The Characteristic Polynomial on Compact Groups with Haar Measure : Some Equalities in Law
نویسندگان
چکیده
Abstract. 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 ([3]) 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.
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