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Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Constantin Teleman, Christopher Woodward (2003)

Annales de l’institut Fourier

The set of conjugacy classes appearing in a product of conjugacy classes in a compact, 1 -connected Lie group K 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 G / P , where G is the complexification of K and P is a maximal parabolic subgroup. This generalizes the results for S U ( n ) of Agnihotri and the second author and Belkale on...

Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Arzu Boysal, Michèle Vergne (2009)

Annales de l’institut Fourier

Let P ( s ) be a family of rational polytopes parametrized by inequations. It is known that the volume of P ( s ) is a locally polynomial function of the parameters. Similarly, the number of integral points in P ( s ) is a locally quasi-polynomial function of the parameters. Paul-Émile Paradan proved a jump formula for this function, when crossing a wall. In this article, we give an algebraic proof of this formula. Furthermore, we give a residue formula for the jump, which enables us to compute it.

Parameter spaces for quadrics

Anders Thorup (1996)

Banach Center Publications

The parameter spaces for quadrics are reviewed. In addition, an explicit formula for the number of quadrics tangent to given linear subspaces is presented.

Parity sheaves, moment graphs and the p -smooth locus of Schubert varieties

Peter Fiebig, Geordie Williamson (2014)

Annales de l’institut Fourier

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 p -smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.

Partial flag varieties and preprojective algebras

Christof Geiß, Bernard Leclerc, Jan Schröer (2008)

Annales de l’institut Fourier

Let Λ be a preprojective algebra of type A , D , E , and let G be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ -module, and we introduce a mutation operation between complete rigid modules in Sub Q . This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to  G .

Periodic integrals and tautological systems

Bong H. Lian, Ruifang Song, Shing-Tung Yau (2013)

Journal of the European Mathematical Society

We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.

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