The Classical Groups

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 (1939), which 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.

Weyl (1939a) gave an informal talk about the topic of his book. There was a second edition in 1946.

Contents

Chapter I defines invariants and other basic ideas and describes the relation to Felix Klein's Erlangen program in geometry.

Chapter II describes the invariants of the special and general linear group of a vector space V on the polynomials over a sum of copies of V and its dual. It uses the Capelli identity to find an explicit set of generators for the invariants.

Chapter III studies the group ring of a finite group and its decomposition into a sum of matrix algebras.

Chapter IV discusses Schur–Weyl duality between representations of the symmetric and general linear groups.

Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal and symplectic groups, showing that the ring of invariants is generated by the obvious ones.

Chapter VII describes the Weyl character formula for the characters of representations of the classical groups.

Chapter VIII on invariant theory proves Hilbert's theorem that invariants of the special linear group are finitely generated.

Chapter IX and X give some supplements to the previous chapters.

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