# Solutions to the XXX type Bethe ansatz equations and flag varieties

Open Mathematics (2003)

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

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topE. 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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- [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 Zbl0877.33013
- [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 Zbl0860.17022

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