Explanation
The irreducible representations of a simply-connected compact Lie group are indexed by their highest weights. These weights are the lattice points in an orthant Q+ in the weight lattice of the Lie group consisting of the dominant integral weights. It can be proved that there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram, such that any dominant integral weight is a non-negative integer linear combinations of the fundamental weights. The corresponding irreducible representations are the fundamental representations of the Lie group. From the expansion of a dominant weight in terms of the fundamental weights one can take a corresponding tensor product of the fundamental representations and extract one copy of the irreducible representation corresponding to that dominant weight.
Read more about this topic: Fundamental Representation
Famous quotes containing the word explanation:
“My companion assumes to know my mood and habit of thought, and we go on from explanation to explanation, until all is said that words can, and we leave matters just as they were at first, because of that vicious assumption.”
—Ralph Waldo Emerson (1803–1882)
“How strange a scene is this in which we are such shifting figures, pictures, shadows. The mystery of our existence—I have no faith in any attempted explanation of it. It is all a dark, unfathomed profound.”
—Rutherford Birchard Hayes (1822–1893)
“The explanation of the propensity of the English people to portrait painting is to be found in their relish for a Fact. Let a man do the grandest things, fight the greatest battles, or be distinguished by the most brilliant personal heroism, yet the English people would prefer his portrait to a painting of the great deed. The likeness they can judge of; his existence is a Fact. But the truth of the picture of his deeds they cannot judge of, for they have no imagination.”
—Benjamin Haydon (1786–1846)