On a Conjectured Formula for Quiver Varieties
Anders Skovsted Buch
DOI: 10.1023/A:1011245531325
Abstract
In A.S. Buch and W. Fulton [ Invent. Math. 135 (1999), 665-687] a formula for the cohomology class of a quiver variety is proved. This formula writes the cohomology class of a quiver variety as a linear combination of products of Schur polynomials. In the same paper it is conjectured that all of the coefficients in this linear combination are non-negative, and given by a generalized Littlewood-Richardson rule, which states that the coefficients count certain sequences of tableaux called factor sequences. In this paper I prove some special cases of this conjecture. I also prove that the general conjecture follows from a stronger but simpler statement, for which substantial computer evidence has been obtained. Finally I will prove a useful criterion for recognizing factor sequences.
Pages: 151–172
Keywords: quiver varieties; Littlewood-Richardson rule; Schur functions; Young tableaux
Full Text: PDF
References
1. A.S. Buch, “Chern class formulas for degeneracy loci,” in Proc. Formal Power Series and Alg. Comb., Fields Institute, Toronto, 1998, pp. 103-113.
2. A.S. Buch, “Combinatorics of degeneracy loci,” Ph.D. Thesis, The University of Chicago, 1999.
3. A.S. Buch, “Stanley symmetric functions and quiver varieties,” Journal of Algebra. 235 (2001), 243-260.
4. A.S. Buch and W. Fulton, “Chern class formulas for quiver varieties,” Invent. Math. 135 (1999), 665-687.
5. S. Fomin and C. Greene, “Noncommutative Schur functions and their applications,” Discrete Math. 193 (1998), 179-200.
6. W. Fulton, Young Tableaux, Cambridge University Press, 1997.
7. W. Fulton, “Universal Schubert polynomials,” Duke Math. J. 96 (1999), 575-594.
8. A.M. Garsia and S.C. Milne, “Method for constructing bijections for classical partition identities,” in Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 2026-2028.
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13. R.P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge University Press, 1999.
2. A.S. Buch, “Combinatorics of degeneracy loci,” Ph.D. Thesis, The University of Chicago, 1999.
3. A.S. Buch, “Stanley symmetric functions and quiver varieties,” Journal of Algebra. 235 (2001), 243-260.
4. A.S. Buch and W. Fulton, “Chern class formulas for quiver varieties,” Invent. Math. 135 (1999), 665-687.
5. S. Fomin and C. Greene, “Noncommutative Schur functions and their applications,” Discrete Math. 193 (1998), 179-200.
6. W. Fulton, Young Tableaux, Cambridge University Press, 1997.
7. W. Fulton, “Universal Schubert polynomials,” Duke Math. J. 96 (1999), 575-594.
8. A.M. Garsia and S.C. Milne, “Method for constructing bijections for classical partition identities,” in Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 2026-2028.
9. A. Lascoux and M.-P. Sch\ddot utzenberger, “Le mono\ddot ide plaxique,” in Noncommutative Structures in Algebra and Geometric Combinatorics (Naples, 1978), Vol. 109 of Quad. “Ricerca Sci.”. Rome, 1981, pp. 129-156.
10. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
11. B.E. Sagan, The Symmetric Group, Brooks/Cole Advanced Books & Software, 1991.
12. M.-P. Sch\ddot utzenberger, “La correspondance de Robinson,” in Lecture Notes in Math. Vol. 579, pp. 59-113, 1977.
13. R.P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge University Press, 1999.