Solutions to the XXX type Bethe ansatz equations and flag varieties

E. Mukhin; A. Varchenko

Open Mathematics (2003)

  • Volume: 1, Issue: 2, page 238-271
  • ISSN: 2391-5455

Abstract

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We consider a version of the A N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl N+1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B N and C N. Populations of B N and C N type are isomorphic to the flag varieties of C N and B N types respectively.

How to cite

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E. Mukhin, and A. Varchenko. "Solutions to the XXX type Bethe ansatz equations and flag varieties." Open Mathematics 1.2 (2003): 238-271. <http://eudml.org/doc/268726>.

@article{E2003,
abstract = {We consider a version of the A N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl N+1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B N and C N. Populations of B N and C N type are isomorphic to the flag varieties of C N and B N types respectively.},
author = {E. Mukhin, A. Varchenko},
journal = {Open Mathematics},
keywords = {82B23; 14C17; 17B37},
language = {eng},
number = {2},
pages = {238-271},
title = {Solutions to the XXX type Bethe ansatz equations and flag varieties},
url = {http://eudml.org/doc/268726},
volume = {1},
year = {2003},
}

TY - JOUR
AU - E. Mukhin
AU - A. Varchenko
TI - Solutions to the XXX type Bethe ansatz equations and flag varieties
JO - Open Mathematics
PY - 2003
VL - 1
IS - 2
SP - 238
EP - 271
AB - We consider a version of the A N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl N+1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B N and C N. Populations of B N and C N type are isomorphic to the flag varieties of C N and B N types respectively.
LA - eng
KW - 82B23; 14C17; 17B37
UR - http://eudml.org/doc/268726
ER -

References

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  1. [1] N.M. Bogoliubov, A.G. Izergin, V.E. Korepin: Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. Zbl0787.47006
  2. [2] W. Fulton: Intersection Theory, Springer-Verlag, 1984. Zbl0541.14005
  3. [3] L.D. Faddeev: “Lectures on quantum inverse scattering method”, In: X-C. Song. ed.: Integrable Systems, Nankai Lectures on Math Phys. (1987), World Scientific, Singapore, 1990, pp. 23–70. 
  4. [4] L.D. Faddeev and L.A. Takhtajan: “Quantum inverse problem method and the Heisenberg XXZ model”, Russian Math. Survey, Vol. 34, (1979), pp. 11–68. 
  5. [5] Ph. Griffiths, J. Harris: Principles of algebraic geometry, A Whiley-Interscience Publication, 1994. 
  6. [6] E. Mukhin and A. Varchenko: “Critical Points of Master Functions and Flag Varieties”, math.QA/0209017, (2002), pp. 1–49. 
  7. [7] E. Mukhin and A. Varchenko: “The quantized Knizhnik-Zamolodchikov equation in tensor products of irreducible sl(2)-modules”, In: Calogero-Moser-Sutherland Models workshop, Springer, 2000, pp. 347–384. 
  8. [8] E. Mukhin and A. Varchenko: “Populations of solutions of the XXX Bethe equations associated to Kac-Moody algebras”, math.QA/0212092, (2002), pp. 1–7. 
  9. [9] V. Tarasov and A. Varchenko: “Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras”, Invent. math., Vol. 128, (1997), pp. 501–588. http://dx.doi.org/10.1007/s002220050151 Zbl0877.33013
  10. [10] V. Tarasov and A. Varchenko: “Completeness of Bethe vectors and Difference equations with Regular Singular points”, International Mathematics Research Notices, Vol. 13, (1995), pp. 637–669. http://dx.doi.org/10.1155/S1073792895000377 Zbl0860.17022

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