arXiv subjects
Publications and source records attributed to E. Mukhin.
We establish the method of Bethe ansatz for the XXZ type model obtained from the R-matrix associated to quantum toroidal gl(1). We do that by using shuffle realizations of the modules and by showing that the Hamiltonian of the model is obtained from a simple multiplication operator by taking an appropriate quotient. We expect this approach to be applicable to a wide variety of models.
We give lower bounds for the numbers of real solutions in problems appearing in Schubert calculus in the Grassmannian Gr(n,d) related to osculating flags. It is known that such solutions are related to Bethe vectors in the Gaudin model associated to gl(n). The Gaudin Hamiltonians are selfadjoint with respect to a nondegenerate indefinite Hermitian form. Our bound comes from the computation of the signature of that form.
We introduce families of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra of the symmetric group. Bethe subalgebras are deformations of the Gelfand-Zetlin subalgebra. We describe various properties of Bethe subalgebras.
We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y(gl_N)$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra $B^q \subset Y(gl_N)$ on a $gl_N$-weight subspace $V(b)_λ\subset V(b)$ of weight $λ$. Here the Bethe algebra depends on the parameters $q=(q_1,...,q_N)$. We identify the $B^q$-module $V(b)_λ$ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. If $q=(1,...,1)$, we study the action of $B^{q=1}$ on a space $V(b)^{sing}_λ$ of singular vectors of a certain weight. Again, we identify the $B^{q=1}$-module $V(b)^{sing}_λ$ 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.
We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.
We define and study representations of quantum toroidal $gl_n$ with natural bases labeled by plane partitions with various conditions. As an application, we give an explicit description of a family of highest weight representations of quantum affine $gl_n$ with generic level.
Let g be a simple Lie algebra. We consider the category O-hat of those modules over the affine quantum group Uq(g-hat) whose Uq(g)-weights have finite multiplicity and lie in a finite union of cones generated by negative roots. We show that many properties of the category of the finite-dimensional representations naturally extend to the category O-hat. In particular, we develop the theory of q-characters and define the minimal affinizations of parabolic Verma modules. In types ABCFG we classify these minimal affinizations and conjecture a Weyl denominator type formula for their characters.
We study polynomials p(x) satisfying a differential equation of the form p"(x)-h'(x)p'(x)+H(x)p(x)=0, where h(x)=x^3/3+ax. We prove a conjecture of A. Eremenko and A. Gabrielov.
We use the theory of q-characters to establish a number of short exact sequences in the category of finite-dimensional representations of the quantum affine groups of types A and B. That allows us to introduce a set of 3-term recurrence relations which contains the celebrated T-system as a special case.
In third paper of the series we construct a large family of representations of the quantum toroidal $\gl_1$ algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand-Zetlin type basis for a class of irreducible lowest weight $\gl_\infty$-modules.
We give a set of sufficient conditions for a Laurent polynomial to be the q-character of a finite-dimensional irreducible representation of a quantum affine group. We use this result to obtain an explicit path description of q-characters for a class of modules in type B. In particular, this proves a conjecture of Kuniba-Ohta-Suzuki.
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.
We begin a study of the representation theory of quantum continuous $\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. 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 $\mathcal E$ are labeled by a continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal E$ has many properties familiar from the representation theory of $\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from $\mathcal E$ to spherical double affine Hecke algebras $S\ddot H_N$ 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.
We construct a family of irreducible representations of the quantum continuous $gl_\infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.
Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master function and show that the action of this algebra on itself is isomorphic to the action of the Bethe algebra on a suitable subspace of the multiplicity space. As a byproduct we prove that the Bethe vectors corresponding to different critical points of the master function are linearly independent and, in particular, nonzero.
We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of N-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. A new object is the Bethe vector averaging maps.
We identify the Bethe algebra of the Gaudin model associated to gl(N) acting on a suitable representation with the center of the rational Cherednik algebra and with the algebra of regular functions on the Calogero-Moser space.
We show that the Gaudin Hamiltonians H_1,...,H_n generate the Bethe algebra of the n-fold tensor power of the vector representation of gl_N. Surprisingly the formula for the generators of the Bethe algebra in terms of the Gaudin Hamiltonians does not depend on N. Moreover, this formula coincides with Wilson's formula for the stationary Baker-Akhiezer function on the adelic Grassmannian.