Frobenius determinant theorem

In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in (Dedekind 1968), with an English translation in (Curtis 2003, p. 51)).

If one takes the multiplication table of a finite group G and replaces each entry g with the variable x_g, and subsequently takes the determinant, then the determinant factors as a product of n irreducible polynomials, where n is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising conjecture, and it became known as the Frobenius determinant theorem.

Formal statement

Let a finite group ${\displaystyle G}$ have elements ${\displaystyle g_{1},g_{2},\dots ,g_{n}}$, and let ${\displaystyle x_{g_{i}}}$ be associated with each element of ${\displaystyle G}$. Define the matrix ${\displaystyle X_{G}}$ with entries ${\displaystyle a_{ij}=x_{g_{i}g_{j}}}$. Then

${\displaystyle \det X_{G}=\prod _{j=1}^{r}P_{j}(x_{g_{1}},x_{g_{2}},\dots ,x_{g_{n}})^{\deg P_{j}}}$

where the ${\displaystyle P_{j}}$'s are pairwise non-proportional irreducible polynomials and ${\displaystyle r}$ is the number of conjugacy classes of G.^[1]