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 xg, 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 [math]\displaystyle{ G }[/math] have elements [math]\displaystyle{ g_1, g_2,\dots,g_n }[/math], and let [math]\displaystyle{ x_{g_i} }[/math] be associated with each element of [math]\displaystyle{ G }[/math]. Define the matrix [math]\displaystyle{ X_G }[/math] with entries [math]\displaystyle{ a_{ij}=x_{g_i g_j} }[/math]. Then
- [math]\displaystyle{ \det X_G = \prod_{j=1}^r P_j(x_{g_1},x_{g_2},\dots,x_{g_n})^{\deg P_j} }[/math]
where the [math]\displaystyle{ P_{j} }[/math]'s are pairwise non-proportional irreducible polynomials and [math]\displaystyle{ r }[/math] 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, https://books.google.com/books?isbn=0821826778 Review
- Dedekind, Richard (1968) [1931], 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
 |  |
