Analytic Bethe ansatz for fundamental representations of Yangians

Abstract

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian Y (X r ) or its quantum affine analogue U q (X (1) r ) for X r = B r , C r and D r . Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of Y (X r ). Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the T -system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semistandard Young tableaux.

Introduction to Yangian Symmetry in Integrable Field Theory

“…Drinfeld also provided another realization of Y (g) in [34], analogous to a Cartan-Weyl basis, which is often used to study Y (g) representation theory, and a set of polynomials in correspondence with (finite-dimensional) Y (g) reps which can be used to classify them (allowing one, for example, to deduce the existence of the fundamental Y (g) irreps discussed in this chapter). For connections between these two approaches see [35]; for that between Y (g) and the Bethe ansatz see [17,36]; for that with Hecke algebras see [3,37]; for that with separation-of-variables techniques see [38]. We noted earlier that Y (g) may be thought of as a deformation of a polynomial algebra, with parameter z: the analogue of the full loop algebra, with powers of z −1 as well, is the 'quantum double' of Y (g) [39,40].…”

Quantum Integrable Models and Discrete Classical Hirota Equations

“…[38], [44]). It yields T a 1 (u) in terms of elliptic polynomials Q t with roots constrained by the nested Bethe ansatz equations.…”

Paths, tableaux andq-characters of quantum affine algebras: theCncase

“…For C n and D n , we still conjecture (Conjecture 2.2) that χ λ/µ,a is the q-character of a certain, but not necessarily irreducible, representation of U q (ĝ). However, the tableaux description for χ λ/µ,a is known only for the basic cases, (λ, µ) = ((1 i ), φ) and (λ, µ) = ((i), φ) [22,21,14].The main purpose of the paper is to give the tableaux description of χ λ/µ,a in the C n case.Let us preview our results and explain what makes the tableaux description more complicated for C n and D n than A n and B n . To obtain the tableaux description of χ λ/µ,a , we apply the paths method of [16].…”

“…We determine the tableaux description by the horizontal, vertical and "extra" rules for skew diagrams of at most three rows and of at most two columns. The one-row and one-column cases are already given by [22]. …”

“…Let λ/µ be a skew diagram of one column (i.e., l(λ ′ ) = 1). Let Tab(C n , λ/µ) be the set of all the HV -tableaux T (actually, the horizontal rule (H) is not required) with the following condition: The following theorem is due to [22]. We reproduce it using the paths description.…”