Solutions to the XXX type Bethe ansatz equations and flag varieties
E. Mukhin ; A. Varchenko
Open Mathematics, Tome 1 (2003), p. 238-271 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2003-01-01
EUDML-ID : urn:eudml:doc:268726 
@article{bwmeta1.element.doi-10_2478_BF02476011,
     author = {E. Mukhin and A. Varchenko},
     title = {Solutions to the XXX type Bethe ansatz equations and flag varieties},
     journal = {Open Mathematics},
     volume = {1},
     year = {2003},
     pages = {238-271},
     zbl = {1029.82008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02476011}
}

E. Mukhin; A. Varchenko. Solutions to the XXX type Bethe ansatz equations and flag varieties. Open Mathematics, Tome 1 (2003) pp. 238-271. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02476011/

[1] N.M. Bogoliubov, A.G. Izergin, V.E. Korepin: Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. | Zbl 0787.47006

[2] W. Fulton: Intersection Theory, Springer-Verlag, 1984. | Zbl 0541.14005

[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] 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] Ph. Griffiths, J. Harris: Principles of algebraic geometry, A Whiley-Interscience Publication, 1994.

[6] E. Mukhin and A. Varchenko: “Critical Points of Master Functions and Flag Varieties”, math.QA/0209017, (2002), pp. 1–49.

[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] 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] 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 | Zbl 0877.33013

[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 | Zbl 0860.17022