Abstract
We study general aspects of the reductive dual pair correspondence, also known as Howe duality. We make an explicit and systematic treatment, where we first derive the oscillator realizations of all irreducible dual pairs: (GL(M, ℝ), GL(N, ℝ)), (GL(M, ℂ), GL(N, ℂ)), (U∗(2M), U∗(2N)), (U (M+, M−), U (N+, N−)), (O(N+, N−), Sp (2M, ℝ)), (O(N, ℂ), Sp(2M, ℂ)) and (O∗(2N ), Sp(M+, M−)). Then, we decompose the Fock space into irreducible representations of each group in the dual pairs for the cases where one member of the pair is compact as well as the first non-trivial cases of where it is non-compact. We discuss the relevance of these representations in several physical applications throughout this analysis. In particular, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established.
Highlights
The reductive dual pair correspondence, known as Howe duality [1, 2], provides a useful mathematical framework to study a physical system, allowing for a straightforward analysis of its symmetries and spectrum
We study general aspects of the reductive dual pair correspondence, known as Howe duality
Many times when the dual pair correspondence makes its occurrence in the physics literature, its role often goes unnoticed or is not being emphasized, albeit it is acting as an underlying governing principle
Summary
The reductive dual pair correspondence, known as Howe duality [1, 2], provides a useful mathematical framework to study a physical system, allowing for a straightforward analysis of its symmetries and spectrum. Oscillator representations were studied in mathematics by Segal [23], Shale [24] and by Weil [25] (often referred to as Weil representations) Based on these earlier works (and others that we omit to mention), Howe eventually came up with the reductive dual pair correspondence in 1976 (published much later in [1, 2]). In 2003, Vasiliev generalized his theory to any dimensions using vector oscillators [52], and revisited its representation theory in [53] In both of these works, the dual pair correspondence played a crucial role, and since it has been used several times within the context of higher spin theories.. Physical applications will be visited in depth in a sequel, we include brief comments on them throughout the current paper, as summarized below
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