Q-operator and factorised separation chain for Jack polynomials

V B Kuznetsov; V V Mangazeev; E K Sklyanin

doi:10.1016/S0019-3577(03)90057-7

Q-operator and factorised separation chain for Jack polynomials

V B Kuznetsov, V V Mangazeev, E K Sklyanin

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P-lambda((1/g))(x(1,...,)x(n)) are eigenfunctions of a one-parameter family of integral operators Q(z). The operators Q(z) are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q(zk) we construct an integral operator S,, factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S-n admits a factorisation described in terms of restricted Jack polynomials P-lambda((1/g))(x(1,...,)x(k), 1,...,1). Using the operator Q(z) for z = O we give a simple derivation of a previously known integral representation for Jack polynomials.

Original language	English
Pages (from-to)	451-482
Number of pages	32
Journal	Indagationes Mathematicae
Volume	14
Issue number	3-4
DOIs	https://doi.org/10.1016/S0019-3577(03)90057-7
Publication status	Published - 15 Dec 2003

Keywords

Jack polynomials
integral operators
SYMMETRIC FUNCTIONS
SINGULAR VECTORS
TODA CHAIN
VARIABLES
SYSTEM
MODEL
ALGEBRA

Access to Document

10.1016/S0019-3577(03)90057-7

http://dx.doi.org/10.1016/S0019-3577(03)90057-7

Cite this

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title = "Q-operator and factorised separation chain for Jack polynomials",

abstract = "Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P-lambda((1/g))(x(1,...,)x(n)) are eigenfunctions of a one-parameter family of integral operators Q(z). The operators Q(z) are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q(zk) we construct an integral operator S,, factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S-n admits a factorisation described in terms of restricted Jack polynomials P-lambda((1/g))(x(1,...,)x(k), 1,...,1). Using the operator Q(z) for z = O we give a simple derivation of a previously known integral representation for Jack polynomials.",

keywords = "Jack polynomials, integral operators, SYMMETRIC FUNCTIONS, SINGULAR VECTORS, TODA CHAIN, VARIABLES, SYSTEM, MODEL, ALGEBRA",

author = "Kuznetsov, {V B} and Mangazeev, {V V} and Sklyanin, {E K}",

year = "2003",

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doi = "10.1016/S0019-3577(03)90057-7",

language = "English",

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journal = "Indagationes Mathematicae",

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TY - JOUR

T1 - Q-operator and factorised separation chain for Jack polynomials

AU - Kuznetsov, V B

AU - Mangazeev, V V

AU - Sklyanin, E K

PY - 2003/12/15

Y1 - 2003/12/15

N2 - Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P-lambda((1/g))(x(1,...,)x(n)) are eigenfunctions of a one-parameter family of integral operators Q(z). The operators Q(z) are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q(zk) we construct an integral operator S,, factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S-n admits a factorisation described in terms of restricted Jack polynomials P-lambda((1/g))(x(1,...,)x(k), 1,...,1). Using the operator Q(z) for z = O we give a simple derivation of a previously known integral representation for Jack polynomials.

AB - Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P-lambda((1/g))(x(1,...,)x(n)) are eigenfunctions of a one-parameter family of integral operators Q(z). The operators Q(z) are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q(zk) we construct an integral operator S,, factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S-n admits a factorisation described in terms of restricted Jack polynomials P-lambda((1/g))(x(1,...,)x(k), 1,...,1). Using the operator Q(z) for z = O we give a simple derivation of a previously known integral representation for Jack polynomials.

KW - Jack polynomials

KW - integral operators

KW - SYMMETRIC FUNCTIONS

KW - SINGULAR VECTORS

KW - TODA CHAIN

KW - VARIABLES

KW - SYSTEM

KW - MODEL

KW - ALGEBRA

U2 - 10.1016/S0019-3577(03)90057-7

DO - 10.1016/S0019-3577(03)90057-7

M3 - Article

SN - 0019-3577

VL - 14

SP - 451

EP - 482

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 3-4

ER -