### Abstract

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

Original language | English (US) |
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Title of host publication | Seminaire de Probabilites XLIII |

Pages | 351-377 |

Number of pages | 27 |

Volume | 2006 |

DOIs | |

State | Published - 2011 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2006 |

ISSN (Print) | 00758434 |

### Fingerprint

### Keywords

- Characteristic polynomials
- Correlation kernel
- Decomposition of Haar measure
- Ewens sampling formula
- Random matrices

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Seminaire de Probabilites XLIII*(Vol. 2006, pp. 351-377). (Lecture Notes in Mathematics; Vol. 2006). https://doi.org/10.1007/978-3-642-15217-7_15

**Ewens measures on compact groups and hypergeometric kernels.** / Bourgade, Paul; Nikeghbali, Ashkan; Rouault, Alain.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Seminaire de Probabilites XLIII.*vol. 2006, Lecture Notes in Mathematics, vol. 2006, pp. 351-377. https://doi.org/10.1007/978-3-642-15217-7_15

}

TY - CHAP

T1 - Ewens measures on compact groups and hypergeometric kernels

AU - Bourgade, Paul

AU - Nikeghbali, Ashkan

AU - Rouault, Alain

PY - 2011

Y1 - 2011

N2 - On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

AB - On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

KW - Characteristic polynomials

KW - Correlation kernel

KW - Decomposition of Haar measure

KW - Ewens sampling formula

KW - Random matrices

UR - http://www.scopus.com/inward/record.url?scp=78049445753&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78049445753&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-15217-7_15

DO - 10.1007/978-3-642-15217-7_15

M3 - Chapter

AN - SCOPUS:78049445753

SN - 9783642152160

VL - 2006

T3 - Lecture Notes in Mathematics

SP - 351

EP - 377

BT - Seminaire de Probabilites XLIII

ER -