论文标题
对角线$ p $ - permuntunt函子，半径和功能等效性
Diagonal $p$-permutation functors, semisimplicity, and functorial equivalence of blocks
论文作者
论文摘要
让$ k $为特征$ p> 0 $的代数封闭的字段，让$ r $成为一个交换戒指，让$ \ mathbb {f} $是特征性0的代数封闭的字段。我们考虑$ r $ linear类别$ \ mathcal {f} f}^f}^f}^uppp__ {rpp_k} $ - 我们首先表明类别$ \ MATHCAL {f}^δ_ {\ Mathbb {f} pp_k} $是半imple，我们对其简单对象进行了参数化以及对其评估的描述。接下来，对于任何一对$（g，b）$的有限组$ g $和一个块idempotent $ b $的$ kg $，我们将对角$ p $ -p $ -p $ -permuntion fuction $ rt^δ_{g，b} $ in $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ rpp} {f}^Δ_{rpp_ {rpp_ {rpp_k_k} $。我们发现函数$ \ mathbb {f} t^δ__{g，b} $的分解是$ \ mathcal {f}^δ_ {\ mathbb {f} pp_k} $中简单函数的直接总和。这导致了nilpotent块的表征，其相关函数在$ \ mathcal {f}^δ_ {\ Mathbb {f} pp_k} $中。 Finally, for such pairs $(G,b)$ of a finite group and a block idempotent, we introduce the notion of functorial equivalence over $R$, which (in the case $R=\mathbb{Z}$) is slightly weaker than $p$-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite $p$-group $D$, there is only a finite number of对$（g，b）$，其中$ g $是一个有限的组，每块$ b $ hobs Idempotent为$ kg $，缺陷同构至$ d $，最多超过$ \ mathbb {f} $。
Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring, and let $\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider the $R$-linear category $\mathcal{F}^Δ_{Rpp_k}$ of diagonal $p$-permutation functors over $R$. We first show that the category $\mathcal{F}^Δ_{\mathbb{F}pp_k}$ is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations. Next, to any pair $(G,b)$ of a finite group $G$ and a block idempotent $b$ of $kG$, we associate a diagonal $p$-permutation functor $RT^Δ_{G,b}$ in $\mathcal{F}^Δ_{Rpp_k}$. We find the decomposition of the functor $\mathbb{F}T^Δ_{G,b}$ as a direct sum of simple functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. This leads to a characterization of nilpotent blocks in terms of their associated functors in $\mathcal{F}^Δ_{\mathbb{F}pp_k}$. Finally, for such pairs $(G,b)$ of a finite group and a block idempotent, we introduce the notion of functorial equivalence over $R$, which (in the case $R=\mathbb{Z}$) is slightly weaker than $p$-permutation equivalence, and we prove a corresponding finiteness theorem: for a given finite $p$-group $D$, there is only a finite number of pairs $(G,b)$, where $G$ is a finite group and $b$ a block idempotent of $kG$ with defect isomorphic to $D$, up to functorial equivalence over $\mathbb{F}$.
