$N \leftrightarrow -N$ duality of SU(N) for stable sequences of representations

TL;DR

This paper explores a duality in SU(N) representations that relates representations with Young diagrams depending on N to those with -N, impacting the understanding of Casimir eigenvalues and representation decompositions.

Contribution

It generalizes the N ↔ -N duality for SU(N) representation dimensions and Casimir eigenvalues, including representations with N-dependent Young diagrams.

Findings

Established a duality relation for N-dependent Young diagram representations

Analyzed implications for Casimir eigenvalues and universal decomposition hypotheses

Extended duality to include the adjoint representation and related structures

Abstract

We generalize $N \leftrightarrow -N$ duality of dimension formulae of $SU(N)$ representations on a (class of) representations with $N$-dependent Young diagrams (which include the adjoint representation), and on eigenvalues of the Casimir operator for those representations. We discuss the consequences for the hypothesis of universal decomposition of powers of the adjoint representation into Casimir subspaces.