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 series generalization, the Schur functions.

First we define some notation. Let $\lambda$ be a partition of $n$, and let $T$ be a filling of the Young diagram for $\lambda$. Then by $x^T$ we mean the monomial

$$x^T=\prod _{i=1}^{\mathrm{\infty }}{x}_i^{c_i(T)},$$

where $c_i(T)$ is the number of times the number $i$ appears in the filling $T$. Since $T$ only has finitely many boxes, the product is finite. For example, let $\lambda =(3,3,2,2)$, and let $T$ be the filling