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Characters and Representations After 1897

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The Mathematics of Frobenius in Context

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Abstract

Frobenius’ papers of 1896–1897 marked the beginning of a new theory, a theory that continued to evolve in various directions for over a half-century. Frobenius himself, along with Burnside, made significant contributions to the theory after 1897, and many new ideas, viewpoints, and directions were introduced by Frobenius’ student Issai Schur (1875–1941), and then by Schur’s student Richard Brauer (1901–1977). In this chapter, these later developments will be sketched, with particular emphasis on matters that relate to the presentation in the previous sections.

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Notes

  1. 1.

    Since \(\hat{\sigma }(R) =\hat{\sigma } (x)\) with \(x\) specialized to x R  = 1, x S  = 0, S ≠ R, it is easy to see that \(\hat{\sigma }(z) =\hat{\sigma } (x)\hat{\sigma }(y)\) implies that \(\hat{\sigma }(AB) =\hat{\sigma } (A)\hat{\sigma }(B)\); that \(\hat{\sigma }(E) = I\) is immediate from the definition. Frobenius’ entire definition can be articulated without group matrices as follows: Extend σ to all of \(\mathfrak{H}\) by setting σ(R) = 0 for all \(R\notin \mathfrak{G}\). Then it is easy to check that \(\hat{\sigma }(S) =\big (\sigma _{ij}(S)\big)\), where \(\sigma _{ij}(S) =\sigma (A_{i}^{-1}SA_{j})\) for all \(S \in \mathfrak{H}\).

  2. 2.

    In his “Representation” paper, Frobenius had shown that if μ is any representation of \(\mathfrak{H}\), then the elementary divisors of \(\mu (x)\) are all linear [213, p. 87].

  3. 3.

    See Frobenius’ 1903 paper [222], which is discussed further on.

  4. 4.

    Letter dated 23 December 1893, and located in the archives of the Niedersächsiche Staats- und Universitätsbibliothek, Göttingen (Cod.  Ms. Philos. 205. Nr. 16). Although Frobenius was critical of Dedekind’s penchant for what he regarded as unnecessary abstraction, his admiration for Dedekind is also evident throughout these passages. The book that Weber planned became his classic Lehrbuch der Algebra [582, 583]. The edition of Dirichlet’s Vorlesungen über Zahlentheorie to which Frobenius referred was the forthcoming fourth edition of 1894.

  5. 5.

    The value χ (κ)(E) could also have been computed using, e.g., the coefficient C 3, 1, 0, 6 of \(x_{1}^{3}x_{2}x_{4}^{6}\), which equals − 3, for then sgnΔ(3, 1, 0, 6) = sgn − 540 =  − 1.

  6. 6.

    Young interpreted the product PQ of two permutations as Q followed by P, hence reading from right to left, whereas Frobenius adopted the reverse convention. I have followed Young’s convention in presenting Frobenius’ results.

  7. 7.

    On this matter, see Section 11.5 of my book [276].

  8. 8.

    Letter dated 15 July 1896.

  9. 9.

    Frobenius used the word “group” instead of “hypercomplex system” and in general used group-related terms to describe properties of hypercomplex systems.

  10. 10.

    Burnside of course meant “odd composite order,” since groups of prime order are simple.

  11. 11.

    If r is one of the \(\varphi (q - 1)\) primitive roots of q, then m is defined by r m ≡ p (mod q).

  12. 12.

    Using Burnside’s p a q b theorem, P. Hall generalized it to prove that \(\mathfrak{H}\) is solvable if it has the property that for every representation of its order as a product of two relatively prime integers, subgroups of those orders exist. Hall’s proof involved the notion of a Hall subgroup, which played a significant role in later work, including Feit and Thompson’s “odd-order paper” [162] discussed in the following paragraph.

  13. 13.

    A proof without characters of Burnside’s theorem on groups of order p a q b can be gleaned from the Feit–Thompson paper [162]. Relatively short proofs without characters were given in the 1970s by Goldschmidt [251] for p, q odd and by Matsuyama [436] for p = 2. These proofs utilize some of the modern ideas and results in group theory and are not as elementary as Burnside’s proof.

  14. 14.

    See the accounts by Aschbacher [9] and Gorenstein [252].

  15. 15.

    A more detailed discussion of Schur’s dissertation and its role in the history of the representation theory of Lie groups can be found in my book [276]. See especially Section 3 of Chapter 10.

  16. 16.

    In addition to these “absolute invariants,” more general invariants satisfying \(I(b^{\prime}) = {(\det \,A)}^{w}I(b)\), where w is a nonnegative integer, were also considered.

  17. 17.

    This and the following quotations are my translation of portions of the conclusion of Frobenius’ evaluation as transcribed by Biermann [22, p. 127].

  18. 18.

    Quoted by Biermann [22, p. 135]. Later, in a 1914 memorandum supporting Schur for a position, Frobenius expressed similar sentiments about Schur’s work in general: “As only a few other mathematicians do, he practices the Abelian art of correctly formulating problems, suitably transforming them, cleverly dismantling them, and then conquering them one by one” [22, pp. 139, 223]. After his death, Frobenius’ words were quoted by Planck when Schur was admitted to membership in the Berlin Academy of Sciences in 1922. See Schur, Gesammelte Abhandlungen 2, p. 414.

  19. 19.

    These words were written in a memorandum supporting Schur for a professorship [22, p. 224], but they are not at all an exaggeration of Schur’s accomplishments vis à vis the efforts of the Klein school.

  20. 20.

    The group \(\mathfrak{M}\) can be identified with the second cohomology group of \(\mathfrak{H}\) over \(\mathbb{C}\), but group cohomology did not exist at this time. On this and other anticipations of modern theories by Schur, see [406, p. 101]. See also the discussion in Section 15.6 below of Brauer’s work on Schur’s index theory and its connections with Galois cohomology.

  21. 21.

    Schur, Abhandlungen 1, 346–441.

  22. 22.

    For an exposition of Schur’s proof and its role in deriving Schur’s orthogonality relations for irreducible representations, see Curtis’ book [109, pp. 140ff.].

  23. 23.

    Frobenius used the word Kärrnerarbeit. See his memoranda of 1902 and 1914 to the Prussian Ministry of Culture regarding the possible appointment of Hilbert to a professorship in Berlin [22, pp. 209–210, 222–223].

  24. 24.

    The finiteness problem for a given type of invariant is to prove that there is a finite number of such invariants such that any invariant of the given type is expressible as a polynomial function of these.

  25. 25.

    The joint work is contained in Schur, Abhandlungen 2, 334–358, and Ostrowski, Papers 2, 127–151, and is discussed in my book on the history of Lie group theory [276, Ch. 10, §4].

  26. 26.

    I am grateful to the late Mrs. Brauer and to Walter Feit and Jonathan Alperin for kindly making Richard Brauer’s notes from these lectures available to me. A later version of Schur’s lectures was published by Grunsky [527].

  27. 27.

    I have discussed Cayley’s counting problem and its extensions by Molien and Schur (alluded to below) in [272] and [276, Ch. 7,§4, Ch. 10,§5].

  28. 28.

    For the full story, see my paper [274], and especially Chapters 11–12 of my book [276].

  29. 29.

    For a clear and detailed discussion of the results in Schur’s paper [524], including sketches of some of the proofs, sometimes along more modern lines, see Curtis’ book [109, pp. 157ff.].

  30. 30.

    See Section 5 of Loewy’s paper [422]. Maschke’s theorem is actually valid for any field with characteristic not dividing the order of the group—a fact that is immediately clear from the proof Schur gave using his lemma [523, §3]—but Loewy was not interested in fields of finite characteristic [422, p. 59n].

  31. 31.

    For Schur’s other arithmetic researches on representation theory, see his Abhandlungen 1, 251–265 (1908), 295–311 (1909), 451–463 (1911).

  32. 32.

    See Encyclopedic Dictionary of Mathematics, 2nd edn., Art. 190.N.

  33. 33.

    The following presentation of Brauer’s work draws heavily on the articles on Brauer by Feit [160] and (especially) Green [256]. Not long after I had written it, Curtis’ book Pioneers of Representation Theory [109] appeared. The final two chapters provide a much more extensive and mathematically detailed treatment of Brauer’s work. In particular, Brauer’s theory of blocks and its application to the theory of finite groups [109, Ch. VII, §3] is not covered in my presentation.

  34. 34.

    Wedderburn’s work and its historical background have been treated in detail by Parshall [461].

  35. 35.

    See the personal reminiscences of Alfred Brauer, Richard Brauer’s brother, on p. vii of Schur’s Abhandlungen 1.

  36. 36.

    See Section 59 of the 1923 edition [544] or Section 69 of the second edition of 1927 [545].

  37. 37.

    According to Brauer’s own recollections, Papers 1, p. xviii.

  38. 38.

    If \({\omega }^{{p}^{a} } = 1\), then the binomial expansion implies that \({(\omega -1)}^{{p}^{a} } \equiv 0\;(\mathrm{mod}\,p)\), and so ω = 1 in \(\mathbb{K}_{p}\).

  39. 39.

    After giving his proof, Speiser wrote “damit ist bewiesen, dass wir in den irreduziblen algebraischen Darstellungen alle irreduziblen Darstellungen im GF(p n) gefunden haben” [1923:167; 1927:223].

  40. 40.

    In 1947, Brauer used his induction theorem (discussed below) to show that \(\varepsilon\) can be taken as a primitive mth root of unity, where m is the least common multiple of the orders of elements in \(\mathfrak{H}\). This proof did not use Brauer characters. See Brauer’s Collected Papers 1, 553.

  41. 41.

    For a lucid historical overview of the history of class field theory and references to the primary and secondary literature, see Conrad’s exposition [105].

  42. 42.

    At this time, Artin still spoke of “substitutions” rather than automorphisms.

  43. 43.

    Seven years later, in his second paper on generalized L-functions [8], Artin showed how to deal as well with the case of ramified primes \(\mathfrak{p}\).

  44. 44.

    See the footnote to Theorem 9.18.

  45. 45.

    See following Theorem 9.20.

  46. 46.

    See Brauer’s remarks [34, pp. 502–503, 503, n. 3].

  47. 47.

    Recall that \(\mathfrak{H}^{\prime}\) is generated by all products of the form \(H_{1}H_{2}H_{1}^{-1}H_{2}^{-1}\), with \(H_{1},H_{2} \in \mathfrak{H}\). Since χ i is a Dedekind character, \(\chi _{i}(H_{1}H_{2}H_{1}^{-1}H_{2}^{-1}) =\chi _{i}(H_{1})\chi _{i}(H_{2})\chi _{i}{(H_{1})}^{-1}\chi _{i}{(H_{2})}^{-1} = 1\) because complex numbers commute.

  48. 48.

    A p-primary subgroup is a subgroup every element of which has order a power of p. Since \(\mathfrak{H}\) is finite, the p-primary subgroup has order that is a power of p.

  49. 49.

    Gelbart [250] has written an illuminating expository account of the Langlands program that indicates how Artin’s conjecture fits into it. See especially pp. 203–204 and 208–209. Langlands’ expository article [396] also conveys an idea of the role in the theory of numbers played by representation theory.

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    Thomas Hawkins

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Hawkins, T. (2013). Characters and Representations After 1897. 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_15

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