Article Dans Une Revue Linear Algebra and its Applications Année : 2016

Range-compatible homomorphisms on spaces of symmetric or alternating matrices

Résumé

Let U and V be finite-dimensional vector spaces over an arbitrary field K, and S be a linear subspace of the space L(U,V) of all linear maps from U to V. A map F:SV is called range-compatible when it satisfies F(s)im(s) for all sS. Among the range-compatible maps are the so-called local ones, that is the maps of the form ss(x) for a fixed vector x of U. In recent works, we have classified the range-compatible group homomorphisms on S when the codimension of S in L(U,V) is small. In the present article, we study the special case when S is a linear subspace of the space Sn(K) of all n by n symmetric matrices: we prove that if the codimension of S in Sn(K) is less than or equal to n2, then every range-compatible homomorphism on S is local provided that K does not have characteristic 2. With the same assumption on the codimension of S, we also classify the range-compatible homomorphisms on S when K has characteristic 2. Finally, we prove that if S is a linear subspace of the space An(K) of all n by n alternating matrices with entries in K, and the codimension of S is less than or equal to n3, then every range-compatible homomorphism on S is local.

Dates et versions

hal-01690932 , version 1 (23-01-2018)

Identifiants

Citer

Clément de Seguins Pazzis, Clément de Seguins Pazzis. Range-compatible homomorphisms on spaces of symmetric or alternating matrices. Linear Algebra and its Applications, 2016, 503, pp.135 - 163. ⟨10.1016/j.laa.2016.03.047⟩. ⟨hal-01690932⟩
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