On polynomial representations of finite groups

TL;DR

This paper extends Frobenius' polynomial method to develop new character theories for finite groups using partition algebras, leading to novel invariants and polynomial representations that unify various aspects of group theory.

Contribution

It introduces a generalized polynomial framework for finite group characters, establishing equivalences with Frobenius polynomials and deriving new invariants and conjectures.

Findings

New character theories via partition algebras

Finite simple groups determined by invariants p_{ijl}

Degrees of irreducible characters as polynomial solutions

Abstract

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.