# Proof by characters of the orthogonal-orthogonal duality and relations of Casimir invariants

## TL;DR

This paper proves the orthogonal-orthogonal duality theorem using character methods, providing new derivations of Casimir invariant relations and exploring potential applications in physical systems beyond existing literature.

## Contribution

It introduces a novel character-based proof of the duality theorem and derives linear relations between Casimir invariants using Young diagram geometry.

## Key findings

- New proof of orthogonal-orthogonal duality using characters
- Derived relations between Casimir invariants
- Potential applications in various physical systems

## Abstract

The theorem of orthogonal-orthogonal duality of Rowe, Repka, and Carvalho is proven by a method based on characters that is very different from theirs and akin to Helmers's half a century earlier proof of the analogous sympletic-symplectic duality. I demonstrate how three duality theorems listed by Rowe, Repka, and Carvalho allow very brief derivations of linear relations between the Casimir invariants of the connected representations based on the geometry of their Young diagrams, and discuss for which physical systems other than such already considered in the literature an analysis in terms of the orthogonal-orthogonal duality might be useful.