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. His proof of the theorem sparked a new branch of mathematics known as representation theory of finite groups.

Formal statement

Let a finite group $G$ have elements $g_{1},g_{2},\dots ,g_{n}$, 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

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

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

Examples

If $G=\mathbb {Z} /2\mathbb {Z} =\langle g\mid g^{2}=1\rangle$, the matrix would be

$X_{G}={\begin{bmatrix}x_{1_{G}}&x_{g}\\x_{g}&x_{1_{G}}\end{bmatrix}}.$

The determinant of this matrix is

$\det X_{G}=(x_{1_{G}}-x_{g})(x_{1_{G}}+x_{g}).$

The number of irreducible polynomial factors is two, which is equal to the number of conjugacy classes of $\mathbb {Z} /2\mathbb {Z}$.

If $G=S_{3}$, the symmetric group of order 3, the matrix would be

$X_{G}={\begin{bmatrix}x_{e}&x_{(12)}&x_{(23)}&x_{(31)}&x_{(123)}&x_{(321)}\\x_{(12)}&x_{e}&x_{(321)}&x_{(123)}&x_{(31)}&x_{(23)}\\x_{(23)}&x_{(123)}&x_{e}&x_{(321)}&x_{(12)}&x_{(31)}\\x_{(31)}&x_{(321)}&x_{(123)}&x_{e}&x_{(23)}&x_{(12)}\\x_{(123)}&x_{(23)}&x_{(31)}&x_{(12)}&x_{(321)}&x_{e}\\x_{(321)}&x_{(31)}&x_{(12)}&x_{(23)}&x_{e}&x_{(123)}\end{bmatrix}}.$

The determinant of this matrix factors out as

$\det X_{G}=\left(\sum _{\sigma \in S_{3}}x_{\sigma }\right)\left(\sum _{\sigma \in S_{3}}{\text{sign}}(\sigma )x_{\sigma }\right)\left(F(x_{e},x_{(123)},x_{(321)})-F(x_{(12)},x_{(23)},x_{(31)})\right)^{2}$

where $F(a,b,c)=a^{2}+b^{2}+c^{2}-ab-bc-ca$. The number of irreducible polynomial factors is three, which is equal to the number of conjugacy clsses of $S_{3}$. The degree-2 polynomial factor has multiplicity 2.

Proof

This proof is based on the one given by Evan Chen, which involves representation theory. It relies on the following lemma.

Lemma—Let $Y$ be an $n\times n$ matrix whose entries are independent variables $y_{ij}$. Then $\det Y$ is an irreducible polynomial.

Let $V=(V,\rho )=\mathbb {C} [G]$ be the regular representation of group $G$. Consider the linear map

$T=\sum _{g\in G}x_{g}\rho (g)$,

whose matrix is given by $X_{G}$. We wish to examine $\det T$.

By Maschke's theorem, $\mathbb {C} [G]$ is a semisimple algebra, so it is possible to break down $V$ into a direct sum of irreducible representations,

$V=\bigoplus _{i=1}^{r}V_{i}^{\oplus \dim V_{i}}$

where each $V_{i}$ is an irreducible representation of $V$. This lets us write

$\det T=\prod _{i=1}^{r}\left(\det(T|_{V_{i}})\right)^{\dim V_{i}},$

where each $\det(T|_{V_{i}})$ is a polynomial factor of $\det T$.

A result from character theory states that the number of nonisomorphic irreps of regular representation $V$ equals the number of conjugacy classes of $G$. This explains why the number of polynomial factors is equal to the number of conjugacy classes.

Furthermore, $\dim V_{i}$ is both the degree and multiplicity of the polynomial $\det(T|_{V_{i}})$, which explains why the degree and multiplicity of each polynomial factor are equal.

To complete the proof, we wish to show that polynomials $\det(T|_{V_{i}})$ are irreducible and not proportional to each other.

Proof of irreducibility: By Jacobson density theorem, for any matrix $M\in {\text{Mat}}(V_{i})$, there exists a particular choice of complex numbers for each $x_{g}\in G$ such that

$M=\sum _{g\in G}x_{g}\rho _{i}(g)=T|_{V_{i}}(\{x_{g}\})$

This shows that $T|_{V_{i}}$, when viewed as a matrix with polynomial entries, must have linearly independent entries. Thus, by letting each of these entries be an independent variable $y_{ij}$, it follows by Lemma above that $\det T|_{V_{i}}$ is an irreducible polynomial.

Proof of non-proportionality: This follows by noticing that we can read off the character $\chi _{V_{i}}$ from the coefficients of $\det T|_{V_{i}}$, using the fact that for all $g\in G$, the coefficient of $x_{g}x_{1_{G}}^{k-1}$ in $\det T|_{V_{i}}$ is equal to $\chi _{V_{i}}(g)$. Since characters are linearly independent to each other, it follows that $\det T|_{V_{i}}$ is not proportional to any other polynomial factor.

References

• Chen, Evan. "An Infinitely Large Napkin" (PDF). Retrieved 3 September 2025.
• Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, doi:10.1090/S0273-0979-00-00867-3, ISBN 978-0-8218-2677-5, MR 1715145 Review
• Dedekind, Richard (1968) [1931], Fricke, Robert; Noether, Emmy; Ore, öystein (eds.), Gesammelte mathematische Werke. Bände I–III, New York: Chelsea Publishing Co., JFM 56.0024.05, MR 0237282
• Etingof, Pavel (2005). "Lectures on Representation Theory" (PDF).
• Frobenius, Ferdinand Georg (1968), Serre, J.-P. (ed.), Gesammelte Abhandlungen. Bände I, II, III, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04120-7, MR 0235974