Frobenius determinant theorem

In mathematics, the Frobenius determinant theorem is a discovery 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 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 fact, and this theorem became known as the Frobenius determinant theorem.

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 r is the number of conjugacy classes of G.

References

• 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. Lectures on Representation Theory.
• 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