939 research outputs found
Reality property of discrete Wronski map with imaginary step
For a set of quasi-exponentials with real exponents, we consider the discrete
Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We
prove that if the coefficients of the discrete Wronskian are real and for every
its roots the imaginary part is at most |h|, then the complex span of this set
of quasi-exponentials has a basis consisting of quasi-exponentials with real
coefficients. This result is a generalization of the statement of the B. and M.
Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe
ansatz for the XXX model.Comment: Latex, 9 page
On reality property of Wronski maps
We prove that if all roots of the discrete Wronskian with step 1 of a set of
quasi-exponentials with real bases are real, simple and differ by at least 1,
then the complex span of this set of quasi-exponentials has a basis consisting
of quasi-exponentials with real coefficients. This result generalizes the B.
and M.Shapiro conjecture about spaces of polynomials.
The proof is based on the Bethe ansatz method for the XXX model.Comment: Latex, 20 page
Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)
We consider a tensor product V(b)= \otimes_{i=1}^n\C^N(b_i) of the Yangian
evaluation vector representations. We consider the action of the
commutative Bethe subalgebra on a -weight subspace
of weight . Here the Bethe algebra depends
on the parameters . We identify the -module
with the regular representation of the algebra of functions on a
fiber of a suitable discrete Wronski map. If , we study the action
of on a space of singular vectors of a certain
weight. Again, we identify the -module with the
regular representation of the algebra of functions on a fiber of another
suitable discrete Wronski map.
These results we announced earlier in relation with a description of the
quantum equivariant cohomology of the cotangent bundle of a partial flag
variety and a description of commutative subalgebras of the group algebra of a
symmetric group.Comment: Latex, 23 pages, misprints correcte
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