Symmetry operators, polarizations, and a generalized Capelli identity
Titel:
Symmetry operators, polarizations, and a generalized Capelli identity
Auteur:
Williamson, S. G.
Verschenen in:
Linear & multilinear algebra
Paginering:
Jaargang 10 (1981) nr. 2 pagina's 93-102
Jaar:
1981-04
Inhoud:
The classical Capelli identity [1,5, p. 88, 7, p. 39], a fundamental tool in the theory of invariants, shows that, in the case of the symmetric group, the alternating symmetry operator acting on multilinear forms is the determinant of polarizations in the variables of the forms [5, p. 88]. Rota [5, p. 88] has posed the question as to whether or not other symmetry operators of the symmetric group are expressible explicitly in a similar manner in terms of polarizations of variables of the forms. In this paper we answer this question in the affirmative by showing that the symmetry operator Tm associated with any representation M of the symmetric group is a Schur matrix function [6] of polarizations in the variables of the form. In the particular case where M is of degree one we have either M = sgn, the "sign" or alternating character, or M = id, the trivial character. In the former case we obtain the classical result expressing the alternating symmetry operator as a determinant of polarizations. In the latter case, the completely symmetric operator (p(1, 2, …m) becomes a permanent of polarizations. These results are obtained by proving a generalization of Capelli's identity.
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