Highest weight representations of the quantum algebra U_h(gl_\infty)

A class of highest weight irreducible representations of the quantum algebra U_h(gl_\infty) is constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written.Comment: 7 pages, PlainTe

Novel classical ground state of a many body system in arbitrary dimensions

The classical ground state of a D- dimensional many body system with two and three body interactions is studied as a function of the strength of the three body interaction. We prove exactly that beyond a critical strength of the three body interaction, the classical ground state of the system is one in which all the particles are on a line. The positions of the particles in this string configuration are uniquely determined by the zeros of the Hermite polynomials.Comment: 4 pages, RevTeX, no figure; version to appear in Physical Review Letter

Basic Representations of A_{2l}^(2) and D_{l+1}^(2) and the Polynomial Solutions to the Reduced BKP Hierarchies

Basic representations of A_{2l}^(2) and D_{l+1}^(2) are studied. The weight vectors are represented in terms of Schur's $Q$-functions. The method to get the polynomial solutions to the reduced BKP hierarchies is shown to be equivalent to a certain rule in Maya game.Comment: January 1994, 11 page

A survey of Hirota's difference equations

A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.Comment: LaTeX, 43 pages, LaTeX figures (with emlines2.sty

Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson Rule

The basic representation of \A is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.Comment: February 1995, 7pages, Using AMS-Te

The Bloch-Okounkov correlation functions of classical type

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc, and it can also be interpreted as correlation functions on integrable gl_\infty-modules of level one. Such gl_\infty-correlation functions at higher levels were then calculated by Cheng and Wang. In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of gl_\infty of type B,C,D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B,C,D and some fermionic type q-identities.Comment: v2, very minor changes, Latex, 41 pages, to appear in Commun. Math. Phy

The Bloch-Okounkov correlation functions, a classical half-integral case

Bloch and Okounkov's correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of \hgl_\infty-modules of level one. Recent works have calculated these character functions for higher levels for \hgl_\infty and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain $q$-dimension formulas for integral modules of type $D$ at half-integral level.Comment: v2: minor changes to the introduction; accepted for publication in Letters in Mathematical Physic

Intertwining operators and Hirota bilinear equations

An interpretation of Hirota bilinear relations for classical $\tau$ functions is given in terms of intertwining operators. Noncommutative example of $U_q(sl_2)$ is presented.Comment: Latex, 13 pages, no figures. Contribution to the Proceedings of Alushta Conference, June 199

Eigensystem and Full Character Formula of the W_{1+infinity} Algebra with c=1

By using the free field realizations, we analyze the representation theory of the W_{1+infinity} algebra with c=1. The eigenvectors for the Cartan subalgebra of W_{1+infinity} are parametrized by the Young diagrams, and explicitly written down by W_{1+infinity} generators. Moreover, their eigenvalues and full character formula are also obtained.Comment: 12 pages, YITP/K-1049, SULDP-1993-1, RIMS-959, Plain TEX, ( New references

Spectral Decomposition of Path Space in Solvable Lattice Model

We give the {\it spectral decomposition} of the path space of the U_q(\hatsl) vertex model with respect to the local energy functions. The result suggests the hidden Yangian module structure on the \hatsl level $l$ integrable modules, which is consistent with the earlier work [1] in the level one case. Also we prove the fermionic character formula of the \hatsl level $l$ integrable representations in consequence.Comment: 27 pages, Plain Tex, epsf.tex, 7 figures; minor revision. identical with the version to be published in Commun.Math.Phy