Relation To Representation Theory
The Schur polynomials occur in the representation theory of the symmetric groups, general linear groups, and unitary groups, and in fact this is how they arose. The Weyl character formula implies that the Schur polynomials are the characters of finite dimensional irreducible representations of the general linear groups, and helps to generalize Schur's work to other compact and semisimple Lie groups.
Several expressions arise for this relation, one of the most important being the expansion of the Schur functions sλ in terms of the symmetric power functions . If we write χλ
ρ for the character of the representation of the symmetric group indexed by the partition λ evaluated at elements of cycle type indexed by the partition ρ, then
where ρ = (1r1, 2r2, 3r3, ...) means that the partition ρ has rk parts of length k.
Read more about this topic: Schur Polynomial
Famous quotes containing the words relation to, relation and/or theory:
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“Parents ought, through their own behavior and the values by which they live, to provide direction for their children. But they need to rid themselves of the idea that there are surefire methods which, when well applied, will produce certain predictable results. Whatever we do with and for our children ought to flow from our understanding of and our feelings for the particular situation and the relation we wish to exist between us and our child.”
—Bruno Bettelheim (20th century)
“Osteopath—One who argues that all human ills are caused by the pressure of hard bone upon soft tissue. The proof of his theory is to be found in the heads of those who believe it.”
—H.L. (Henry Lewis)