A generalization of Fulton's conjecture for arbitrary groups
We prove a generalization of Fulton's conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.
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arXiv subjects
Prakash Belkale
Publications and source records attributed to Prakash Belkale.
Orthogonal bundles, theta characteristics and the symplectic strange duality
A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the moduli space of curves (for Hitchin's connection). The symplectic strange duality conjecture, conjectured by Beauville is shown to hold for all curves of genus at least two, by using Abe's proof of the conjecture for generic curves, and the above monodromy result.
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Strange duality and the Hitchin/WZW connection
For a compact Riemann surface X of positive genus, the space of sections of certain theta bundle on moduli of bundles of rank r and level k admits a natural map to (the dual of) a similar space of sections of rank k and level r (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as X varies in a family. We prove that this map is (projectively) flat.
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Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups
We consider the eigenvalue problem and the associated intersection theory of homogenous spaces for the symplectic and odd orthogonal groups. We solve the Horn and saturation problems for these classical groups.
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The strange duality conjecture for generic curves
For X a compact Riemann surface of positive genus, the strange duality conjecture predicts that the space of sections of certain theta bundle on moduli of bundles of rank r and level k is naturally dual to a similar space of sections of rank k and level r. We prove this conjecture for X generic in the moduli space of curves of a given genus.
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Geometric Proof of a Conjecture of Fulton
We give a geometric proof of a conjecture of W. Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).
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Geometric Proofs of Horn and Saturation Conjectures
We provide a geometric proof of the Schubert calculus interpretation of the Horn conjecture, and show how the saturation conjecture follows from it. The geometric proof gives a strengthening of Horn and saturation conjectures. We also establish transversality theorems for Schubert calculus in non-zero characteristic. Some parts of the version posted in Nov 2002 (concerning explicit invariants constructed from Schubert calculus) have been removed from this version. They have appeared separately in IMRN 2004, no. 69, pages 3709--3721 " Invariant theory of GL(n) and Intersection theory of Grassmannians"
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Quantum Generalization of the Horn Conjecture
We prove a theorem which implies a quantum (multiplicative) analogue of the Horn conjecture, and also of the saturation conjecture. We obtain transversality statements for quantum schubert calculus in any characteristic and also determine the smallest power of q in an arbitrary (small quantum) product of Schubert varieties in a Grassmannian.
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Multiplicity of critical points of master functions and Schubert calculus
In [MV], some correspondences were defined between critical points of master functions associated to sl_{N+1} and subspaces of C[x] with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus.
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Eigenvalue problem and a new product in cohomology of flag varieties
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).
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Irredundance in Eigenvalue Problems
We show the irredundancy, necessity and sufficiency of inequalities corresponding to intersections whose ``Witten Bundle'' is polyrigid for the product of unitary matrices problem.
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Periods and Igusa Zeta functions
We show that coefficients in the Laurent series of Igusa Zeta functions are periods. This will be used in a subsequent paper (by P. Brosnan) to show that certain numbers occurring in study of Feynman amplitudes (upto gamma factors) are periods.
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Transformation formulas in Quantum Cohomology
The article discusses an action of the center of G on the quantum cohomology of G/P's constructed geometrically. It is shown how to recover Bertram's Quantum Schubert Calculus from this action, and also a refinement of a formula of Fulton and Woodward for the terms of the smallest order in the product of two Schubert varieties in the Quantum cohomology of a Grassmannian. The results date to July 2001, and are needed to pose a Quantum analogue of the Horn and Saturation Conjectures.
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Matroids, motives and conjecture of Kontsevich
Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes essentially generate all arithmetic of schemes of finite type over the integers.
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