Frobenius determinant theorem

In mathematics, the Frobenius determinant theorem states that if one takes the multiplication table of a finite group G and replaces each entry g with a variable x_g, and subsequently takes the determinant, then the resulting multivariable polynomial factors as a product of n irreducible polynomials, where n is the number of conjugacy classes of G. Moreover, in the factorization, each of these irreducibles appears raised to a power equal to its degree.

This was first stated as a conjecture in a letter of 1896 by the mathematician Richard Dedekind to F. G. Frobenius, who proved it by methods which began a new branch of mathematics, the representation theory of finite groups. See Template:Harv, with English translation Template:Harv.

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 $G=\mathbb{Z}/2\mathbb{Z}=\{e,g\}$ with $g^2=e$ with n = 2 conjugacy classes, the matrix would be:

$X_G=\left[\begin{array}{cc}{x}_e& x_g\\ x_g& x_e\end{array}\right].$

The determinant of this matrix has n = 2 irreducible factors of degree 1, each with multiplicity 1:

$det{X}_G=(x_e-x_g)(x_e+x_g).$

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 classes 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:

Let $V=(V,\rho )=ℂ[G]$ 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 $detT$.

By Maschke's theorem, $ℂ[G]$ 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:

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

where each $det(T{|}_{V_i})$ is a polynomial factor of $detT$.

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 $detT{|}_{V_i}$ 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 $detT{|}_{V_i}$, using the fact that for all $g\in G$, the coefficient of $x_g{x}_{1_G}^{k-1}$ in $detT{|}_{V_i}$ is equal to ${\chi }_{V_i}(g)$. Since characters are linearly independent to each other, it follows that $detT{|}_{V_i}$ is not proportional to any other polynomial factor.

References

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