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 theory, point and/or view:
“Don’t confuse hypothesis and theory. The former is a possible explanation; the latter, the correct one. The establishment of theory is the very purpose of science.”
—Martin H. Fischer (1879–1962)
“There could be no fairer destiny for any physical theory than that it should point the way to a more comprehensive theory in which it lives on as a limiting case.”
—Albert Einstein (1879–1955)
“What is the worst of woes that wait on age?
What stamps the wrinkle deeper on the brow?
To view each loved one blotted from life’s page,
And be alone on earth, as I am now.”
—George Gordon Noel Byron (1788–1824)