N. V. Tsilevich

Quantum inverse scattering method for the q-boson model and symmetric functions

The purpose of this paper is to show that the quantum inverse scattering method for the so-called q-boson model has a nice interpretation in terms of the algebra of symmetric functions. In particular, in the case of the phase model (corresponding to q = 0) the creation operator coincides (modulo a scalar factor) with the operator of multiplication by the generating function of complete homogeneous symmetric functions, and the wave functions are expressed via the Schur functions sλ(x). The general case of the q-boson model is related in a similar way to the Hall-Littlewood symmetric functions Pλ(x;q2).

Quasi-invariance of the gamma process and multiplicative properties of the Poisson-Dirichlet measures

Abstract In this paper we describe new fundamental properties of the law P Γ of the classical gamma process and related properties of the Poisson-Dirichlet measures PD( θ ). We prove the quasi-invariance of the measure P Γ with respect to an infinite-dimensional multiplicative group (the fact first discovered in [4]) and the Markov-Krein identity as corollaries of the formula for the Laplace transform of P Γ . The quasi-invariance of the measure P Γ allows us to obtain new quasi-invariance properties of the measure PD( θ ). The corresponding invariance properties hold for σ -finite analogues of P Γ and PD( θ ). We also show that the measure P Γ can be considered as a limit of measures corresponding to the α -stable Levy processes when the parameter α tends to zero. Our approach is based on considering simultaneously the gamma process (especially its Laplace transform) and its simplicial part — the Poisson-Dirichlet measures.

THE MARKOV-KREIN CORRESPONDENCE IN SEVERAL DIMENSIONS

AbstractGiven a probability distribution

Stationary Random Partitions of Positive Integers

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Induced representations of the infinite symmetric group and their spectral theory

on a space X, let M=M

An Infinite-Dimensional Analogue of the Lebesgue Measure and Distinguished Properties of the Gamma Process

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