论文标题
Cameron和Kiyota对具有规定价值的尖锐字符的猜想
A conjecture of Cameron and Kiyota on sharp characters with prescribed values
论文作者
论文摘要
令$χ$为有限组$ g $的虚拟（广义）字符，$ l = l = l（χ）$是$ g- \ lbrace 1 \ rbrace $的$χ$的图像。这对$（g，χ）$据说是$ l $的尖锐，如果$ | g | = \ prod _ {l \ in L}（χ（1） - L）$。如果$ g $的主要特征不是$χ$的不可约组成部分，则对$ $（g，χ）$称为归一化。在本文中，我们首先为卡梅伦和基约塔以1988美元的价格提出的猜想提供了一些反例。该猜想指出，如果$（g，χ）$是尖锐的，并且$ | l | \ geq 2 $，则内部产品$（χ，χ）_g $由$ L $唯一确定。然后，我们证明，如果$（g，χ）$归一化，$χ$是$ g $的字符，而$ l $至少包含至少一个不合理的值，那么这个猜想是正确的。
Let $ χ$ be a virtual (generalized) character of a finite group $ G $ and $ L=L(χ)$ be the image of $ χ$ on $ G-\lbrace 1 \rbrace $. The pair $ (G, χ) $ is said to be sharp of type $ L $ if $|G|=\prod _{ l \in L} (χ(1) - l) $. If the principal character of $G$ is not an irreducible constituent of $χ$, the pair $(G,χ)$ is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in $1988$. This conjecture states that if $(G,χ)$ is sharp and $|L|\geq 2$, then the inner product $(χ,χ)_G$ is uniquely determined by $ L $. We then prove that this conjecture is true in the case that $(G,χ) $ is normalized, $χ$ is a character of $ G $, and $ L $ contains at least an irrational value.
