Frobenius determinant theorem explained

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, with an English translation in).

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

G have elements g _{1,} g _{2,...,g} _{n} , and let x _{g _{i}} be associated with each element of G . Define the matrix X _{G} with entries a _{ij} = x _{g _{i} g _{j}} . Then\det X _{G} = \prod ^{r} _{j =1} P _{j(x} ^{g _{1}} ,x _{g _{2}} , ... ,x _{g _{n}} ) ^{\deg P _{j}}

where the

P _{j} 's are pairwise non-proportional irreducible polynomials and r is the number of conjugacy classes of G.

References

• Review
  • Web site: Etingof . Pavel . 2005. Lectures on Representation Theory.