Solutions to the XXX type Bethe ansatz equations and flag varieties

• Autorzy: E. Mukhin , A. Varchenko
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

algebraic Bethe anzatz; Schubert Calculus; descrete Wronskian; 82B23; 14C17; 17B37

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2003
• Tom: 1
• Numer: 2
• Strony: 238-271

Daty

wydano

2003-06-01

online

2003-06-01

Twórcy

autor

E. Mukhin

• Indiana University-Purdue University Indianapolis

autor

A. Varchenko

• University of North Carolina at Chapel Hill

Bibliografia

• [1] N.M. Bogoliubov, A.G. Izergin, V.E. Korepin: Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993.
• [2] W. Fulton: Intersection Theory, Springer-Verlag, 1984.
• [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
• [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

Identyfikatory

DOI

10.2478/BF02476011