On the linear preservers of Schur matrix functionals

2018/07/17 by Clément de Seguins Pazzis, Pazzis, Clément de Seguins · 2 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1807.06264

Abstract

Let \mathbbF be a field and f : \mathfrakS_n → \mathbbF ∖ \0\ be an arbitrary map. The Schur matrix functional associated to f is defined as M ∈ M_n(\mathbbF) ↦ \widetildef(M):=∑_σ∈ \mathfrakS_n f(σ) ∏_j=1ⁿ m_σ(j),j. Typical examples of such functionals are the determinant (where f is the signature morphism) and the permanent (where f is constant with value 1). Given two such maps f and g, we study the endomorphisms U of the vector space M_n(\mathbbF) that satisfy \widetildeg(U(M))=\widetildef(M) for all M ∈ M_n(\mathbbF). In particular, we give a closed form for the linear preservers of the functional \widetildef when f is central, and as a special case we extend to an arbitrary field Botta's characterization of the linear preservers of the permanent.

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