论文标题
在$ \ mathbb {f} _1 $上方的颤抖表示形式上
On quiver representations over $\mathbb{F}_1$
论文作者
论文摘要
我们研究Quiver $ Q $ Q $的表示类别$ \ textrm {rep}（q，q，\ mathbb {f} _1）$上的“带有一个元素的字段”，由$ \ mathbb {f} _1 _1 $表示，$ \ \ \\ textrm {rep {rep} $ {q，q，\ bbb { $ q $ $ \ $ \ m m i \ mathbb {f} _1 $的表示通常反映了$ \ mathbb {f} _q $的组合，但要显示一些微妙的 - 例如，我们证明，连接的Quiver $ q $是$ \ nathbb {f} $ if $ q y如果是$ \ mathbb {f} $ q y if $ q的有限表示类型。然后，对于每个表示$ \ MATHBB {V} $ $ Q $ of $ \ MATHBB {F} _1 $我们将系数Quiver $γ_\ Mathbb {V} $关联，其信息具有与$ \ Mathbb {v} $相同的信息。这使我们能够在$ \ mathbb {f} _1 $上翻译表示形式，纯粹是在相关系数Quivers的组合方面转换。我们还探讨了$ Q $ $ \ m athbb {f} _1 $ $ q $的不可分解表示的增长 - 与字段上的表示形式也有相似之处，但存在一些微妙的差异。最后，我们将$ n $ -loop颤抖的nilpotent表示类别的霍尔代数链接在$ \ mathbb {f} _1 $上，与szczesny介绍的偏斜形状的Hopf Algebra。
We study the category $\textrm{Rep}(Q,\mathbb{F}_1)$ of representations of a quiver $Q$ over "the field with one element", denoted by $\mathbb{F}_1$, and the Hall algebra of $\textrm{Rep}(Q,\mathbb{F}_1)$. Representations of $Q$ over $\mathbb{F}_1$ often reflect combinatorics of those over $\mathbb{F}_q$, but show some subtleties - for example, we prove that a connected quiver $Q$ is of finite representation type over $\mathbb{F}_1$ if and only if $Q$ is a tree. Then, to each representation $\mathbb{V}$ of $Q$ over $\mathbb{F}_1$ we associate a coefficient quiver $Γ_\mathbb{V}$ possessing the same information as $\mathbb{V}$. This allows us to translate representations over $\mathbb{F}_1$ purely in terms of combinatorics of associated coefficient quivers. We also explore the growth of indecomposable representations of $Q$ over $\mathbb{F}_1$ - there are also similarities to representations over a field, but with some subtle differences. Finally, we link the Hall algebra of the category of nilpotent representations of an $n$-loop quiver over $\mathbb{F}_1$ with the Hopf algebra of skew shapes introduced by Szczesny.
