Adams, J. F. The fundamental representations of \(E_ 8\). (English) Zbl 0576.22017 Algebraic topology, Proc. Conf. in Honor of P. Hilton, St. John’s/Can. 1983, Contemp. Math. 37, 1-10 (1985). [For the entire collection see Zbl 0549.00016.] First paragraph of the article: ”This paper is a snippet of a book I have in preparation about the exceptional Lie groups. This snippet argues that in order to understand the representations of \(E_ 8\), one should begin by understanding just three of them; one is the adjoint representation, and it goes on to construct the other two”. These three representations \(\alpha\), \(\beta\) and \(\gamma\) correspond to the endpoints of the Witt-Dynkin diagram, in particular \(\alpha\) to the end of the ”longest” arm, i.e. the adjoint representation of the simply connected form \(E_ 8\). Let G be a simple, compact, simply connected Lie group (e.g. \(E_ 8)\). For a finite dimensional G-module M let top(M) be the smallest G-submodule which has the same extreme weights and multiplicities as M. Then with the aid of this top-functor the author states that the exterior powers of \(\alpha\), \(\beta\) and \(\gamma\) generate the representation ring \(R(E_ 8)\). Moreover, \(\beta\) is uniquely and explicitely contained in the symmetric square \(\sigma^ 2(\alpha)\subset \alpha \otimes \alpha\) and similarly \(\gamma\) is contained in \(\alpha\) \(\otimes \beta\). Reviewer: H.Leptin Cited in 1 Document MSC: 22E46 Semisimple Lie groups and their representations Keywords:exceptional Lie groups; representations of \(E_ 8\) Citations:Zbl 0549.00016 PDFBibTeX XML