Q-operator and factorised separation chain for Jack polynomials

Vadim B. Kuznetsov; Vladimir V. Mangazeev; Evgeny K. Sklyanin

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

Q-operator and factorised separation chain for Jack polynomials

Vadim B. Kuznetsov^*, Vladimir V. Mangazeev, Evgeny K. Sklyanin

^*Corresponding author for this work

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

35 Citations (Scopus)

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_λ^(1/g) (x₁, ...,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_n 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_λ^(1/g) (x₁, ...,x_k, 1, ..., 1). Using the operator Q_z for z = 0 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

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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 Pλ(1/g) (x1, ...,xn) are eigenfunctions of a one-parameter family of integral operators Qz. The operators Qz are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Qzk we construct an integral operator Sn factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator Sn admits a factorisation described in terms of restricted Jack polynomials Pλ(1/g) (x1, ...,xk, 1, ..., 1). Using the operator Qz for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.",

keywords = "Integral operators, Jack polynomials",

author = "Kuznetsov, \{Vadim B.\} and Mangazeev, \{Vladimir V.\} and Sklyanin, \{Evgeny K.\}",

year = "2003",

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day = "15",

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, Vadim B.

AU - Mangazeev, Vladimir V.

AU - Sklyanin, Evgeny 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 Pλ(1/g) (x1, ...,xn) are eigenfunctions of a one-parameter family of integral operators Qz. The operators Qz are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Qzk we construct an integral operator Sn factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator Sn admits a factorisation described in terms of restricted Jack polynomials Pλ(1/g) (x1, ...,xk, 1, ..., 1). Using the operator Qz for z = 0 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 Pλ(1/g) (x1, ...,xn) are eigenfunctions of a one-parameter family of integral operators Qz. The operators Qz are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Qzk we construct an integral operator Sn factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator Sn admits a factorisation described in terms of restricted Jack polynomials Pλ(1/g) (x1, ...,xk, 1, ..., 1). Using the operator Qz for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.

KW - Integral operators

KW - Jack polynomials

UR - https://www.scopus.com/pages/publications/1042279863

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 -

Q-operator and factorised separation chain for Jack polynomials

Abstract

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