Triality, exceptional Lie algebras and Deligne dimension formulas.
\textit{P. Deligne} [C. R. Acad. Sci., Paris, Sér. I 322, 321--326 (1996; Zbl 0910.22008)] observed some uniform representation-theoretic behaviour shared by exceptional Lie groups and found a small family of representations which behave in the same way. In particular, the dimensions of these representations are rational functions involving only linear factors. Using a computer program, \textit{A. M. Cohen} and \textit{R. de Man} [C. R. Acad. Sci., Paris, Sér. I 322, 427--432 (1996; Zbl 0849.22017)] found some computational evidence and extended the Deligne formulas to \({\mathfrak g}^{\otimes 3}\) and \({\mathfrak g}^{\otimes 4}\) where \({\mathfrak g}\) is the corresponding Lie algebra and \textit{P. Deligne} and \textit{R. de Man} [C. R. Acad. Sci., Paris, Sér. I 323, 577--582 (1996; Zbl 0910.22009)] refined the framework for these representations. In the paper under review the authors extend the above results in a computer free way. They give additional dimension formulas for the exceptional series as well as beautiful uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic chart. The authors use a version of the Vinberg construction of the exceptional Lie algebras in terms of normed division algebras. They also use the triality model of the magic square of Freudenthal, which they review, and present a simplified proof of its validity. The paper concludes with discussions on how to obtain other series of Lie algebras equipped with series of representations of dimensions which are rational functions involving linear factors only.
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