### Abstract

This Note presents some equalities in law for Z_{N} : = det (Id - G), where G is an element of a subgroup of the set of unitary matrices of size N, endowed with its unique probability Haar measure. Indeed, under some general conditions, Z_{N} can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways: either by a recursive decomposition of the Haar measure (Section 1) or by previous results by Killip and Nenciu (2004) on orthogonal polynomials with respect to some measure on the unit circle (Section 2). This latter method leads naturally to a study of determinants of a class of principal submatrices (Section 3). To cite this article: P. Bourgade et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Original language | English (US) |
---|---|

Pages (from-to) | 229-232 |

Number of pages | 4 |

Journal | Comptes Rendus Mathematique |

Volume | 345 |

Issue number | 4 |

DOIs | |

State | Published - Aug 15 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Comptes Rendus Mathematique*,

*345*(4), 229-232. https://doi.org/10.1016/j.crma.2007.06.023

**The characteristic polynomial on compact groups with Haar measure : some equalities in law.** / Bourgade, Paul; Nikeghbali, Ashkan; Rouault, Alain.

Research output: Contribution to journal › Article

*Comptes Rendus Mathematique*, vol. 345, no. 4, pp. 229-232. https://doi.org/10.1016/j.crma.2007.06.023

}

TY - JOUR

T1 - The characteristic polynomial on compact groups with Haar measure

T2 - some equalities in law

AU - Bourgade, Paul

AU - Nikeghbali, Ashkan

AU - Rouault, Alain

PY - 2007/8/15

Y1 - 2007/8/15

N2 - This Note presents some equalities in law for ZN : = det (Id - G), where G is an element of a subgroup of the set of unitary matrices of size N, endowed with its unique probability Haar measure. Indeed, under some general conditions, ZN can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways: either by a recursive decomposition of the Haar measure (Section 1) or by previous results by Killip and Nenciu (2004) on orthogonal polynomials with respect to some measure on the unit circle (Section 2). This latter method leads naturally to a study of determinants of a class of principal submatrices (Section 3). To cite this article: P. Bourgade et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

AB - This Note presents some equalities in law for ZN : = det (Id - G), where G is an element of a subgroup of the set of unitary matrices of size N, endowed with its unique probability Haar measure. Indeed, under some general conditions, ZN can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways: either by a recursive decomposition of the Haar measure (Section 1) or by previous results by Killip and Nenciu (2004) on orthogonal polynomials with respect to some measure on the unit circle (Section 2). This latter method leads naturally to a study of determinants of a class of principal submatrices (Section 3). To cite this article: P. Bourgade et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

UR - http://www.scopus.com/inward/record.url?scp=34547966112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547966112&partnerID=8YFLogxK

U2 - 10.1016/j.crma.2007.06.023

DO - 10.1016/j.crma.2007.06.023

M3 - Article

AN - SCOPUS:34547966112

VL - 345

SP - 229

EP - 232

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 4

ER -