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XII.—Clebsch-Aronhold Symbols and the Theory of Symmetric Functions*

  • H. W. Turnbull (a1) and A. H. Wallace (a1)
Synopsis

A square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.

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References
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Aronhold, , 1858. “Theorie der homogenen Funktionen dritten Grades von drei Veränderlichen”, Journ. für Math., LV, 97191.
Clebsch, , 1861 a. “Über eine Transformation der homogenen Funktionen dritter Ordnung mit vier Veränderlichen”, Journ. für Math., LVIII, 109126.
Clebsch, , 1861 b. “Über symbolische Darstellung algebraischer Formen”, Journ. für Math., LIX, 162.
Elliott, , 1913. Algebra of Quantics, Oxford.
Hammond, , 1882. “On the Calculating of Symmetric Functions”, Proc. London Math. Soc., ser. 1, XIII, 79.
Littlewood, , 1940. The Theory of Group Characters, Oxford.
MacMahon, , 1915. Combinatory Analysis, Cambridge.
MacMahon, , 1924. “Researches in the Theory of Determinants”, Trans. Cambridge Phil. Soc., XXIII, 89135.
MacMahon, , 1927. “The Structure of a Determinant”, Journ. London Math. Soc., 11, 273286.
Muir, , 1897. “A Relation between Permanents and Determinants”, Proc. Roy. Soc. Edin., XXII, 134136.
Rutherford, , 1948. Substitutional Analysis, Edinburgh.
Sylvester, , 1851. “On the Relation between the Minor Determinants of Linearly Equivalent Quadratic Functions”, Phil. Mag., ser. 4, 1, 295–305, 415; Coll. Math. Papers, 1, 241–250, 251.
Sylvester, , 1852. “On the Principles of the Calculus of Forms“, Cambridge and Dublin Math. Journ., VII, 5297; Coll. Math. Papers, 1, 284–327.
Turnbull, , 1932. “The Invariant Theory of a General Bilinear Form”, Proc. London Math. Soc., ser. 2, XXXIII, 119.
Turnbull, , 1945. The Theory of Determinants, Matrices and Lnvariants (2nd Ed.), London and Glasgow.
Young, , 1929. “On Quantitative Substitutional Analysis” (Fourth Memoir), Proc. London Math. Soc., ser. 2, xxxi, 253288.
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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