$D^\Delta$-对双集范畴
The $D^Δ$-pair biset category
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中文总结 AI 辅助
本文定义了以$D^\Delta$-对为对象的双集范畴，证明对角$p$-置换函子范畴等价于该范畴的自然商范畴上的线性函子范畴，为研究对角$p$-置换函子提供了新途径。
中文摘要 AI 辅助
设$p$为素数，$D^\Delta$-对由有限$p$-群及其$p'$-自同构组成。本文引入对角$D^\Delta$-对双集，定义以$D^\Delta$-对为对象、以对角$D^\Delta$-对双集的格罗滕迪克群为态射群的范畴。主要结果表明，在合适的系数环上，对角$p$-置换函子范畴等价于该新范畴的一个自然商范畴上的线性函子范畴，由此可仅通过有限$p$-群及其$p'$-阶自同构构建的范畴来研究对角$p$-置换函子。
英文摘要
Let $p$ be a prime number. A $D^Δ$-pair is a pair consisting of a finite $p$-group and a $p'$-automorphism of the group. In this paper, we introduce diagonal $D^Δ$-pair bisets and a category whose objects are $D^Δ$-pairs and whose morphism groups are Grothendieck groups of diagonal $D^Δ$-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal $p$-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal $p$-permutation functors can be studied through a category built only from finite $p$-groups and their automorphisms of $p'$-order.
