Шур полиномдары және олардың қолданылуы

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.

The Schur polynomials occur in the representation theory of the symmetric groups, general linear groups, and unitary groups. 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 χ 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.

A proof of this can be found in R. Stanley's Enumerative Combinatorics Volume 2, Corollary 7.17.5. The integers χ can be computed using the Murnaghan–Nakayama rule.

There are several approaches to prove Schur positivity of a given symmetric function F.
If F is described in a combinatorial manner, a direct approach is to produce a bijection with semi standard Young tableaux. The Edelman–Greene correspondence and the Robinson–Schensted–Knuth correspondence are examples of such bijections. A bijection with more structure is a proof using so called crystals. This method can be described as defining a certain graph structure described with local rules on the underlying combinatorial objects. A similar idea is the notion of dual equivalence. This approach also uses a graph structure, but on the objects representing the expansion in the fundamental quasisymmetric basis. It is closely related to the RSK correspondence.