A reduced subduction graph and higher multiplicity in S_n transformation coefficients

TL;DR

This paper introduces a simplified subduction graph approach and new rules to efficiently compute transformation coefficients between standard and split bases in symmetric group representations, especially for higher multiplicities.

Contribution

It provides selection and identity rules that reduce computational complexity and demonstrates an orthonormalized solution for a high-multiplicity case in $S_{10}$ decomposition.

Findings

Reduced subduction graph simplifies calculations

New rules decrease unknowns and equations

Explicit solution for multiplicity-three case in $S_{10}$

Abstract

Transformation coefficients between {\it standard} bases for irreducible representations of the symmetric group $S_n$ and {\it split} bases adapted to the $S_{n_1} \times S_{n_2} \subset S_n$ subgroup ($n_1 +n_2 = n$) are considered. We first provide a \emph{selection rule} and an \emph{identity rule} for the subduction coefficients which allow to decrease the number of unknowns and equations arising from the linear method by Pan and Chen. Then, using the {\it reduced subduction graph} approach, we may look at higher multiplicity instances. As a significant example, an orthonormalized solution for the first multiplicity-three case, which occurs in the decomposition of the irreducible representation $[4,3,2,1]$ of $S_{10}$ into $[3,2,1] \otimes [3,1]$ of $S_6 \times S_4$, is presented and discussed.