Weyl’s Great Papers of 1925 and 1926
At the end of the last chapter we noted that Weyl’s first two accomplishments in the study of the representations of Lie groups were his characterization of the irreducible representations of the special linear group in terms of symmetry classes of tensors and his establishment of the complete reducibility theorem for this group. These results were important to him because they provided the proper (group-theoretic) foundation for the calculus of tensors. Impressive as these results are, they are but a part of the contents of his first detailed paper on the representation of semisimple groups, published in Mathematische Zeitschrift with the title, “Theory of the Representation of Continuous Semisimple Groups by Linear Transformations” , The following chapter is devoted to a discussion of this remarkable paper (Sections 1–5) and two related matters: Cartan’s response to it (Section 6) and, in Section 7, Weyl’s paper with F. Peter , in which the details of an idea already presented in the 1925 paper are worked out. The Peter-Weyl paper effectively launched what is now called harmonic analysis on groups. Together these two papers constitute an important landmark in the history of Lie’s theory and, more generally, in the history of mathematics.
KeywordsIrreducible Representation Irreducible Character Regular Representation Semisimple Group Special Linear Group
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