Symmetric functions | Representation theory of finite groups | Invariant theory | Homogeneous polynomials | Orthogonal polynomials

Schur polynomial

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. (Wikipedia).

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Related pages

Schur functor | Weyl character formula | Crystal base | Giambelli's formula | Lie group | Murnaghan–Nakayama rule | Unitary group | K-theory | Representation theory of the symmetric group | Littlewood–Richardson rule | Stanley symmetric function | Kostka number | Symmetric polynomial | LLT polynomial | Determinant | Complete homogeneous symmetric polynomial | Schubert polynomial | General linear group | Transposition (mathematics) | Representation theory | Elementary symmetric polynomial | Hall–Littlewood polynomials | Mathematics | Robinson–Schensted–Knuth correspondence | Young tableau | Pieri's formula | Lindström–Gessel–Viennot lemma | Basis (linear algebra) | Hook length formula | Issai Schur | Gröbner basis | Irreducible representation