A note on some new hook-content identities

Abstract

Based on the work of A. Vershik[1], we introduce two new combinatorial identities. We show how these identities can be used to prove a new hook-content identity. The main motivation for deriving this identity was a particular optimization problem in the field of quantum information processing.for the representation of GL(d) × S n , where V λ is either zero or a polynomial irreducible representation of GL(d), S λ is an irreducible representation of S n and λ runs over the partitions of n and is conveniently represented by Young diagram. Both the symmetric group and GL(d) (and especially its compact subgroups U (d) and SU (d)) are of paramount importance in theoretical physics, especially in quantum mechanics. For example, S n is a fundamental symmetry of systems of identical particles and unitary groups represent the set of reversible (finite-dimensional) transformations. Therefore, it is not surprising that physics community keeps a steady interest in the representation theory of these groups and in the Shur-Weyl duality, from the early work of Weyl [3] up to the most recent applications in quantum computing and quantum information processing. For example, Equation (1) denotes a subsystem decomposition (induced by the symmetry of the system-environment interaction) in which one can identify error free subsystems [4,5], or the relevant subsystems for quantum estimation [6]. These are just a couple of examples of a much wider variety of applications (see e.g. Ref.[7] for a review). In the light of this discussion it is clear that the dimensions of the irreducible spaces V λ and S λ , are, more often than not, a crucial piece of information. The value of dim(S λ ) and dim(V λ ) are given by the hook length formula [8] and the hook-content formula [9] respectively.

Quantum information processing and composite quantum fields

“…Some beautiful symmetric group identities have been found in the subject of quantum information processing [1]. A combinatoric proof has been given [2].…”