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
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