Range-compatible homomorphisms on spaces of symmetric or alternating matrices

Pazzis, Clément de Seguins; de Seguins Pazzis, Clément

Pazzis, Clément de Seguins ; de Seguins Pazzis, Clément

HAL, hal-01690932 / Harvested from HAL

Text on HAL

Résumé

Let

$U$ and

$V$ be finite-dimensional vector spaces over an arbitrary field

$\mathbb{K}$ , and

$\mathcal{S}$ be a linear subspace of the space

$\mathcal{L}(U,V)$ of all linear maps from

$U$ to

$V$ . A map

$F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies

$F(s) \in \mathrm{im}(s)$ for all

$s \in \mathcal{S}$ . Among the range-compatible maps are the so-called local ones, that is the maps of the form

$s \mapsto s(x)$ for a fixed vector

$x$ of

$U$ . In recent works, we have classified the range-compatible group homomorphisms on

$\mathcal{S}$ when the codimension of

$\mathcal{S}$ in

$\mathcal{L}(U,V)$ is small. In the present article, we study the special case when

$\mathcal{S}$ is a linear subspace of the space

$S_n(\mathbb{K})$ of all

$n$ by

$n$ symmetric matrices: we prove that if the codimension of

$\mathcal{S}$ in

$S_n(\mathbb{K})$ is less than or equal to

$n-2$ , then every range-compatible homomorphism on

$\mathcal{S}$ is local provided that

$\mathbb{K}$ does not have characteristic

$2$ . With the same assumption on the codimension of

$\mathcal{S}$ , we also classify the range-compatible homomorphisms on

$\mathcal{S}$ when

$\mathbb{K}$ has characteristic

$2$ . Finally, we prove that if

$\mathcal{S}$ is a linear subspace of the space

$A_n(\mathbb{K})$ of all

$n$ by

$n$ alternating matrices with entries in

$\mathbb{K}$ , and the codimension of

$\mathcal{S}$ is less than or equal to

$n-3$ , then every range-compatible homomorphism on

$\mathcal{S}$ is local.

Publié le : 2016-08-04
Classification: [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]

@article{hal-01690932,
     author = {Pazzis, Cl\'ement de Seguins and de Seguins Pazzis, Cl\'ement},
     title = {Range-compatible homomorphisms on spaces of symmetric or alternating matrices},
     journal = {HAL},
     volume = {2016},
     number = {0},
     year = {2016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-01690932}
}

Pazzis, Clément de Seguins; de Seguins Pazzis, Clément. Range-compatible homomorphisms on spaces of symmetric or alternating matrices. HAL, Tome 2016 (2016) no. 0, . http://gdmltest.u-ga.fr/item/hal-01690932/