Specifically, given g ∈ G, ρg is the linear map on V determined by its action on the basis by right translation by g−1, i.e. Alternatively, these representations can be defined on the K-vector space W of all functions G → K. It is in this form that the regular representation is generalized to topological groups such as Lie groups.
Recall that the character of the regular representation χ(g) is the number of fixed points of g acting on the regular representation V. It means the number of fixed points χ(g) is zero when g is not id and |G| otherwise.
To put the construction more abstractly, the group ring K[G] is considered as a module over itself.
The modular case, when the characteristic of K does divide |G|, is harder mainly because with K[G] not semisimple, a representation can fail to be irreducible without splitting as a direct sum.
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 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.
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.
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[G] 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.
For a topological group G, the regular representation in the above sense should be replaced by a suitable space of functions on G, with G acting by translation.
If G is a Lie group but not compact nor abelian, this is a difficult matter of harmonic analysis.
The locally compact abelian case is part of the Pontryagin duality theory.
Given an algebra over a field A, it doesn't immediately make sense to ask about the relation between A as left-module over itself, and as right-module.
They have been shown to be related to topological quantum field theory in 1 + 1 dimensions by a particular instance of the cobordism hypothesis.