On the linear equation method for the subduction problem in symmetric groups

TL;DR

This paper introduces a graph-based approach to efficiently solve the subduction problem in symmetric groups by analyzing transformation matrices between different bases, improving computational methods.

Contribution

It presents the subduction graph concept and an improved algorithm utilizing graph search for solving the subduction problem in symmetric groups.

Findings

The subduction graph effectively describes the combinatorial structure of the problem.

An improved algorithm based on graph searching enhances solution efficiency.

Matrix forms for multiplicity separations are expressed via Sylvester matrices.

Abstract

We focus on the tranformation matrices between the standard Young-Yamanouchi basis of an irreducible representation for the symmetric group S_n and the split basis adapted to the direct product subgroups S_{n_1} \times S_{n-n_1} . We introduce the concept of subduction graph and we show that it conveniently describes the combinatorial structure of the equation system arisen from the linear equation method. Thus we can outline an improved algorithm to solve the subduction problem in symmetric groups by a graph searching process. We conclude observing that the general matrix form for multiplicity separations, resulting from orthonormalization, can be expressed in terms of Sylvester matrices relative to a suitable inner product in the multiplicity space.