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Solutions to the XXX type Bethe ansatz equations and flag varieties

100%

Mukhin E., Varchenko A.

Open Mathematics

2003

tom 1

nr 2

238-271

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.

Miura opers and critical points of master functions

100%

Mukhin E., Varchenko A.

Open Mathematics

2005

tom 3

nr 2

155-182

Critical points of a master function associated to a simple Lie algebra $$\mathfrak{g}$$ come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra $$^t \mathfrak{g}$$ . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population.

Ograniczanie wyników

2 Open Mathematics

2 Mukhin E.

2 Varchenko A.

1 2005

1 2003

Wyniki wyszukiwania

Solutions to the XXX type Bethe ansatz equations and flag varieties

Miura opers and critical points of master functions