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)).

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

$\det X_G=\prod _{j=1}^r{P}_j(x_{g_1},x_{g_2},\dots ,x_{g_n}{)}^{\mathrm{deg}{P}_j}$

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

Examples

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

$X_G=\left[\begin{array}{cc}{x}_{1_G}& x_g\\ x_g& x_{1_G}\end{array}\right].$

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 $\mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}$.

If $G=S_3$, the symmetric group of order 3, the matrix would be

$X_G=\left[\begin{array}{cccccc}{x}_e& x_{(12)}& x_{(23)}& x_{(31)}& x_{(123)}& x_{(321)}\\ x_{(12)}& x_e& x_{(321)}& x_{(123)}& x_{(31)}& x_{(23)}\\ x_{(23)}& x_{(123)}& x_e& x_{(321)}& x_{(12)}& x_{(31)}\\ x_{(31)}& x_{(321)}& x_{(123)}& x_e& x_{(23)}& x_{(12)}\\ x_{(123)}& x_{(23)}& x_{(31)}& x_{(12)}& x_{(321)}& x_e\\ x_{(321)}& x_{(31)}& x_{(12)}& x_{(23)}& x_e& x_{(123)}\end{array}\right].$

The determinant of this matrix factors out as

$\det X_G=\left(\sum _{\sigma \in S_3}{x}_{\sigma }\right)(\sum _{\sigma \in S_3}\text{sign}(\sigma )x_{\sigma }){(F(x_e,x_{(123)},x_{(321)})-F(x_{(12)},x_{(23)},x_{(31)}))}^2$

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,

$V=\underset{i=1}{\overset{r}{⨁}}{V}_i^{\oplus \dim V_i}$

where each $V_i$ is an irreducible representation of $V$. This lets us write

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

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". https://venhance.github.io/napkin/Napkin.pdf. 
• 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, https://books.google.com/books?isbn=0821826778  Review
• Dedekind, Richard (1968), Fricke, Robert; Noether, Emmy; Ore, öystein, eds., Gesammelte mathematische Werke. Bände I–III, New York: Chelsea Publishing Co. 
• Etingof, Pavel (2005). "Lectures on Representation Theory". http://www-math.mit.edu/~etingof/cltrunc.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