A Remark on Random Vectors and Irreducible Representations

Alexander Kushkuley

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arXiv · 2110.15504

A Remark on Random Vectors and Irreducible Representations

Alexander Kushkuley

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Abstract

The expectation of a squared scalar product of two random independent unit vectors that are uniformly distributed on a unit sphere in $\mathbb{R}^n $ is equal to $1/n$. We show that this is a characteristic property of random unit vectors defined on invariant probability subspaces of irreducible representations of compact Lie groups. We also discuss a relation of this fact to some properties of random invariant tensors

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Alexander Kushkuley. 2021-10-29. A Remark on Random Vectors and Irreducible Representations. https://arxiv.org/abs/2110.15504

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