Clebcsh-Gordan coefficients in the symmric group $S_n$

TL;DR

This paper develops an algorithm to evaluate Clebcsh-Gordan coefficients for the symmetric group $S_n$, enabling applications in mathematical physics by combining these with inner product coefficients.

Contribution

It introduces a build-up algorithm based on subgroup chains to compute the outer product coefficients of $S_n$ representations, a novel approach in this context.

Findings

Algorithm successfully computes Clebcsh-Gordan coefficients for $S_n$

Applications demonstrated in mathematical physics contexts

Enhanced understanding of symmetric group representation products

Abstract

The coefficients of fractional parentage (CFP) or Clebcsh-Gordan coefficients of the outer product of representations of the symmetric group $S_n$ are evaluated using an build up algorithm defined in terms of the chain involving the chain $S_{n-1}\subset S_n$ and $S_{n-2}\subset S_n$ chains. Some applications in mathematical physics are considered by combining them with the Clebcsh-Gordan (C-G) coefficients of the inner product of representations of the symmetric group $S_n$