The Classical Groups

The Classical Groups

Author	Hermann Weyl
Publication date	1939
ISBN	9780691057569

In Weyl's wonderful and terrible¹ book The Classical Groups [W] one may discern two main themes: first, the study of the polynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classical group action; second, the isotypic decomposition of the full tensor algebra for such an action.

¹Most people who know the book feel the material in it is wonderful. Many also feel the presentation is terrible. (The author is not among these latter.)

Howe (1989, p.539)

The Classical Groups: Their Invariants and Representations is a mathematics book by Hermann Weyl published in 1939. The book describes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.

The second edition was published in 1946. Weyl gave an informal talk about the topic of his book in 1939.

Reception

Roger Howe called the book "wonderful and terrible".

References

• Howe, Roger (1988), "The classical groups and invariants of binary forms", in Wells, R. O. Jr. (ed.), The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., vol. 48, Providence, R.I.: American Mathematical Society, pp.  133–166, ISBN   978-0-8218-1482-6, MR   0974333
• Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American Mathematical Society , 313 (2), American Mathematical Society: 539–570, doi: 10.2307/2001418 , ISSN   0002-9947, JSTOR   2001418, MR   0986027
• Jacobson, Nathan (1940), "Book Review: The Classical Groups", Bulletin of the American Mathematical Society , 46 (7): 592–595, doi: 10.1090/S0002-9904-1940-07236-2 , ISSN   0002-9904, MR   1564136
• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN   978-0-691-05756-9, MR   0000255
• Weyl, Hermann (1939a), "Invariants" , Duke Mathematical Journal , 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN   0012-7094, MR   0000030