Scalar products in models with a GL(3) trigonometric R-matrix: Highest coefficient

We study quantum integrable models with a GL(3) trigonometric R-matrix solvable by the nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of bilinear combinations of the highest coefficients. We show that there exist two different highest coefficients in the models with a GL(3) trigonometric R-matrix. We obtain various representations for the highest coefficients in terms of sums over partitions. We also prove several important properties of the highest coefficients, which are necessary for evaluating the scalar products.     

Determinant Representations for Scalar Products in the Algebraic Bethe Ansatz

Theoretical and Mathematical Physics - We study integrable models with gl(2|1) symmetry that are solvable by the nested algebraic Bethe ansatz. We obtain a new determinant representation for scalar...     

On representations of GL(n) distinguished by GL(n − 1) over a p-adic field

For a nonarchimedean local field F, let GL(n):= GL(n, F) and GL(n?1) be embedded in GL(n) via g ? ( _{0 1} ^{g 0} ). Let π be an irreducible admissible representation of GL(n) for n ≥ 3. We prove that π is GL(n ? 1)-distinguished if and only if the Langlands parameter L(π) associated to π by the Local Langlands Correspondence has a subrepresentation L(11 _n?2) of dimension n?2 corresponding to the trivial representation of GL(n?2) such that the two-dimensional quotient L(π)/L(11 _n?2) corresponds either to an infinite-dimensional representation or the one-dimensional representations $\nu ^{ \pm (\tfrac{{n - 2}}{2})} $ of GL(2). We also prove that, for a parabolic subgroup P of GL(n) and an irreducible admissible representation ρ of the Levi subgroup of P, $\dim _\mathbb{C} (Hom_{GL(n - 1)} [ind_P^{GL(n)} (\rho ),\mathbb{I}_{n - 1} ]) \leqslant 2$ . For the standard Borel subgroup B _n of GL(n) and characters x _i of GL(1), we classify all representations ξ of the form $ind_{B_n }^{GL(n)} (\chi _1 \otimes \cdots \otimes \chi _n )$ for which $\dim _\mathbb{C} (Hom_{GL(n - 1)} [\xi ,\mathbb{I}_{n - 1} ]) = 2$ .     

Quantum volterra model and the universalR-matrix

We prove that an integrable system solved by the quantum inverse scattering method can be described by a purely algebraic object (universal R-matrix) and a proper algebraic representation. For the quantum Volterra model, we construct the L-operator and the fundamental R-matrix from the universal R-matrix for the quantum affine algebra U_q(ŝl₂) and the q-oscillator representation for it. Thus, there is an equivalence between an integrable system with the symmetry algebra A and the representation of this algebra. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 384–396, December, 1997.     

Algebraic Bethe Ansatz for a seven-vertex model

The paper is devoted to construction of algebraic Bethe Ansatz for a seven-vertex model. R-matrix of the system is obtained by means of twist from the six-vertex model considered by us earlier. The presence of the seventh nonzero element in the R-matrix complicates the situation. In particular, commutation relations of elements of the monodromy matrix become more complicated in comparison with the six-vertex model. We construct algebraic Bethe Ansatz by introducing a new operator that is the difference of two operators on the main diagonal of the monodromy matrix. The eigenstates and spectrum of the system are found. This is the first step on the way of comparison of systems with six-and seven-vertex R-matrices, respectively. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 178–186.     

Nested Bethe Ansatz for the RTT Algebra of sp(4) Type

Theoretical and Mathematical Physics - We show how to formulate the algebraic nested Bethe ansatz for RTT algebras with an R-matrix of the sp(4) type. We obtain the Bethe vectors and Bethe...     

Explicit formulas for the waldspurger and bessel models

This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by Waldspurger. Ifn=2, they were considered by Novodvorsky and Piatetski-Shapiro, who called them Bessel models. In the works of these authors, uniqueness of the models is established; in this paper functional equations and explicit formulas for them are obtained. As a global application, the Bessel period of the Eisenstein series on SO(2n+1) formed with a cuspidal automorphic representation π on GL(n) is computed—it is shown to be a product of L-series. This generalizes work of Böcherer and Mizumoto forn=2 and base field ?, and puts it in a representation-theoretic context. In an appendix by M. Furusawa, a new Rankin-Selberg integral is given for the standardL-function on SO(2n+1)×GL(n). The local analysis of the integral is carried out using the formulas of the paper.     

Generating function for scalar products in the algebraic Bethe ansatz

Theoretical and Mathematical Physics - We construct a family of determinant representations for scalar products of Bethe vectors in models with $$ \mathfrak{gl} (3)$$ symmetry. This family is...     

Theta-lifting and geometric quantization for GL(n, ?)

In this paper, we verify Vogan??s conjecture on quantization in the representation theory for G = GL(n,?). Also we get some relationship between the induction of orbits and Howe??s ??-lifting of unitary representations.     

Higher-order analogues of the unitarity condition for quantum <Emphasis Type="Italic">R</Emphasis>-matrices

We derive a family of nth-order identities for quantum R-matrices of the Baxter–Belavin type in the fundamental representation. The set of identities includes the unitarity condition as the simplest case (n = 2). Our study is inspired by the fact that the third-order identity provides commutativity of the Knizhnik–Zamolodchikov–Bernard connections. On the other hand, the same identity yields the R-matrix-valued Lax pairs for classical integrable systems of Calogero type, whose construction uses the interpretation of the quantum R-matrix as a matrix generalization of the Kronecker function. We present a proof of the higher-order scalar identities for the Kronecker functions, which is then naturally generalized to R-matrix identities.     

Line graph of combinations of generalized Bethe trees: Eigenvalues and energy

We characterize the eigenvalues and energy of the line graph L(G) whenever G is (i) a generalized Bethe tree, (ii) a Bethe tree, (iii) a tree defined by generalized Bethe trees attached to a path, (iv) a tree defined by generalized Bethe trees having a common root, (v) a graph defined by copies of a generalized Bethe tree attached to a cycle and (vi) a graph defined by copies of a star attached to a cycle; in this case, explicit formulas for the eigenvalues and energy of L(G) are derived.     

Spin chain related to the quantum superalgebra slq(1¦1)

We consider an integrable system with R-matrix related to the algebra sl _q(1 | 1). The Hamiltonian of the system is constructed, and its spectrum is found by means of the algebraic Bethe ansatz. The symmetry algebra of the chain is written out. The partition function of the model on the lattice with domain wall boundary conditions is calculated. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 146–162.     

The Yangian double of the Lie superalgebra A(m, n)

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. Normally ordered bases in the Yangian and its dual in the quantum double are introduced. We calculate the pairing between the elements of these bases and obtain a formula for the universal R-matrix of the Yangian double as well as a formula for the universal R-matrix (introduced by Drinfeld) of the Yangian.     

Soliton Theory, Symmetric Functions and Matrix Integrals

We consider a certain scalar product of symmetric functions which is parameterized by a function r and an integer n. On the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of this product with the help of multi-integrals. This gives links between a theory of symmetric functions, soliton theory and models of random matrices (such as a model of normal matrices).     

On Shalika models and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

We use modular symbols to construct p-adic L-functions for cohomological cuspidal automorphic representations on GL(2n), which admit a Shalika model. Our construction differs from former ones in that it systematically makes use of the representation theory of p-adic groups.     

Restriction of Representations of GL (<Emphasis Type="Italic">n</Emphasis> + 1, <Emphasis Type="Italic">ℂ</Emphasis>) to GL (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">ℂ</Emphasis>) and Action of the Lie Overalgebra

Consider a restriction of an irreducible finite dimensional holomorphic representation of \(\text {GL}(n + 1,\mathbb {C})\) to the subgroup \(\text {GL}(n,\mathbb {C})\). We write explicitly formulas for generators of the Lie algebra \(\mathfrak {g}\mathfrak {l}(n + 1)\) in the direct sum of representations of \(\text {GL}(n,\mathbb {C})\). Nontrivial generators act as differential-difference operators, the differential part has order n ??1, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of \(\text {GL}(n,\mathbb {C})\).     

GL3-invariant solutions of the Yang-Baxter equation and associated quantum systems

GL₃-invariant, finite-dimensional solutions of the Yang-Baxter equations acting in the tensor product of two irreducible representations of the group GL₃ are investigated. A number of relations are obtained for the transfer matrices which demonstrate the connection of representation theory and the Bethe Ansatz in GL₃-invariant models. Some of the most interesting quantum and classical integrable systems connected with GL₃-invariant solutions of the Yang-Baxter equation are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 92–121, 1982.     

“Twisted” rational <Emphasis Type="Italic">r</Emphasis>-matrices and the algebraic Bethe ansatz: Applications to generalized Gaudin models,Bose–Hubbard dimers,and Jaynes–Cummings–Dicke-type models

We construct quantum integrable systems associated with the Lie algebra gl(n) and non-skew-symmetric “shifted and twisted” rational r-matrices. The obtained models include Gaudin-type models with and without an external magnetic field, n-level (n?1)-mode Jaynes–Cummings–Dicke-type models in the Λ-configuration, a vector generalization of Bose–Hubbard dimers, etc. We diagonalize quantum Hamiltonians of the constructed integrable models using a nested Bethe ansatz.     

Irreducible representations of the full linear group

We obtain some results on Young diagrams. Based on these results, we construct bases for Young symmetry classes of tensors. Using these bases, we obtain a complete reduction of the representation A ??^mA [A∈GL(n,$\text{C}$] and irreducible matrix representations of the full linear group.