Schur polynomial

Summary

The Schur polynomial $s_\lambda(x_1, \ldots, x_n)$ is a symmetric polynomial attached to an integer partition $\lambda$, see the definition. It is a quotient of two alternating determinants, and it is also a sum over semistandard Young tableaux of shape $\lambda$. For partitions $\lambda$ of $d$ with at most $n$ parts, the $s_\lambda$ form a basis of the degree-$d$ symmetric polynomials in $n$ variables. They are the characters of the irreducible polynomial representations of $GL_n(\C)$, and their products are described by the Littlewood–Richardson rule.