If one takes the multiplication table of a finite groupG and replaces each entry g with the variable xg, 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. His proof of the theorem sparked a new branch of mathematics known as representation theory of finite groups.[1]
Formal statement
Let a finite group have elements , and let be associated with each element of . Define the matrix with entries . Then
where the 's are pairwise non-proportional irreducible polynomials and is the number of conjugacy classes of G.[2]
Examples
If , the matrix would be
The determinant of this matrix is
The number of irreducible polynomial factors is two, which is equal to the number of conjugacy classes of .
where . The number of irreducible polynomial factors is three, which is equal to the number of conjugacy clsses of . The degree-2 polynomial factor has multiplicity 2.[3]
Proof
This proof is based on the one given by Evan Chen, which involves representation theory.[3] It relies on the following lemma.
Lemma—Let be an matrix whose entries are independent variables . Then is an irreducible polynomial.
where each is an irreducible representation of . This lets us write
where each is a polynomial factor of .
A result from character theory states that the number of nonisomorphic irreps of regular representation equals the number of conjugacy classes of . This explains why the number of polynomial factors is equal to the number of conjugacy classes.
Furthermore, is both the degree and multiplicity of the polynomial , which explains why the degree and multiplicity of each polynomial factor are equal.
To complete the proof, we wish to show that polynomials are irreducible and not proportional to each other.
Proof of irreducibility: By Jacobson density theorem, for any matrix , there exists a particular choice of complex numbers for each such that
This shows that , when viewed as a matrix with polynomial entries, must have linearly independent entries. Thus, by letting each of these entries be an independent variable , it follows by Lemma above that is an irreducible polynomial.
Proof of non-proportionality: This follows by noticing that we can read off the character from the coefficients of , using the fact that for all , the coefficient of in is equal to . Since characters are linearly independent to each other, it follows that is not proportional to any other polynomial factor.