Structure and properties of the algebra of partially transposed permutation operators

TL;DR

This paper investigates the algebraic structure of partially transposed permutation operators in tensor spaces, revealing its semi-simplicity and classifying all irreducible representations using algebraic methods.

Contribution

It provides a complete algebraic analysis of the algebra of partially transposed permutation operators, including classification of irreducible representations and their matrix elements.

Findings

The algebra is semi-simple.

Two types of irreducible representations are identified.

Explicit expressions for matrix elements are derived.

Abstract

We consider the structure of algebra of operators, acting in $n-$fold tensor product space, which are partially transposed on the last term. Using purely algebraical methods we show that this algebra is semi-simple and then, considering its regular representation, we derive basic properties of the algebra. In particular, we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations. It appears that there are two types of irreducible representations of the algebra. The first one is strictly connected with the representations of the group $S(n-1)$ induced by irreducible representations of the group $S(n-2)$. The second type is structurally connected with irreducible representations of the group $S(n-1)$.