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Immanant conversion on symmetric matrices. (English) Zbl 1294.15008

Summary: Let \(\sigma_n(\mathbb{C})\) denote the space of all \(n\times n\) symmetric matrices over the complex field \(\mathbb{C}\). – The main objective of this paper is to prove that the maps \(\Phi:\Sigma_n(\mathbb{C})\to \Sigma_n(\mathbb{C})\) satisfying for any fixed irreducible characters \(\chi,\chi': S_n\to \mathbb{C}\) the condition \(d_\chi(A+\alpha B)= d_{\chi'}(\Phi(A)+ \alpha\Phi(B))\) for all matrices \(A, B\in\Sigma_n(\mathbb{C})\) and all scalars \(\alpha\in\mathbb{C}\) are automatically linear and bijective. As a corollary of the above result we characterize all such maps \(\Phi\) acting on \(\Sigma_n(\mathbb{C})\).

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A86 Linear preserver problems
15B57 Hermitian, skew-Hermitian, and related matrices
15A04 Linear transformations, semilinear transformations
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References:

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