Dual Pair Correspondence in Physics: Oscillator Realizations and Representations

TL;DR

This paper systematically derives oscillator realizations of all irreducible dual pairs in Howe duality, decomposes Fock spaces into irreducible representations, and explores their physical relevance and Casimir relations.

Contribution

It provides explicit oscillator realizations and decompositions for all irreducible dual pairs, including non-compact cases, and establishes Casimir operator relations.

Findings

Explicit oscillator realizations for all dual pairs.

Decomposition of Fock space into irreducible representations.

Closed-form Casimir operator relations.

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: $(GL(M,\mathbb R), GL(N,\mathbb R))$, $(GL(M,\mathbb C), GL(N,\mathbb C))$, $(U^*(2M), U^*(2N))$, $(U(M_+,M_-), U(N_+,N_-))$, $(O(N_+,N_-),Sp(2M,\mathbb R))$, $(O(N,\mathbb C), Sp(2M,\mathbb C))$ 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.