Schur polynomial

A Schur polynomial is a special symmetric polynomial associated to a partition of an integer, or equivalently to a Young diagram. Schur polynomials also have a power seriesMathworldPlanetmath generalizationPlanetmathPlanetmath, the Schur functions.

First we define some notation. Let λ be a partition of n, and let T be a filling of the Young diagram for λ. Then by xT we mean the monomialPlanetmathPlanetmathPlanetmath


where ci(T) is the number of times the number i appears in the filling T. Since T only has finitely many boxes, the productPlanetmathPlanetmath is finite. For example, let λ=(3,3,2,2), and let T be the filling