Skip to main content

Decomposable Tensors in Exterior Powers

  • Chapter
  • First Online:
  • 702 Accesses

Part of the book series: IMPA Monographs ((IMPA,volume 4))

Abstract

The geometrical picture presented in this chapter is essentially a reworking, within a finite-dimensional context, of the idea of writing Plücker equations for the infinite Grassmannian parametrizing the solutions of the KP hierarchy, classically due to Sato, Date, Jimbo, Kashiwara and Miwa [18, 19, 133, 134]. All this can be told in a purely algebraic way from the point of view of vertex operators [5, 73, 78, 79].

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
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

Learn about institutional subscriptions

References

  1. J. Abbott, A.M. Bigatti, G. Lagorio, CoCoA–5: a system for doing computations in commutative algebra (2015). Available via http://cocoa.dima.unige.it

  2. E. Arbarello, Sketches of KdV, in Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000). Contemporary Mathematics, vol. 312 (American Mathematical Society, Providence, 2002), pp. 9–69

    Google Scholar 

  3. E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Operators approach to the Kadomtsev-Petviashvili equation, Transformation groups for soliton equations III. J. Phys. Soc. Jpn. 50, 3806–3812 (1981)

    MathSciNet  MATH  Google Scholar 

  4. E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations. VI. KP hierarchies of orthogonal and symplectic type. J. Phys. Soc. Jpn. 50, 3813–3818 (1981)

    MathSciNet  MATH  Google Scholar 

  5. I. Frenkel, I. Penkov, V. Serganova, A categorification of the boson–fermion correspondence via representation theory of \(\mathfrak{s}l(\infty )\) (2014). arXiv:1405.7553

    Google Scholar 

  6. L. Gatto, P. Salehyan, The boson–fermion correspondence from linear ODEs. J. Algebra 415, 162–183 (2014). http://www.sciencedirect.com/science/article/pii/S0021869314003263. doi:10.1016/j.jalgebra.2014.05.030

    Google Scholar 

  7. L. Gatto, P. Salehyan, Vertex operators arising from linear ODEs (2013). arXiv:1310.5132v1

    Google Scholar 

  8. L. Gatto, P. Salehyan, On the Plücker equations defining the Grassmann Cone (2016). arXiv:1603.00510v1

    Google Scholar 

  9. M. Gekhtman, A. Kasman, On KP generators and the geometry of the HBDE. J. Geom. Phys. 56 (2), 282–309 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley Classics Library, New York, 1994)

    Book  MATH  Google Scholar 

  11. J. Harris, Algebraic Geometry. GTM, vol. 133 (Springer, New York, 1992)

    Google Scholar 

  12. W. Hodge, D. Pedoe, Methods of Algebraic Geometry, vol. II, Book III (Cambridge University Press, Cambridge, 1947)

    MATH  Google Scholar 

  13. N. Jing, N. Rozhkovskaya, Vertex Operators Arising from Jacobi–Trudy Identities (2015). arXiv:1411.4725v2

    Google Scholar 

  14. V.G. Kac, Vertex Algebras for Beginners. University Lecture Series, vol. 10 (American Mathematical Society, Providence, 1996)

    Google Scholar 

  15. V.G. Kac, A.K. Raina, N. Rozhkovskaya, Highest Weight Representations of Infinite Dimensional Lie Algebras. Advanced Series in Mathematical Physics, vol. 29, 2nd edn. (World Scientific, Singapore, 2013)

    Google Scholar 

  16. V.G. Kac, M. Wakimoto, Exceptional hierarchies of soliton equations, in Theta Functions, Bowdoin 1987, ed. by L. Ehrenpreis, R.C. Gunning. Proceedings of Symposia in Pure Mathematics, vol. 49, Part 1, AMS, Providence, Rhode Island (1989), pp. 191–238

    Google Scholar 

  17. A. Kasman, Glimpses on solitons. J. Am. Math. Soc. 20 (4), 1079–1089 (2007)

    Article  Google Scholar 

  18. A. Kasman, K. Pedings, A. Reiszl, T. Shiota, Universality of rank 6 Plücker relations and Grassmann cone preserving maps. Proc. Am. Math. Soc. 136 (1), 77–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. I.G. Macdonald, Symmetric Functions and Hall Polynomials (Clarendon Press, Oxford, 1979)

    MATH  Google Scholar 

  20. M. Marcus, Finite Dimensional Multilinear Algebra, Part II (Marcel Dekker, New York, 1975)

    MATH  Google Scholar 

  21. S. Mukai, Curves and Grassmannians, in Algebraic Geometry and Related Topics (Inchon, 1992). Lecture Notes Algebraic Geometry, vol. I (Cambridge University Press, Cambridge, 1993), pp. 19–40

    Google Scholar 

  22. M. Sato, Soliton equations as dynamical systems on infinite dimensional grassmann manifolds. RIMS Kokioroku 439, 30–46 (1981)

    Google Scholar 

  23. M. Sato, The KP hierarchy and infinite-dimensional Grassmann manifolds, in Theta Functions, Bowdoin 1987, ed. by L. Ehrenpreis, R.C. Gunning. Proceedings of Symposia in Pure Mathematics, vol. 49, Part 1, AMS, Providence, Rhode Island (1989), pp. 51–66

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Gatto, L., Salehyan, P. (2016). Decomposable Tensors in Exterior Powers. In: Hasse-Schmidt Derivations on Grassmann Algebras. IMPA Monographs, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-31842-4_6

Download citation

Publish with us

Policies and ethics