Explicit constructions of unitary transformations between equivalent irreducible representations

TL;DR

This paper presents an explicit algorithm for constructing unitary transformations between equivalent irreducible representations of finite groups, generalizing classical orthogonality relations and providing specific forms for symmetric groups.

Contribution

It introduces a new algorithm for explicitly constructing similarity transformations between equivalent irreps, extending classical orthogonality relations, with applications to symmetric groups.

Findings

Derived a generalized orthogonality relation for finite group irreps.

Provided explicit unitary matrices for conjugated Young-Yamanouchi representations.

Offered a practical algorithm assuming access to matrix elements of irreps.

Abstract

Irreducible representations (irreps) of a finite group $G$ are equivalent if there exists a similarity transformation between them. In this paper, we describe an explicit algorithm for constructing this transformation between a pair of equivalent irreps, assuming we are given an algorithm to compute the matrix elements of these irreps. Along the way, we derive a generalization of the classical orthogonality relations for matrix elements of irreps of finite groups. We give an explicit form of such unitary matrices for the important case of conjugated Young-Yamanouchi representations, when our group $G$ is symmetric group $S(N)$.