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References

  1. S. Abeasis and A. Del Fra, Degenerations for the representations of an equioriented quiver of type Am, Boll. Un. Mat. Ital. Suppl. (1980), no. 2, 157–171. MR 84e:16019

    Google Scholar 

  2. M. Brion, Lectures on the geometry of flag varieties, this volume.

    Google Scholar 

  3. _____, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), no. 1, 137–159, Special issue in celebration of Claudio Procesi’s 60th birthday. MR 2003m:14017

    Google Scholar 

  4. A.S. Buch, Grothendieck classes of quiver varieties, Duke Math. J. 115 (2002), no. 1, 75–103. MR 2003m:14018

    Article  Google Scholar 

  5. _____, A Littlewood-Richardson rule for the K-theory of Grassmannians, Acta Math. 189 (2002), no. 1, 37–78. MR 2003j:14062

    Google Scholar 

  6. _____, Alternating signs of quiver coefficients, preprint, 2003.

    Google Scholar 

  7. A.S. Buch and W. Fulton, Chern class formulas for quiver varieties, Invent. Math. 135 (1999), no. 3, 665–687. MR 2000f:14087

    Article  Google Scholar 

  8. A.S. Buch, A. Kresch, H. Tamvakis, and A. Yong, Grothendieck polynomials and quiver formulas, To appear in Amer. J. Math., 2003.

    Google Scholar 

  9. S. Fomin and A.N. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, Proc. Formal Power Series and Alg. Comb. (1994), 183–190.

    Google Scholar 

  10. _____, The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 123–143.

    Article  Google Scholar 

  11. W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 85k:14004

    Google Scholar 

  12. _____, Flags, Schubert polynomials. degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420. MR 93e:14007

    Article  Google Scholar 

  13. _____, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997, With applications to representation theory and geometry. MR 99f:05119

    Google Scholar 

  14. W. Fulton and A. Lascoux, A Pieri formula in the Grothendieck ring of a flag bundle, Duke Math. J. 76 (1994), no. 3, 711–729. MR 96j:14036

    Article  Google Scholar 

  15. W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998, Appendix J by the authors in collaboration with I. Ciocan-Fontanine. MR 99m:14092

    Google Scholar 

  16. A. Knutson, E. Miller, and M. Shimozono, Four positive formulas for type A quiver polynomials, preprint, 2003.

    Google Scholar 

  17. A. Knutson and R. Vakil, manuscript in preparation.

    Google Scholar 

  18. V. Lakshmibai and P. Magyar, Degeneracy schemes, quiver schemes, and Schubert varieties, Internat. Math. Res. Notices (1998), no. 12, 627–640. MR 99g:14065

    Article  Google Scholar 

  19. A. Lascoux, Anneau de Grothendieck de la variété de drapeaux, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhüuser Boston, Boston, MA, 1990, pp. 1–34. MR 92j:14064

    Google Scholar 

  20. _____, Transition on Grothendieck polynomials, Physics and combinatorics, 2000 (Nagoya), World Sci. Publishing, River Edge, NJ, 2001, pp. 164–179. MR 2002k:14082

    Google Scholar 

  21. A. Lascoux and M.-P. Schützenberger, Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 11, 629–633. MR 84b:14030

    Google Scholar 

  22. C. Lenart, Combinatorial aspects of the K-theory of Grassmannians, Ann. Comb. 4 (2000), no. 1, 67–82. MR 2001j:05124

    Google Scholar 

  23. D.E. Littlewood and A.R. Richardson, Group characters and algebra, Phil. Trans. R. Soc., A 233 (1934), 99–141.

    Google Scholar 

  24. E. Miller, Alternating formulae for K-theoretic quiver polynomials, preprint, 2003.

    Google Scholar 

  25. P. Pragacz, Enumerative geometry of degeneracy loci, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 3, 413–454. MR 90e:14004

    Google Scholar 

  26. A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math. 80 (1985), no. 2, 283–294. MR MR788411 (87d:14044)

    Article  Google Scholar 

  27. R. Vakil, A geometric Littlewood-Richardson rule, preprint, 2003.

    Google Scholar 

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Buch, A.S. (2005). Combinatorial K-theory. In: Pragacz, P. (eds) Topics in Cohomological Studies of Algebraic Varieties. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7342-3_3

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