Frobenius determinant theorem - Alchetron, the free social encyclopedia

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 , … , g n , and let x g i be associated with each element of G . Define the matrix X G with entries a i j = 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

where r is the number of conjugacy classes of G.

References

Frobenius determinant theorem Wikipedia

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