A mapping from the unitary to doubly stochastic matrices and symbols on a finite set

TL;DR

This paper demonstrates that the transformation from unitary matrices to doubly stochastic matrices, via the squared modulus of entries, is a submersion for almost all unitaries, using operator symbols on finite sets.

Contribution

It establishes the submersion property of the mapping from unitary to doubly stochastic matrices for almost all unitaries, providing detailed proofs within the operator symbols framework.

Findings

The mapping is a submersion for almost all unitary matrices.

The proof employs operator symbols on finite sets.

Provides detailed proofs of previous results.

Abstract

We prove that the mapping from the unitary to the doubly stochastic matrices that maps a unitary matrix (u_{kl}) to the doubly stochastic matrix (|u_{kl}|^2) is a submersion for almost all unitary matrices. The proof uses the framework of operator symbols on a finite set. We give detailed proofs of the results announced in our earlier preprint.