论文标题
积极特征的新的不可压缩的对称张量类别
New incompressible symmetric tensor categories in positive characteristic
论文作者
论文摘要
我们提出了一种在代数封闭的字段$ \ bf k $上构建对称刚性karoubian类别的亚伯式信封的方法。如果$ {\ rm char}（{\ bf k}）= p> 0 $，我们使用此方法来构建概括$ {\ rm ver} _ {p^n} $，$ {\ rm ver} _ {p^n}^+$ in Incompressiable Abelian Symetrian Symbletic symetric Tensor的tensor tensor tecriencoriand arx in arx in arx in arx in ar arx in arx in arx and arx类别$ P = 2 $，Gelfand-Kazhdan和Georgiev-Mathieu $ n = 1 $。即，$ {\ rm ver} _ {p^n} $是$ sl_2（\ bf k）$的类别的abelian信封，$ n $ th $ th steinberg模块，$ n $ the steinberg模块，$ {\ rm ver} _ _ p^n} _2^+$ ns subcation subcatory是subcatory（ k）$ - 模块。我们表明，$ {\ rm ver} _ {p^n} $是特征零中Verlinde编织的张量类别的特征$ p $的减少，这解释了符号。我们详细研究了这些类别的结构，尤其表明它们将它们分类为真实的环形环$ \ Mathbb {z} [2 \ cos（2π/p^n）] $，以及该$ {\ rm ver} _ {p^n} $嵌入$ {\ rm ver} _ {我们推测，$ \ bf k $超过$ \ bf k $的每个中度增长类别都接纳了嵌套序列$ {\ rm ver} _ {\ rm ver} _ {p} _ {p} {p} {p} \ subset \ subset {\ rm rm rm ver {$ ver {$ cy的$ {p^\ infty} $的纤维函数。这将在特征零中提供DELIGNE定理的类似物，并提供Arxiv：1503.01492的结果的概括，该概念表明该猜想对融合类别有，然后将光纤函子降落在$ {\ rm ver} _p $中。
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to construct generalizations ${\rm Ver}_{p^n}$, ${\rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${\rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(\bf k)$ by the $n$-th Steinberg module, and ${\rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(\bf k)$-modules. We show that ${\rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $\mathbb{Z}[2\cos(2π/p^n)]$, and that ${\rm Ver}_{p^n}$ embeds into ${\rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $\bf k$ admits a fiber functor to the union ${\rm Ver}_{p^\infty}$ of the nested sequence ${\rm Ver}_{p}\subset {\rm Ver}_{p^2}\subset\cdots$. This would provide an analog of Deligne's theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${\rm Ver}_p$.
