Radial Laplacian on rotation groups

TL;DR

This paper derives the explicit form of the radial Laplacian on rotation groups, analyzes its eigenvalues using Weyl character formulas, and presents the material accessibly for non-experts in Lie groups.

Contribution

It provides a clear derivation of the radial Laplacian on rotation groups and computes its eigenvalues using character theory, making advanced Lie group concepts more accessible.

Findings

Derived the explicit expression of the radial Laplacian.

Computed eigenvalues of the Laplacian using Weyl character formula.

Presented a synthetic, accessible overview of the Laplacian on rotation groups.

Abstract

The Laplacian on the rotation group is invariant by conjugation. Hence, it maps class functions to class functions. A maximal torus consists of block diagonal matrices whose blocks are planar rotations. Class functions are determined by their values of this maximal torus. Hence, the Laplacian induces a second order operator on the maximal torus called the radial Laplacian. In this paper, we derive the expression of the radial Laplacian. Then, we use it to find the eigenvalues of the Laplacian, using that characters are class functions whose expressions are given by the Weyl character formula. Although this material is familiar to Lie-group experts, we gather it here in a synthetic and accessible way which may be useful to non experts who need to work with these concepts.