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Polarization of symmetric functions. (Polarisation de fonctions symétriques.) (French) Zbl 1304.05148

Algèbre et théorie des nombres 2012/2. Besançon: Presses Universitaires de Franche-Comté. Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2012/2, 113-122 (2012).
The aim of this paper is to give a proof of a theorem stated without proof by K. Lønsted and S. L. Kleiman [Compos. Math. 38, 83–111 (1979; Zbl 0406.14017)]. The theorem states that for all \(a \in A\) and for all integers \(j\) such that \(1\leq j \leq \sigma_j(a)\), \[ \Sigma_R^G(A) = R[\bigcup_{1\leq j \leq n}] \sigma_j(a), \] where \( \sigma_j = S_j(g_1a,\dots,g_na)\) is the \(j\)-th elementary symmetric polynomial in \(n\) variables, \(R\) is a commutative ring, \(A\) an \(R\)-algebra, and \(G\) a finite group of \(R\)-automorphisms of \(A\). \(A^G\) is the \(R\)-subalgebra of \(A\), invariant under the action of \(G\) on \(A\) and \( \Sigma_R^G\) is an \(R\)-subalgebra of \(A^G\).
The proof employs the tool of polarization of symmetric polynomials (as defined, for example, in the well-known work of H. Weyl [The classical groups, their invariants and representations. Princeton, New Jersey: Univ. Press et al. (1939; Zbl 0020.20601)]. These functions have the virtue of being invariants of products of symmetric groups. The theorem for which they will enable a proof has important consequences in algebraic geometry.
For the entire collection see [Zbl 1258.11002].

MSC:

05E05 Symmetric functions and generalizations
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