Geodesic
Quantum generalization of the Horn conjecture
Belkale, Prakash1
1Department of Mathematics, University of North Carolina–Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, North Carolina 27599
Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 365-408

Voir la notice de l'article provenant de la source American Mathematical Society

The following results are presented in this paper: (1) a quantum (multiplicative) generalization of the Horn conjecture which gives a recursive characterization of the possible eigenvalues of a product of unitary matrices, (2) the saturation conjecture for the fusion structure coefficients for SL$(n)$, (3) transversality statements for quantum Schubert calculus in any characteristic for the ordinary Grassmannians, (4) determination of the smallest power of $q$ in an arbitrary (small quantum) product of Schubert varieties in an ordinary Grassmannian.
DOI : 10.1090/S0894-0347-07-00584-X

Belkale, Prakash  1

1 Department of Mathematics, University of North Carolina–Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, North Carolina 27599 
Belkale, Prakash. Quantum generalization of the Horn conjecture. Journal of the American Mathematical Society, Tome 21 (2008) no. 2, pp. 365-408. doi: 10.1090/S0894-0347-07-00584-X
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[1] Agnihotri, S., Woodward, C. Eigenvalues of products of unitary matrices and quantum Schubert calculus Math. Res. Lett. 1998 817 836

[2] Beauville, Arnaud Conformal blocks, fusion rules and the Verlinde formula 1996 75 96

[3] Belkale, Prakash Local systems on ℙ¹-𝕊 for 𝕊 a finite set Compositio Math. 2001 67 86

[4] Belkale, Prakash Transformation formulas in quantum cohomology Compos. Math. 2004 778 792

[5] Belkale, Prakash Geometric proofs of Horn and saturation conjectures J. Algebraic Geom. 2006 133 173

[6] Bertram, Aaron Quantum Schubert calculus Adv. Math. 1997 289 305

[7] Biswas, Indranil A criterion for the existence of a parabolic stable bundle of rank two over the projective line Internat. J. Math. 1998 523 533

[8] Eisenbud, David Commutative algebra 1995

[9] Fulton, W., Pandharipande, R. Notes on stable maps and quantum cohomology 1997 45 96

[10] Fulton, W., Woodward, C. On the quantum product of Schubert classes J. Algebraic Geom. 2004 641 661

[11] Fulton, William Eigenvalues, invariant factors, highest weights, and Schubert calculus Bull. Amer. Math. Soc. (N.S.) 2000 209 249

[12] Fulton, William Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients Linear Algebra Appl. 2000 23 36

[13] Gepner, Doron Fusion rings and geometry Comm. Math. Phys. 1991 381 411

[14] Hartshorne, Robin Algebraic geometry 1977

[15] Horn, Alfred Eigenvalues of sums of Hermitian matrices Pacific J. Math. 1962 225 241

[16] Kleiman, Steven L. The transversality of a general translate Compositio Math. 1974 287 297

[17] Klyachko, Alexander A. Stable bundles, representation theory and Hermitian operators Selecta Math. (N.S.) 1998 419 445

[18] Knutson, Allen, Tao, Terence The honeycomb model of 𝐺𝐿_{𝑛}(𝐶) tensor products. I. Proof of the saturation conjecture J. Amer. Math. Soc. 1999 1055 1090

[19] Kollár, János Rational curves on algebraic varieties 1996

[20] Kontsevich, M., Manin, Yu. Gromov-Witten classes, quantum cohomology, and enumerative geometry [ MR1291244 (95i:14049)] 1997 607 653

[21] Le Potier, J. Lectures on vector bundles 1997

[22] Matsumura, Hideyuki Commutative ring theory 1989

[23] Mehta, V. B., Seshadri, C. S. Moduli of vector bundles on curves with parabolic structures Math. Ann. 1980 205 239

[24] Mumford, David Lectures on curves on an algebraic surface 1966

[25] Pauly, Christian Espaces de modules de fibrés paraboliques et blocs conformes Duke Math. J. 1996 217 235

[26] Postnikov, Alexander Affine approach to quantum Schubert calculus Duke Math. J. 2005 473 509

[27] Sottile, Frank Pieri’s formula via explicit rational equivalence Canad. J. Math. 1997 1281 1298

[28] Sottile, Frank Real rational curves in Grassmannians J. Amer. Math. Soc. 2000 333 341

[29] Sottile, Frank Elementary transversality in the Schubert calculus in any characteristic Michigan Math. J. 2003 651 666

[30] Vakil, Ravi Schubert induction Ann. of Math. (2) 2006 489 512

[31] Witten, Edward The Verlinde algebra and the cohomology of the Grassmannian 1995 357 422

[32] Yong, Alexander Degree bounds in quantum Schubert calculus Proc. Amer. Math. Soc. 2003 2649 2655

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