A Remark on Random Vectors and Irreducible Representations

TL;DR

This paper explores the relationship between random vectors on invariant subspaces of irreducible representations of compact Lie groups and their scalar product expectations, extending known results about uniform sphere distributions.

Contribution

It demonstrates that the property of expected squared scalar products being 1/n characterizes random vectors on invariant subspaces of irreducible Lie group representations.

Findings

Expected squared scalar product equals 1/n for vectors on invariant subspaces

Characterization of random vectors via irreducible representations

Extension of sphere distribution properties to Lie group contexts

Abstract

It was observed in \cite{Al2} that the expectation of a squared scalar product of two random independent unit vectors that are uniformly distributed on the unit sphere in $\mathbb{R}^n $ is equal to $1/n$. It is shown below that this is a characteristic property of random unit vectors defined on invariant probability subspaces of irreducible real representations of compact Lie groups.