${\mathbb{F}}^2\otimes {\mathbb{F}}^3$中子空间和${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^r$轨道的分类
Classification of subspaces in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3$ and orbits in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^r$
摘要：
本文对有限、实、代数闭域的张量积空间${\mathbb{F}}^2\otimes {\mathbb{F}}^3 \otimes{\mathbb{F}}^r$、$r\geq 1$在两个自然群作用下的元素轨道进行了分类。对于每个轨道，我们确定：一个标准形;张量秩;压缩空间的秩分布;和几何描述。证明是基于对${\mathrm{PG}}({\mathbb{F}}^2\otimes{\mathbb{F}}^3)$中的收缩空间的研究，并且是几何性质的。虽然主要的焦点是在有限域上，但这些技术大多是与域无关的。
英文摘要：
This paper contains the classification of the orbits of elements of the tensor product spaces ${\mathbb{F}}^2\otimes {\mathbb{F}}^3 \otimes{\mathbb{F}}^r$, $r\geq 1$, under the action of two natural groups, for all finite; real; and algebraically closed fields. For each of the orbits we determine: a canonical form; the tensor rank; the rank distribution of the contraction spaces; and a geometric description. The proof is based on the study of the contraction spaces in ${\mathrm{PG}}({\mathbb{F}}^2\otimes{\mathbb{F}}^3)$ and is geometric in nature. Although the main focus is on finite fields, the techniques are mostly field independent.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
