Geometric proofs of Horn and saturation conjectures
Author:
Prakash Belkale
Journal:
J. Algebraic Geom. 15 (2006), 133-173
DOI:
https://doi.org/10.1090/S1056-3911-05-00420-0
Published electronically:
August 23, 2005
MathSciNet review:
2177198
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Abstract |
References |
Additional Information
Abstract: 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 nonzero characteristic.
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bell P. Belkale, Quantum generalization of Horn and Saturation conjectures. preprint, math.AG/0303013.
bro P. Belkale and P. Brosnan. Matroids, Motives and a Conjecture of Kontsevich. Duke Math Journal, vol 116, No. 1 (2003) 147–188.
eisen D. Eisenbud. Commutative Algebra with a view towards algebraic geometry. Graduate Texts in Mathematics 150, Springer-Verlag 1995.
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ful W. Fulton. Eigenvalues of Hermitian matrices. Bull. Amer. Math. Soc. (N.S) 37 (2000) no.3, 209–249.
ful2 W. Fulton. Eigenvalues of Majorized Hermitian Matrices and Littlewood-Richardson Coefficients. Lin. Alg. Appl 319 (2000), 23–36.
int W. Fulton. Intersection Theory. Springer-Verlag Berlin 1998.
young W. Fulton. Young Tableaux. London Math Society, Student Texts, 35, 1997.
EGA A. Grothendieck, J. Dieudonné. Eléments de Géométrie Algébrique I. Grundlehren 166, Springer-Verlag, Heidelberg 1971.
hart R. Hartshorne Algebraic Geometry. Graduate texts in mathematics 52, Springer-Verlag, 1977.
Horn A. Horn. Eigenvalues of sums of Hermitian matrices. Pacific J. Math. (12), 1962, 225–241.
kl S.L. Kleiman. The transversality of a general translate. Compositio Math. 28 (1974), 287–297.
Klyachko A. Klyachko. Stable bundles, representation theory and Hermitian operators. Selecta Math. 4(1998), 419–445.
KT A. Knutson, T. Tao. The Honeycomb model of $\textrm {GL}_ n(C)$ tensor products. I. Proof of the Saturation conjecture. J. Amer. Math. Soc. 12(1999) no.4, 1055–1090.
mat H. Matsumura. Commutative ring theory. Cambridge studies in advanced mathe- matics 8.
MS V. Mehta and C.S. Seshadri. Moduli of vector bundles on curves with parabolic structures. Math Ann. 248 (1980), no 3, 205–239.
purbhoo K. Purbhoo. A vanishing and a nonvanishing condition for Schubert calculus on $G/B$. Preprint, math.CO/0304070.
sottile F. Sottile. Pieri’s Formula via explicit rational equivalence. Canad. J. Math. 49 (1997), 1281–1298.
vakil R. Vakil. Schubert Induction. math.AG/0302296.
Additional Information
Prakash Belkale
Affiliation:
Department of Mathematics, UNC-Chapel Hill, CB #3250, Phillips Hall, Chapel Hill, North Carolina 27599
MR Author ID:
684040
Email:
belkale@email.unc.edu
Received by editor(s):
January 16, 2005
Received by editor(s) in revised form:
May 20, 2005
Published electronically:
August 23, 2005
Additional Notes:
The author was partially supported by NSF grant DMS-0300356
