Tensor powers of adjoint representations ot classical Lie groups

TL;DR

This paper provides explicit constructions for the irreducible components of tensor powers of adjoint representations in classical Lie groups, confirming Vogel's formulae through an alternative method.

Contribution

It introduces simple constructions specific to classical groups for tensor powers of adjoint modules, offering an independent verification of Vogel's general formulae.

Findings

Explicit constructions for tensor square irreducible constituents

Confirmation of Vogel's formulae for classical groups

Alternative approach to existing methods

Abstract

Exploiting particular features of classical groups, simple constructions are given for the irreducible constituents of the tensor square of the adjoint modules and the leading terms in higher tensor powers. This provides an independent confirmation of Vogel's general formulae, and alternative approach to that in some recent papers.