Schur Polynomial

Schur polynomials are a family of symmetric polynomials in n variables, indexed by integer partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. Named after the mathematician Issai Schur, they form a linear basis for the space of all symmetric polynomials of a given degree. In representation theory, Schur polynomials serve as the characters of the irreducible polynomial representations of the general linear group, linking algebraic combinatorics to group representation theory. They are formally defined through Jacobi's bialternant formula, which expresses a Schur polynomial as the ratio of two alternating polynomials, with the denominator equal to the Vandermonde determinant. When two Schur polynomials are multiplied together, the product can be written as a linear combination of other Schur polynomials with non-negative integer coefficients, a rule known as the Littlewood-Richardson rule, and a related family called skew Schur polynomials extends the same structure to pairs of partitions.