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 xg, 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 (Dedekind 1968), with English translation (Curtis 2003, p. 51).
Formal statement Let a finite group G {\displaystyle G} have elements g 1 , g 2 , … , g n {\displaystyle g_{1},g_{2},\dots ,g_{n}} , and let x g i {\displaystyle x_{g_{i}}} be associated with each element of G {\displaystyle G} . Define the matrix X G {\displaystyle X_{G}} with entries a i j = x g i g j {\displaystyle a_{ij}=x_{g_{i}g_{j}}} . Then:
det X G = ∏ j = 1 r P j ( x g 1 , x g 2 , … , x g n ) deg P j {\displaystyle \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 {\displaystyle P_{j}} 's are pairwise non-proportional irreducible polynomials and r {\displaystyle r} is the number of conjugacy classes of G.
Examples If G = Z / 2 Z = { e , g } {\displaystyle G=\mathbb {Z} /2\mathbb {Z} =\{e,g\}} with g 2 = e {\displaystyle g^{2}=e} with n = 2 conjugacy classes, the matrix would be:
X G = [ x e x g x g x e ] . {\displaystyle X_{G}={\begin{bmatrix}x_{e}&x_{g}\\x_{g}&x_{e}\end{bmatrix}}.}
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 ) . {\displaystyle \det X_{G}=(x_{e}-x_{g})(x_{e}+x_{g}).}
If G = S 3 {\displaystyle G=S_{3}} , the symmetric group of order 3, the matrix would be:
X G = [ 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 ) ] . {\displaystyle 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 = ( ∑ σ ∈ S 3 x σ ) ( ∑ σ ∈ S 3 sign ( σ ) x σ ) ( F ( x e , x ( 123 ) , x ( 321 ) ) − F ( x ( 12 ) , x ( 23 ) , x ( 31 ) ) ) 2 {\displaystyle \textstyle \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 − a b − b c − c a {\displaystyle 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 {\displaystyle 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 , ρ ) = C [ G ] {\displaystyle V=(V,\rho )=\mathbb {C} [G]} be the regular representation of group G {\displaystyle G} . Consider the linear map:
T = ∑ g ∈ G x g ρ ( g ) {\displaystyle T=\sum _{g\in G}x_{g}\rho (g)} , whose matrix is given by X G {\displaystyle X_{G}} . We wish to examine det T {\displaystyle \det T} . By Maschke's theorem, C [ G ] {\displaystyle \mathbb {C} [G]} is a semisimple algebra, so it is possible to break down V {\displaystyle V} into a direct sum of irreducible representations,
V = ⨁ i = 1 r V i ⊕ dim V i {\displaystyle V=\bigoplus _{i=1}^{r}V_{i}^{\oplus \dim V_{i}}}
where each V i {\displaystyle V_{i}} is an irreducible representation of V {\displaystyle V} . This lets us write:
det T = ∏ i = 1 r ( det ( T | V i ) ) dim V i , {\displaystyle \det T=\prod _{i=1}^{r}\left(\det(T|_{V_{i}})\right)^{\dim V_{i}},}
where each det ( T | V i ) {\displaystyle \det(T|_{V_{i}})} is a polynomial factor of det T {\displaystyle \det T} . A result from character theory states that the number of nonisomorphic irreps of regular representation V {\displaystyle V} equals the number of conjugacy classes of G {\displaystyle G} . This explains why the number of polynomial factors is equal to the number of conjugacy classes. Furthermore, dim V i {\displaystyle \dim V_{i}} is both the degree and multiplicity of the polynomial det ( T | V i ) {\displaystyle \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 ) {\displaystyle \det(T|_{V_{i}})} are irreducible and not proportional to each other. Proof of irreducibility: By Jacobson density theorem, for any matrix M ∈ Mat ( V i ) {\displaystyle M\in {\text{Mat}}(V_{i})} , there exists a particular choice of complex numbers for each x g ∈ G {\displaystyle x_{g}\in G} such that:
M = ∑ g ∈ G x g ρ i ( g ) = T | V i ( { x g } ) {\displaystyle M=\sum _{g\in G}x_{g}\rho _{i}(g)=T|_{V_{i}}(\{x_{g}\})}
This shows that T | V i {\displaystyle 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 i j {\displaystyle y_{ij}} , it follows by Lemma above that det T | V i {\displaystyle \det T|_{V_{i}}} is an irreducible polynomial. Proof of non-proportionality: This follows by noticing that we can read off the character χ V i {\displaystyle \chi _{V_{i}}} from the coefficients of det T | V i {\displaystyle \det T|_{V_{i}}} , using the fact that for all g ∈ G {\displaystyle g\in G} , the coefficient of x g x 1 G k − 1 {\displaystyle x_{g}x_{1_{G}}^{k-1}} in det T | V i {\displaystyle \det T|_{V_{i}}} is equal to χ V i ( g ) {\displaystyle \chi _{V_{i}}(g)} . Since characters are linearly independent to each other, it follows that det T | V i {\displaystyle \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