Clebsch--Gordan Coefficients of the Quaternion Group

TL;DR

This paper computes the Clebsch--Gordan coefficients for various quaternion groups by deriving eigenvectors of specific matrices related to their irreducible representations.

Contribution

It provides explicit calculations of Clebsch--Gordan coefficients for quaternion groups and their extensions, which were previously not systematically documented.

Findings

Explicit eigenvector formulas for quaternion group representations

Calculation of Clebsch--Gordan coefficients for Q8, Q16, Q32

Extension to a factor group involving Q32

Abstract

The Clebsch--Gordan coefficients of the Kronecker products of the irreducible representations of the Quaternion Group Q8, of the Generalized Quaternion Groups Q16 and Q32, and of the factor group (Q32 X Q32)/{(1,1),(-1,-1)} are computed as eigenvectors of a well-known matrix of triple-products of the irreducible representations.