论文标题
代数的线性退化和一般线性组的某些表示
Linear degenerations of algebras and certain representations of the general linear group
论文作者
论文摘要
令$ \boldsymbolλ\，（= \ mathbb {f}^{n^{3}}）$，其中$ \ mathbb {f} $是一个领域，具有$ | \ m athbb {f} |> 2 $，是$ n $ n $ a $ assension $ n $ assensial $ n $ assensial $ n $ n $ n $ n $ n $ n $ n $ - 向量空间。也让$ g = gl（v）$。关于$ G $ -MODULE的$ \boldsymbolλ$，通过“基础更改”操作〜$ g $在〜$ v $上，我们确定〜$ \boldsymbolλ$的各种$ g $ submodules的组成因子，这些因子与某些重要的代数家庭相对应。这是通过引入线性变性的概念来实现的，该概念使我们能够在代数退化的某些已知结果上获得$ \ mathbb {f} $的类似物。结果，确定了$ gl（v）$ - 〜$ \boldsymbolλ$的结构。
Let $\boldsymbolΛ\,(=\mathbb{F}^{n^{3}})$, where $\mathbb{F}$ is a field with $|\mathbb{F}|>2$, be the space of structure vectors of algebras having the $n$-dimensional $\mathbb{F}$-space $V$ as the underlying vector space. Also let $G=GL(V)$. Regarding $\boldsymbolΛ$ as a $G$-module via the `change of basis' action of~$G$ on~$V$, we determine the composition factors of various $G$-submodules of~$\boldsymbolΛ$ which correspond to certain important families of algebras. This is achieved by introducing the notion of linear degeneration which allows us to obtain analogues over $\mathbb{F}$ of certain known results on degenerations of algebras. As a result, the $GL(V)$-structure of~$\boldsymbolΛ$ is determined.
