Skip to main content

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 227))

  • 5988 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. S. Abeasis and A. Del Fra, Degenerations for the representations of an equioriented quiver of type A m , Boll. Univ. Mat. Ital. Suppl. (1980), no. 2, 157–171.

    Google Scholar 

  2. Allen Knutson, Ezra Miller, and Mark Shimozono, Four positive formulae for type A quiver polynomials. arXiv:math.AG/0308142

    Google Scholar 

  3. Anders Skovsted Buch and William Fulton, Chern class formulas for quiver varieties, Invent. Math. 135 (1999), no. 3, 665–687.

    Article  MathSciNet  Google Scholar 

  4. A. V. Zelevinskiĭ, Two remarks on graded nilpotent classes, Usp. Mat. Nauk 40 (1985), no. 1(241), 199–200.

    Google Scholar 

  5. V. Lakshmibai and Peter Magyar, Degeneracy schemes, quiver schemes, and Schubert varieties, Int. Math. Res. Notices (1998), no. 12, 627–640.

    MathSciNet  Google Scholar 

  6. S. Abeasis, A. Del Fra, and H. Kraft, The geometry of representations of A m , Math. Ann. 256 (1981), no. 3, 401–418.

    Article  MathSciNet  Google Scholar 

  7. Allen Knutson, Ezra Miller, and Mark Shimozono, Four positive formulae for type A quiver polynomials. arXiv:math.AG/0308142 Section 4 and Theorem 6.16

    Google Scholar 

  8. Alexander Yong, On combinatorics of quiver component formulas, preprint, 2003. arXiv:math.CO/0307019

    Google Scholar 

  9. Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, revised edition, Cambridge Studies in Advanced Mathematics Vol. 39, Cambridge University Press, Cambridge, 1998. Theorems 2.1.3, and 2.1.9

    Google Scholar 

  10. William Fulton, Universal Schubert polynomials, Duke Math. J. 96 (1999), no. 3, 575–594.

    Article  MATH  MathSciNet  Google Scholar 

  11. Anders S. Buch, Andrew Kresch, Harry Tamvakis, and Alexander Yong, Schubert polynomials and quiver formulas, Duke Math. J. 122 (2004), no. 1, 125–143.

    MathSciNet  Google Scholar 

  12. Anders S. Buch, Andrew Kresch, Harry Tamvakis, and Alexander Yong, Grothendieck polynomials and quiver formulas, Amer. J. Math., to appear, 2004. arXiv:math.CO/0306389

    Google Scholar 

  13. Anders Skovsted Buch and William Fulton, Chern class formulas for quiver varieties, Invent. Math. 135 (1999), no. 3, 665–687.

    Article  MathSciNet  Google Scholar 

  14. William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci, Springer-Verlag, Berlin, 1998.

    Google Scholar 

  15. Laurent Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs Vol. 6, American Mathematical Society, Providence, RI, 2001, translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3.

    Google Scholar 

  16. Anders S. Buch, László M. Fehér, and Richárd Rimányi, Positivity of quiver coefficients through Thom polynomials, preprint, 2003. http://home.imf.au.dk/abuch/papers/

    Google Scholar 

  17. Alexander Yong, On combinatorics of quiver component formulas, preprint, 2003. arXiv:math.CO/0307019

    Google Scholar 

  18. Anders Skovsted Buch, Stanley symmetric functions and quiver varieties, J. Algebra 235 (2001), no. 1, 243–260.

    Article  MATH  MathSciNet  Google Scholar 

  19. Anders Skovsted Buch, Frank Sottile, and Alexander Yong, Quiver coefficients are Schubert structure constants, preprint, 2003. arXiv:math.CO/0311390

    Google Scholar 

  20. Anders Skovsted Buch, Grothendieck classes of quiver varieties, Duke Math. J. 115 (2002), no. 1, 75–103.

    Article  MATH  MathSciNet  Google Scholar 

  21. Anders Skovsted Buch, Alternating signs of quiver coefficients, preprint, 2003. arXiv:math.CO/0307014

    Google Scholar 

  22. Ezra Miller, Alternating formulas for K-theoretic quiver polynomials, Duke Math J., to appear. arXiv:math.CO/0312250

    Google Scholar 

  23. Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics Vol. 36, Cambridge University Press, Cambridge, 1997, corrected reprint of the 1995 original.

    Google Scholar 

  24. P. Gabriel and A. V. Roiter, Representations of finite-dimensional algebras, Springer-Verlag, Berlin, 1997, translated from the Russian.

    Google Scholar 

  25. László Fehér and Richárd Rimányi, Classes of degeneracy loci for quivers: the Thom polynomial point of view, Duke Math. J. 114 (2002), no. 2, 193–213.

    MathSciNet  Google Scholar 

  26. Grzegorz Bobiński and Grzegorz Zwara, Schubert varieties and representations of Dynkin quivers, Colloq. Math. 94 (2002), no. 2, 285–309.

    MathSciNet  Google Scholar 

  27. Harm Derksen and Jerzy Weyman, Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients, J. Amer. Math. Soc. 13 (2000), no. 3, 467–479 (electronic).

    Article  MathSciNet  Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

(2005). Minors in matrix products. In: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol 227. Springer, New York, NY. https://doi.org/10.1007/0-387-27103-1_17

Download citation

Publish with us

Policies and ethics