论文标题
琐碎源环的正交单位组
The orthogonal unit group of the trivial source ring
论文作者
论文摘要
令$ g $为有限的组，$ p $ a Prime和$（K，\ Mathcal {o}，f）$ a $ p $ - 模块化系统。我们证明，$ \ Mathcal {o} g $的微不足道的源环对{\ em Coolent} $ g $ stable $（χ_p）$的环是同构，其中$χ_p$是$ k [n_g（p）/$ p $，$ p $ $ p $ $ p的虚拟特征某些字符值的平等。我们使用此结果来描述琐碎源环的正交单元作为$ g $融合系统的伯恩赛德环的产物，以及一组连贯的$ g $ g $ - 稳定的元组$（φ_p）$ n_g（p）$ n_g（p）/p \ to f^f^\ f^\ times $。 $ \ Mathcal {o} g $的琐碎源环的正交单位组引起了人们的关注，因为它嵌入了$ \ Mathcal {o} g $的$ p $ permuart自动等量的组中。
Let $G$ be a finite group, $p$ a prime, and $(K,\mathcal{O},F)$ a $p$-modular system. We prove that the trivial source ring of $\mathcal{O} G$ is isomorphic to the ring of {\em coherent} $G$-stable tuples $(χ_P)$, where $χ_P$ is a virtual character of $K[N_G(P)/P]$, $P$ runs through all $p$-subgroups of $G$, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of $G$ with the group of coherent $G$-stable tuples $(φ_P)$ of homomorphisms $N_G(P)/P\to F^\times$. The orthogonal unit group of the trivial source ring of $\mathcal{O} G$ is of interest, since it embeds into the group of $p$-permutation autoequivalences of $\mathcal{O} G$.
