Structure and properties of the algebra of partially transposed permutation operators

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 $S(n-1)$.