Conditional Haar Measures on Classical Compact Groups
نویسنده
چکیده
We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension n. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension p. The developed method leads to the following result: for this conditional measure, writing Z (p) U for the first nonzero derivative of the characteristic polynomial at 1,
منابع مشابه
Measures of maximal entropy
We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.
متن کاملIntegration formulas for Brownian motion on classical compact Lie groups
Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of Invariant and of classical Schur-Weyl dualities based on stochastic calculus.
متن کاملBorel Theorems for Random Matrices from the Classical Compact Symmetric Spaces
Abstract. We study random vectors of the form (Tr(AV ), . . . ,Tr(AV )), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et ...
متن کاملIntegration over Compact Quantum Groups
We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures. Introduction A basic question in functional analysis is to find axioms for quantum groups, which ensure the existence of a Haar measure. In the compact case, this ...
متن کاملA Variance Formula Related to a Quantum Conductance Problem
Let t be a block of an Haar-invariant orthogonal (β = 1), unitary (β = 2) or symplectic (β = 4) matrix from the classical compact groups O(n), U(n) or Sp(n), respectively. We obtain a close form for V ar(tr(t∗t)). The case for β = 2 is related to a quantum conductance problem, and our formula recovers a result obtained by several authors. Moreover, our result shows that the variance has a limit...
متن کامل