Eigenvalue problem and a new product in cohomology of flag varieties
By P. Belkale, Shrawan Kumar. 2004
129 citations recorded by the source corpus.
Let G be a connected semisimple complex algebraic group and let P be a parabolic subgroup. In this paper we define a new (commutative and associative) product on the cohomology of the homogenous spaces G/P and use this to give a more efficient solution of the eigenvalue problem and also for the problem of determining the existence of G-invariants in the tensor product of irreducible representations of G. On the other hand, we show that this new product is intimately connected with the Lie algebra cohomology of the nil-radical of P via some works of Kostant and Kumar. We also initiate a uniform study of the geometric Horn problem for an arbitrary group $G$ by obtaining two (a priori) different sets of necessary recursive conditions to determine when a cohomology product of Schubert classes in G/P is non-zero. Hitherto, this was studied largely only for the group \SL(n).
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6a273425806a2def6d5aede32ff48135fa771ef6
How researchers see this paper
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In Discrete Mathematics and Combinatorics
- B. Leeb
- Hugh Thomas
- John Millson
- Alexander Yong
- Nicolas Ressayre
- Alexander A. Klyachko
- I. Reiten
- Lingyan Guo
In Geometry and Topology
- B. Leeb
- John Millson
- Hugh Thomas
- Alexander Yong
- Nicolas Ressayre
- Alexander A. Klyachko
- Michael Kapovich
- J. Porti
In Mathematical Physics
- Hugh Thomas
- Alexander Yong
- John Millson
- Nicolas Ressayre
- Alexander A. Klyachko
- B. Leeb
- Michael Kapovich
- J. Porti
Citations in the corpus
- A combinatorial rule for (co)minuscule Schubert calculus (2006, background): “In particular, our rule may be interpreted in terms of emerging and classical connections between Schubert calculus and (geometric) representation theory: for example, in connection to the geometric Satake correspondence of Ginzburg, Mirkovi´c-Vilonen and others, see, e.g., [MiVi99], and separately,”
- Geometric invariant theory and the generalized eigenvalue problem (2007, methodology): “In [3], Belkale-Kumar obtained a smaller list which also characterizes this cone in the case when G is diagonally embedded in Ĝ = G .”
- Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity (2003, background): “Remark 1.4. Recently, a smaller system of inequalities for Pn(X) was described by Belkale and Kumar [ BK ]; it is based on a deformation of the (co)homology rings of the generalized Grassmannians.”
- QUANTUM MARGINAL PROBLEM AND REPRESENTATIONS OF THE SYMMETRIC GROUP (2004, background): “In this case Belkale and Kumar [3] have a refined procedure which gives independent inequalities, at least in some examples.”