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Alternative Routes to Representation Theory

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The Mathematics of Frobenius in Context

Abstract

The correspondence between Dedekind and Frobenius makes it clear that if Dedekind had not decided to introduce and study group determinants—a subject with no established tradition and really outside his main interests in algebraic number theory—or if he had decided not to communicate his ideas on group determinants to Frobenius, especially given Frobenius’ complete lack of curiosity about Dedekind’s allusion to a connection between hypercomplex numbers and groups, it is unlikely that Frobenius would be known as the creator of the theory of group characters and representations. This is not to say that the theory would have remained undiscovered for a long time. On the contrary, three lines of mathematical investigation were leading to essentially the same theory that Frobenius had begun to explore: (1) the theory of noncommutative hypercomplex number systems; (2) Lie’s theory of continuous groups; and (3) Felix Klein’s research program on a generalized Galois theory. The main purpose of this chapter is to briefly indicate how these lines of investigation were leading—or in some cases did lead—to the results of Frobenius’ theory.

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Notes

  1. 1.

    See my paper [267] for further details.

  2. 2.

    Actually, Poincaré did not explicitly assume the existence of a two-sided identity, but he—and those who followed him—made an equivalent assumption.

  3. 3.

    On Lie’s early work on continuous groups and its historical background, see Chapters 1–3 of my book [276] on the history of Lie groups.

  4. 4.

    On Killing’s work and its background, see Chapters 4 and 5 of my book [276].

  5. 5.

    In what follows, I have drawn upon N.F. Kanounov’s biography of Molien [335, 14–36].

  6. 6.

    For a discussion of Molien’s actual approach to hypercomplex systems in general and, in particular, an indication of how he proved his theorem on simple systems by means of analogies with Killing’s work, see Section 3 of my paper [267].

  7. 7.

    Here “simply transitive” means simply transitive on an open dense subset of \({\mathbb{C}}^{n}\). This follows directly from the fact that u R is invertible for all (u 1, …, u n ) in an open dense set.

  8. 8.

    The Killing form of a Lie algebra is related to the quadratic form ψ 2(u) by \(K(\mathbf{u},\mathbf{u}) = {[\psi _{1}(\mathbf{u})]}^{2} - 2\psi _{2}(\mathbf{u})\), which reduces to \(K(\mathbf{u},\mathbf{u}) = -2\psi _{2}(\mathbf{u})\) for any Lie algebra satisfying \(\mathfrak{g}^{\prime} = \mathfrak{g}\) and so in particular for semisimple Lie algebras. For a comparative discussion of the contributions of Killing and Cartan to the structure of Lie algebras, see Chapters 5 and 6 of my book [276].

  9. 9.

    Poincaré was familiar with Frobenius’ theory, which in 1912 he deemed the most important advance in the theory of finite groups in many years [486, p. 141], and he realized its connections with the work of Cartan. In 1903 [485, p. 106], he pointed out that “les théories de ces deux savants mathématiciens s’éclairent mutuellement.”

  10. 10.

    For references to the literature on the normal problem see [343, 608]. It should also be noted that Frobenius’ student I. Schur applied Frobenius’ theory of group representations to develop a general theory of projective representations. See Section 15.5.

  11. 11.

    In presenting Molien’s results, his notation has been slightly modified to bring it into line with that of Frobenius. In particular, the letters h, k, l have been given the same significance here as with Frobenius.

  12. 12.

    See the remarks following (13.21).

  13. 13.

    Molien’s letter was published (in Russian translation) by Kanounov [334, p. 57]. The rest of his brief correspondence with Frobenius (in Russian translation) may be found in Kanounov’s biography [335].

  14. 14.

    Molien’s proof is similar to that given later by Burnside [56, p. 299], who states the theorem in terms of a faithful representation \(\mu: \mathfrak{H} \rightarrow \mathfrak{G}\) of an abstract group \(\mathfrak{H}\).

  15. 15.

    Letter to Dedekind dated 24 February 1898.

  16. 16.

    According to Kanounov [333], Molien was refused the professorship at Dorpat as a consequence of the czarist regime’s Russification policy.

  17. 17.

    Letter to Dedekind dated 7 May 1896.

  18. 18.

    See pp. 273–279 of my paper [267] for a discussion of Burnside’s work in general and this point in particular.

  19. 19.

    See in this connection Loewy’s papers [419] and [420, p. 561n], Moore’s paper [448], and Klein’s announcement [344].

  20. 20.

    Picard failed to see that Poincaré’s technique of summing over a group to generate invariants (see below), which he himself had extended to certain countably infinite subgroups of \(\mathrm{PGL}(3, \mathbb{C})\) [472], would yield a completely general proof of Theorem 14.1.

  21. 21.

    See, e.g., Poincaré’s papers [476, p. 97], [475, p. 182] or the discussion in Gray’s book [255].

  22. 22.

    Hurwitz’s paper was mainly concerned with the extension of the technique to continuous groups. His paper played an important role in the extension of Frobenius’ theory of characters and representations to continuous groups as indicated below in Section 15.5.

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    Thomas Hawkins

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Hawkins, T. (2013). Alternative Routes to Representation Theory. In: The Mathematics of Frobenius in Context. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6333-7_14

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