## Asymptotics Of Jack Polynomials As The Number Of Variables Goes To Infinity (1998)

Venue: | Math. Res. Notices |

Citations: | 11 - 5 self |

### BibTeX

@ARTICLE{Okounkov98asymptoticsof,

author = {Andrei Okounkov and Grigori Olshanski},

title = {Asymptotics Of Jack Polynomials As The Number Of Variables Goes To Infinity},

journal = {Math. Res. Notices},

year = {1998},

volume = {13},

pages = {641--682}

}

### OpenURL

### Abstract

In this paper we study the asymptotic behavior of the Jack rational functions P (z 1 ; : : : ; zn ; `) as the number of variables n and the signature grow to infinity. Our results generalize the results of A. Vershik and S. Kerov [VK2] obtained in the Schur function case (` = 1). For ` = 1=2; 2 our results describe approximation of the spherical functions of the infinite-dimensional symmetric spaces U(1)=O(1) and U(21)=Sp(1) by the spherical functions of the corresponding finite-dimensional symmetric spaces. Contents 1.1. Statement of the main result 1.2. Regular and infinitesimally regular sequences 1.3. Extremality of the limit functions 1.4. Related results 1.5. Acknowledgments 2. Jack polynomials and shifted Jack polynomials 2.1. Orthogonality 2.2. Interpolation 2.3. Branching rules 2.4. Binomial formula 2.5. Generating functions 2.6. Partitions and signatures 2.7. Extended symmetric functions 3. Asymptotic properties of Vershik-Kerov sequences of signatures 4. Sufficient conditions of regularity 5. Necessary conditions of regularity 5.1. The "only if " part of Theorem 1.1 5.2. A growth estimate for jf()j, f 2 7. Appendix. A direct proof of the formula (2.10) for generating functions The authors were supported by the Russian Basic Research Foundation grant 95-01-00814. The first author's stay at IAS in Princeton and MSRI in Berkeley was supported by NSF grants DMS--9304580 and DMS--9022140 respectively. Typeset by A M S-T E X 1 A. OKOUNKOV AND G. OLSHANSKI 1.1 Statement of the main result. Jack symmetric functions P (z 1 ; : : : ; z n ; `) 2 Q(`)[z \Sigma1 S(n) which are indexed by decreasing sequences of integers (called signatures) = ( 1 \Delta \Delta \Delta n ) 2 Z are eigenfunctions of the quantum Calogero-Sutherland Hamiltonian [C,Su] (1.1)...