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.

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.^[1]

References

^ Etingof 2005, Theorem 5.4.

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

[1] Etingof 2005, Theorem 5.4.

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