Casimir operators of the exceptional group $F_4$: the chain $B_4\subset F_4\subset D_{13}$

TL;DR

This paper derives explicit expressions for the Casimir operators of the exceptional Lie group $F_4$, utilizing a chain of subgroups to relate its generators and representations, and identifies a basis of operators of specific degrees.

Contribution

It provides a concise formulation of Casimir operators for $F_4$ using subgroup chains, including explicit expressions and a basis of degrees 2, 6, 8, and 12.

Findings

Explicit formulas for Casimir operators of $F_4$

Identification of a basis with degrees 2, 6, 8, 12

Representation of $F_4$ in terms of classical groups

Abstract

Expressions are given for the Casimir operators of the exceptional group $F_4$ in a concise form similar to that used for the classical groups. The chain $B_4\subset F_4\subset D_{13}$ is used to label the generators of $F_4$ in terms of the adjoint and spinor representations of $B_4$ and to express the 26-dimensional representation of $F_4$ in terms of the defining representation of $D_{13}$. Casimir operators of any degree are obtained and it is shown that a basis consists of the operators of degree 2, 6, 8 and 12.