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:
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)
“The theory of rights enables us to rise and overthrow obstacles, but not to found a strong and lasting accord between all the elements which compose the nation.”
—Giuseppe Mazzini (1805–1872)
“But you must pay for conformity. All goes well as long as you run with conformists. But you, who are honest men in other particulars, know, that there is alive somewhere a man whose honesty reaches to this point also, that he shall not kneel to false gods, and, on the day when you meet him, you sink into the class of counterfeits.”
—Ralph Waldo Emerson (1803–1882)
“To see self-sufficiency as the hallmark of maturity conveys a view of adult life that is at odds with the human condition, a view that cannot sustain the kinds of long-term commitments and involvements with other people that are necessary for raising and educating a child or for citizenship in a democratic society.”
—Carol Gilligan (20th century)