论文标题
关于有限组的多项式表示
On polynomial representations of finite groups
论文作者
论文摘要
通过将Frobenius的多项式方法推广到良好的分区代数，我们将为有限的组$ g $开发新的角色理论。对于这些类型的角色理论，得出了一个均匀的定义方程。新字符理论导致小组决定因素的各种因素化。我们将证明这些新字符理论等于通讯良好分区代数的Frobenius多项式。特别是，有限的组$ g $的字符表可以被$ g $的Frobenius多项式代替，这是组决定因素的一种堕落。作为应用程序，我们找到了一个有限组的新系列不变性$ p_ {ijl} $。特别是，有限的简单组由这些不变性决定。随着进一步的应用，所有不可约字符的学位也可以作为$ g $的多项式的解决方案实现，我们可以将精致的McKay猜想和Navarro猜想的一部分结合在不可减至的字符上的Galois猜想中。
By generalizing Frobenius' polynomial method to good partition algebra, we will develop new character theories for a finite group $G$. A uniform defining equations are derived for these kinds of character theories. The new character theories leads to various factorizations of the group determinant. We will show that these new character theories are equivalent to the Frobenius polynomials of the correspondent good partition algebras. In particular, the character table of a finite group $G$ can be replaced by the Frobenius polynomial of $G$ which is a kind of degenerate of the group determinant. As applications, we find a new series of invariants $p_{ijl}$ for a finite group. In particular, a finite simple group is determined by these invariants. As further applications, the degrees of all irreducible characters can also be realized as the solutions of a polynomial of $G$ and we can combine the refined McKay conjecture and part of Galois conjecture of Navarro on degrees of irreducible characters into a new general conjecture.
