We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from [FR2] and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the corresponding normalized R-matrices has a pole.
We consider critical points of master functions associated with integral dominant weights of Kac-Moody algebras and introduce a generating procedure constructing new critical points starting from a given one. The set of all critical points constructed from a given one is called a population. We formulate a conjecture that a population is isomorphic to the flag variety of the Langlands dual Kac-Moody algebra and prove the conjectures for algebras sl N +1 , so 2N +1 , and sp 2N .We show that populations associated with a collection of integral dominant sl N +1weights are in one to one correspondence with intersection points of suitable Schubert cycles in a Grassmannian variety.where κ is the parameter of the KZ equations and A(t; z) is some explicitly written rational function with values in the tensor product [SV].
We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation
g
l
N
[
t
]
\mathfrak {gl}_N[t]
-modules and the scheme-theoretic intersection of suitable Schubert varieties. Moreover, we prove that the multiplicity space as a module over the Bethe algebra is isomorphic to the coregular representation of the scheme-theoretic intersection.
In particular, this result implies the simplicity of the spectrum of the Bethe algebra for real values of evaluation parameters and the transversality of the intersection of the corresponding Schubert varieties.
We consider the XXX -type and Gaudin quantum integrable models associated with the Lie algebra gl N . The models are defined on a tensor product M 1 ⊗ . . . ⊗ M n of irreducible gl N -modules. For each model, there exist N oneparameter families of commuting operators on M 1 ⊗ . . . ⊗ M n , called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.
We begin a study of the representation theory of quantum continuous gl ∞ , which we denote by E. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of E are labeled by a continuous parameter u ∈ C. The representation theory of E has many properties familiar from the representation theory of gl ∞ : vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from E to spherical double affine Hecke algebras SḦN for all N . A key step in this construction is an identification of a natural bases of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the K-theory of Hilbert schemes.
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