag spaces
Let be a semisimple complex algebraic group and its flag variety. Let and let be its enveloping algebra. Let be a Cartan subalgebra of . For , let be the corresponding minimal primitive ideal, let , and let be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the -algebras . When is regular, Hodges has shown that . In this case is generated by the classes corresponding to...
In the current paper we show that the dimension of a family of irreducible reduced curves in a given ample linear system on a toric surface over an algebraically closed field is bounded from above by , where denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality does not imply the nodality of even if belongs to the...
The set of conjugacy classes appearing in a product of conjugacy classes in a compact, -connected Lie group can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety , where is the complexification of and is a maximal parabolic subgroup. This generalizes the results for of Agnihotri and the second author and Belkale on...
The parameter spaces for quadrics are reviewed. In addition, an explicit formula for the number of quadrics tangent to given linear subspaces is presented.
We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the -smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.
Let be a preprojective algebra of type , and let be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories for an injective -module, and we introduce a mutation operation between complete rigid modules in . This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to .