Module Theory Point of View
To put the construction more abstractly, the group ring K 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 representations 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 not semisimple, and a representation can fail to be irreducible without splitting as a direct sum.
Read more about this topic: Regular Representation
Famous quotes containing the words point of view, theory, point and/or view:
““Circumstantial evidence is a very tricky thing,” answered Holmes thoughtfully. “It may seem to point very straight to one thing, but if you shift your own point of view a little, you may find it pointing in an equally uncompromising manner to something entirely different.””
—Sir Arthur Conan Doyle (1859–1930)
“Many people have an oversimplified picture of bonding that could be called the “epoxy” theory of relationships...if you don’t get properly “glued” to your babies at exactly the right time, which only occurs very soon after birth, then you will have missed your chance.”
—Pamela Patrick Novotny (20th century)
“When we are in love, the sentiment is too great to be contained whole within us; it radiates out to our beloved, finds in her a surface which stops it, forces it to return to its point of departure, and it is this rebound of our own tenderness which we call the other’s affection and which charms us more than when it first went out because we do not see that it comes from us.”
—Marcel Proust (1871–1922)
“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)