论文标题
有限的通用线性组的deligne类别和表示，第1部分：通用属性
Deligne categories and representations of the finite general linear group, part 1: universal property
论文作者
论文摘要
我们研究了deligne插值类别$ \ usewiendline {\ mathrm {rep}}（gl_ {t}（\ mathbb {f} _q））$ for $ t \ in \ mathbb {c} $，首先由F. Knop引入。这些类别插值有限的通用线性组$ gl_n（\ mathbb {f} _q）$的有限维数表示的类别。我们通过发电机和关系描述了这一类别中的形态空间。我们表明，此类别的生成对象（表示表示$ \ Mathbb {C} \ Mathbb {f} _Q^n $的$ gl_n（\ Mathbb {f} _Q）$）带有frobenius algebra的结构，该结构具有兼容的$ \ mathbbbbbb {我们调用此类对象$ \ mathbb {f} _q $ - 线性frobenius spaces，并证明$ \ usewiessline {\ mathrm {rep}}}（gl_ {t}（\ mathbb {f} _q _q））是$ \ mather的通用单体类别。分类维度$ t $。在本文的第二部分中，我们证明了$ gl _ {\ infty}（\ Mathbb {f} _Q）$的表示类别的类别的通用属性。
We study the Deligne interpolation categories $\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ for $t\in \mathbb{C}$, first introduced by F. Knop. These categories interpolate the categories of finite dimensional complex representations of the finite general linear group $GL_n(\mathbb{F}_q)$. We describe the morphism spaces in this category via generators and relations. We show that the generating object of this category (an analogue of the representation $\mathbb{C}\mathbb{F}_q^n$ of $GL_n(\mathbb{F}_q)$) carries the structure of a Frobenius algebra with a compatible $\mathbb{F}_q$-linear structure; we call such objects $\mathbb{F}_q$-linear Frobenius spaces, and show that $\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ is the universal symmetric monoidal category generated by such an $\mathbb{F}_q$-linear Frobenius space of categorical dimension $t$. In the second part of the paper, we prove a similar universal property for a category of representations of $GL_{\infty}(\mathbb{F}_q)$.
