In this paper we will study the action of $\mathbb{F}_{q^n}^{2 \times 2}$ on the graph of an $\mathbb{F}_q$-linear function of $\mathbb{F}_{q^n}$ into itself. In particular we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also see some examples for which this does not happen. Moreover, we will establish a connection between such a stabilizer and the right idealizer of the rank-metric code defined by the linear function and give some structural results in the case in which the polynomials are partially scattered.

翻译：本文研究$\mathbb{F}_{q^n}^{2 \times 2}$在$\mathbb{F}_q$-线性函数（将$\mathbb{F}_{q^n}$映射到自身）的图上的作用。特别地，我们将在特定组合假设下证明，其稳定化子（连同矩阵的加法和乘法）构成一个域。我们还将给出一些反例。此外，我们将建立该稳定化子与由该线性函数定义的秩度量码的右理想化子之间的联系，并在多项式为部分散射的情形下给出一些结构性的结果。