AI 中文总结
本文证明由内置换源Morita等价诱导的完美等距的Broué不变量在符号意义下由局部数据决定，符号由源模的秩模p给出，并推出惯性块的Isaacs-Navarro细化猜想。
AI 中文摘要
由Broué引入的完美等距$I$是两个块$b$和$c$之间在有限群块理论中的常见现象。它将$c$的一个不可约特征标$\psi$映射到$b$的一个不可约特征标的$\pm$倍。Broué证明了$\psi$和$I(\psi)$的余次数之比是一个$p$-值为零的有理数，且其在$\mathbb{F}_p$中的类不依赖于$\psi$。Boltje称此元素为$I$的Broué不变量。本文的目标是证明：若$I$来自一个具有内置换源$V$的Morita等价，则$I$的Broué不变量在相差一个符号的意义下由$b$和$c$的局部数据决定。因此，在相差一个符号的意义下，它不依赖于内置换源Morita等价的选择。此外，我们证明该符号因子由$V$的秩模$p$的约化给出。作为推论，我们得到Isaacs--Navarro对Alperin--McKay猜想的细化对于惯性块成立。
英文摘要
A perfect isometry $I$ (introduced by Broué) between two blocks $b$ and $c$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $ψ$ of $c$ to $\pm$ an irreducible character of $b$. Broué proved that the ratio of the codegrees of $ψ$ and $I(ψ)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $ψ$. This element is called the Broué invariant of $I$ by Boltje. The goal of this paper is to show that if $I$ comes from a Morita equivalence with an endopermutation source $V$, then, up to a sign, the Broué invariant of $I$ is determined by local data of $b$ and $c$. Therefore, up to a sign, it is independent of the endopermutation-source Morita equivalence. Moreover, we show that the sign factor is given by the reduction of the rank of $V$ modulo $p$. As a corollary, we obtain that the Isaacs--Navarro refinement of the Alperin--McKay conjecture holds for inertial blocks.
