## Abstract

Let A, B and C = U_{1} A U_{1}^{*} + U_{2} B U_{2}^{*} be hermitian (or real symmetric) matrices, where U_{1} and U_{2} are unitary (respectively orthogonal). Our goal is to develop as concrete as possible expression to the probability distribution of the spectrum of C where U_{i} are drawn from the unitary (respectively orthogonal) group with respect to the Haar measure. While we do this for the unitary group, using representation theory of U_{n}, we can only accomplish the same for the real symmetric case where B has rank 1, by performing explicit calculations. Here is what we do in the unitary case. For a given n by n matrix A over the field of complex numbers we study the operator E_{A}^{N} = ∫ u A u^{*⊗ N} d u, where the integration is taken over the unitarian group with respect to the Haar measure. When A has nonnegative spectrum, we show that E_{A}^{N}, as N tends to infinity, is concentrated around some simple S_{N} × U_{n} submodule of (C^{n})^{⊗ N} determined only by the spectra of A. Along the proof we reprove a generalization of Heckman to the convexity theorem of Horn. The Schur-Weyl duality and the technique of Bernstein polynomials approximations are our tools. Using this we compute the distribution, induced by a sum of two hermitian orbits on the set of hermitian orbits in terms of asymptotic Littlewood-Richardson coefficients times an asymptotic version of the Weyl dimensional formula. More precisely, translation of the lattice permutation and semistandard Young tableaux rules to a set of linear inequalities, enable us to express the density of the distribution above as multiplication of volumes of two concrete polytopes.

Original language | English |
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Pages (from-to) | 268-286 |

Number of pages | 19 |

Journal | Advances in Applied Mathematics |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2006 |

## Keywords

- Eigenvalues
- Hermitian matrices
- Measure
- Unitary representations
- Young tableaux