论文标题
字符的正交歧视因素
On orthogonal discriminants of characters
论文作者
论文摘要
如果在任何模块上的所有非脱位不变的二次形式，则有限群的普通字符$χ$称为正交稳定，如果有任何不变的二次形式，则为字符$χ$具有相同的判别。这是正交判别，$ \ disc（χ）$，$χ$，是字符字段的正方形类别。 根据实验证据，我们猜想正交判别始终是一个奇怪的方形级别。 定义1.4。本说明证明了有限解决组的猜想。对于$ p $ - 组有一个明确的公式 $ \ disc（χ）$读取 $ \ disc（χ）=（-p）^{χ（1）/2} $如果$ p \ equiv 3 \ pmod {4} $和 $ \ disc（χ）=（-1）^{χ（1）/2} $ for $ p = 2 $。
An ordinary character $χ$ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character $χ$ have the same discriminant. This is the orthogonal discriminant, $\disc(χ)$, of $χ$, a square class of the character field. Based on experimental evidence we conjecture that the orthogonal discriminant is always an odd square class in the sense of Definition 1.4. This note proves this conjecture for finite solvable groups. For $p$-group there is an explicit formula for $\disc(χ)$ that reads $\disc(χ) = (-p)^{χ(1)/2}$ if $p\equiv 3 \pmod{4}$ and $\disc (χ) = (-1)^{χ(1)/2}$ for $p=2$.
