Regular Representation - Structure For Finite Cyclic Groups

Structure For Finite Cyclic Groups

For a cyclic group C generated by g of order n, the matrix form of an element of K acting on K 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, g2, ..., gn−1.

When the field K contains a primitive n-th root of unity, one can diagonalise the representation of C by writing down n linearly independent simultaneous eigenvectors for all the n×n circulants. In fact if ζ is any n-th root of unity, the element

1 + ζg + ζ2g2 + ... + ζn−1gn−1

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

ζ−1

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 numbers) 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 determinants 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 Frobenius on group representations 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 acting by multiplication on the basis elements given by g in G. Unless G is abelian, the factorisation must contain non-linear factors corresponding to irreducible representations of G of degree > 1.