The Mathematical Intelligencer

, Volume 12, Issue 4, pp 10–16 | Cite as

Years ago

The one-hundredth anniversary of the death of invariant theory?
  • Karen V. H. Parshall


Linear Transformation British School Binary Quadratic Form Group Representation Theory Quartic Form 
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    James Byrnie Shaw,Synopsis of Linear Associative Algebra: A Report on Its Natural Development and Results Reached up to the Present Time, Washington, D.C.: Carnegie Institution of Washington (1907).Google Scholar
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    David Hubert, Über die Theorie der algebraischen Formen,Mathematische Annalen 36 (1890), 473–534; and “Über die vollen Invariantensysteme,” op. cit., 42 (1893), 313-373. Both of these papers have been translated by Michael Ackermann and commented upon by Robert Hermann inHubert’s Invariant Theory Papers, Lie Groups: History, Frontiers, and Application, vol. 8, Brookline: Math Sci Press (1978). All English quotations are taken from this source.CrossRefMathSciNetGoogle Scholar
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    Hermann Weyl,The Classical Groups: Their Invariants and Representations, Princeton: University Press (1939), 27. Of course, Hilbert also closed his 1893 paper with the remark: “With this, I believe, we have attained the most important general goals of a theory of the function fields formed by the invariants. [Ackermann, trans., p. 301.]”Google Scholar
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    Boole, p. 19. Note here that Boole only tacitly assumed the nonsingularity of his linear transformation, and all of his calculations were explicit. In what follows, I adhere to the historical presentation as much as possible, while still rendering it comprehensible to the modern reader. Thus, I do not use modern formulations and notations when they would trivialize or otherwise distort the nineteenth-century work.Google Scholar
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    Some impetus also came from the number-theoretic work of Gotthold Eisenstein.Google Scholar
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    Ibid., 89. Note that the Hessian is an example of a “covariant,” that is, an expression in the coefficientsand variables of a given form which remains invariant under a linear transformation of the variables of the original form. Clearly, invariants are just special cases of covariants.Google Scholar
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    See, in particular, Siegfried Aronhold, Zur Theorie der homogenen Functionen dritten Grades von drei Variabein,Journal für die reine und angewandte Mathematik 39 (1849), 140–159; Theorie der homogenen Functionen dritten Grades von drei Veränderlichen,op. cit., 55 (1858), 97–191; and Ueber eine fundamentale Begründung der Invariantensysteme,op. cit., 62 (1863), 281–345.Google Scholar
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    Ibid., 269–270.Google Scholar
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    For the exchange of letters between Hilbert and Klein relative to Hubert’s 1890 paper and Gordan’s reaction to it, see (Günther Frei, ed.),Der Briefwechsel David Hubert-Felix Klein (1886-1918), Göttingen: Vandenhoeck & Ruprecht (1985), 61–65. On 24 February, 1890, Gordan wrote to Klein that he was “very dissatisfied [sehr unzufrieden]” with Hilbert’s work because of its merely “existential” as opposed to “constructive” nature.Google Scholar
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    Ackermann, trans., p. 150.Google Scholar
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    Hilbert’s proof of this theorem, which is not unlike that in most standard algebra textbooks, also goes through under the more general hypothesis of a ringR in which every ideal is finitely generated. See, for example, Nathan Jacobson,Basic Algebra II, San Francisco: W. H. Freeman & Co. (1980), 417–418.zbMATHGoogle Scholar
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    Ackermann, trans., 165–183. For a more modern statement, see Weyl, 36; and Peter Hilton and Urs Stammbach,A Course in Homological Algebra, New York: Springer-Verlag (1971), 251–254. It is very important to note that, while Hilbert established the finiteness here, he did not provide any sort of aconstructive method ofproducing syzygies.Google Scholar
  31. 31.
    Among other things in this 1893 paper, Hilbert proved his famous Nullstellensatz (in section three). (See note [4] above.) This paper is generally credited with ushering in modern algebra, although as this article hopefully makes clear relative to invariant theory, these sorts of statements generally beg for deeper historical inquiry.Google Scholar
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    Ackermann, trans., 221–222. Hilbert’s emphasis.Google Scholar
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    Fisher, The last invariant theorists, 243.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 1990

Authors and Affiliations

  • Karen V. H. Parshall
    • 1
  1. 1.Departments of Mathematics and HistoryUniversity of VirginiaCharlottesvilleUSA

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