Editing Regular representation (section)

==Structure for finite cyclic groups==

For a [[cyclic group]] ''C'' generated by ''g'' of order ''n'', the matrix form of an element of ''K''[''C''] acting on ''K''[''C''] by multiplication takes a distinctive form known as a ''[[circulant matrix]]'', in which each row is a shift to the right of the one above (in [[cyclic order]], i.e. with the right-most element appearing on the left), when referred to the natural basis

:1, ''g'', ''g''<sup>2</sup>, ..., ''g''<sup>''n''&minus;1</sup>.

When the field ''K'' contains a [[primitive n-th root of unity|primitive ''n''-th root of unity]], one can [[Diagonalizable matrix|diagonalise]] the representation of ''C'' by writing down ''n'' linearly independent simultaneous [[eigenvector]]s for all the ''n''&times;''n'' circulants. In fact if ζ is any ''n''-th root of unity, the element

:1 + &zeta;''g'' + &zeta;<sup>2</sup>''g''<sup>2</sup> + ... + &zeta;<sup>''n''&minus;1</sup>''g''<sup>''n''&minus;1</sup>

is an eigenvector for the action of ''g'' by multiplication, with eigenvalue

:&zeta;<sup>&minus;1</sup>

and so also an eigenvector of all powers of ''g'', and their linear combinations.

This is the explicit form in this case of the abstract result that over an [[algebraically closed field]] ''K'' (such as the [[complex number]]s) the regular representation of ''G'' is [[completely reducible]], provided that the characteristic of ''K'' (if it is a prime number ''p'') doesn't divide the order of ''G''. That is called ''[[Maschke's theorem]]''. In this case the condition on the characteristic is implied by the existence of a ''primitive'' ''n''-th root of unity, which cannot happen in the case of prime characteristic ''p'' dividing ''n''.

Circulant [[determinant]]s were first encountered in nineteenth century mathematics, and the consequence of their diagonalisation drawn. Namely, the determinant of a circulant is the product of the ''n'' eigenvalues for the ''n'' eigenvectors described above. The basic work of [[Ferdinand Georg Frobenius|Frobenius]] on [[group representation]]s started with the motivation of finding analogous factorisations of the '''group determinants''' for any finite ''G''; that is, the determinants of arbitrary matrices representing elements of ''K''[''G''] acting by multiplication on the basis elements given by ''g'' in ''G''. Unless ''G'' is [[abelian group|abelian]], the factorisation must contain non-linear factors corresponding to [[irreducible representation]]s of ''G'' of degree > 1.