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2 Module theory point of view 3 Topological group case 4 Normal bases in Galois theory 5 More general algebras |
To say that G acts on itself by multiplication is tautological. If we consider this action as a permutation representation it is characterised as having a single orbit and stabilizer the identity subgroup {e} of G. The regular representation of G, for a given field K, is the linear representation made by taking the permutation representation as a set of basis vectors of a vector space over K. The significance is that while the permutation representation doesn't decompose - it is transitive - the regular representation in general breaks up into smaller representations. For example if G is a finite group and K is the complex number field, the regular representation is a direct sum of irreducible representations, in number at least the number of conjugacy classes of G.
To put the construction more abstractly, the group ring K[G] is considered as a module over itself. (There is a choice here of left-action or right-action, but that is not of importance except for notation.) If G is finite and the characteristic of K doesn't divide |G|, this is a semisimple ring and we are looking at its left (right) ring ideals. This theory has been studied in great depth. It is known in particular that the direct sum decomposition of the regular representation contains a representative of every isomorphism class of irreducible linear representation of G over K. You can say that the regular representation is comprehensive for representation theory, in this case. 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 G a topological group, the regular representation in the above sense should be replaced by a suitable space of functions on G, with G acting by translation. See Peter-Weyl theorem for the compact case. 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.
In Galois theory it is shown that for a field L, and a finite group of automorphisms of L, the fixed field K of G has [L:K] = |G|. In fact we can say more: L as K[G]-module is the regular representation. This is the content of the normal basis theorem, a normal basis\' being an element x of L such that the g(x) for g in G are a vector space basis for L over K. Such x exist, and each one gives a K[G]-isomorphism from L to K[G]. From the point of view of algebraic number theory it is of interest to study normal integral bases, where we try to replace L and K by the rings of algebraic integers they contain. One can see already in the case of the Gaussian integers that such bases may not exist: a+bi and a-bi can never form a Z-module basis of Z'[i] because 1 cannot be an integer combination. The reasons are studied in depth in Galois module theory.
The regular representation of a group ring is such that the left-hand and right-hand regular representations give isomorphic modules (and we often need not distinguish the cases). 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. In the group case, the mapping on basis elements g of K[G] defined by taking the inverse element gives an isomorphism of K[G] to its opposite ring. For A general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteeth century. They have been shown to be related to topological quantum field theory in 1+1 dimensions.Significance of the regular representation of a group
Module theory point of view
Topological group case
Normal bases in Galois theory
More general algebras