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    Rowe, David and Repka, Joe 2016. Dual Pairs of Holomorphic Representations of Lie Groups from a Vector-Coherent-State Perspective. Symmetry, Vol. 8, Issue. 3, p. 12.


    Rowe, D J 2015. Applications of the Capelli identities in physics and representation theory. Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue. 5, p. 055203.


    Caracciolo, Sergio Sokal, Alan D. and Sportiello, Andrea 2013. Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians. Advances in Applied Mathematics, Vol. 50, Issue. 4, p. 474.


    Chervov, A. Falqui, G. and Rubtsov, V. 2009. Algebraic properties of Manin matrices 1. Advances in Applied Mathematics, Vol. 43, Issue. 3, p. 239.


    Howe, Roger and Umeda, T�ru 1991. The Capelli identity, the double commutant theorem, and multiplicity-free actions. Mathematische Annalen, Vol. 290, Issue. 1, p. 565.


    Kostant, Bertram and Sahi, Siddhartha 1991. The Capelli identity, tube domains, and the generalized Laplace transform. Advances in Mathematics, Vol. 87, Issue. 1, p. 71.


    Wallace, Andrew H. 1953. A Note on the Capelli Operators associated with a Symmetric Matrix. Proceedings of the Edinburgh Mathematical Society, Vol. 9, Issue. 01, p. 7.


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  • Proceedings of the Edinburgh Mathematical Society, Volume 8, Issue 2
  • October 1948, pp. 76-86

Symmetric Determinants and the Cayley and Capelli Operator

  • H. W. Turnbull (a1)
  • DOI: http://dx.doi.org/10.1017/S0013091500024822
  • Published online: 01 January 2009
Abstract

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

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Capelli (1882, 1886), Math. Annalen 29, 331338.

H. S. White , Bulletin American Math. Soc. 2 (1895) 136138:

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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