Schur polynomial

Remarks

The numerator is an alternating polynomial, so it is divisible by the Vandermonde determinant, and the quotient is a symmetric polynomial in $\Z[x_1, \ldots, x_n]$. It is homogeneous of degree $\abs{\lambda} = \lambda_1 + \cdots + \lambda_n$.

Example: for $n = 2$ and $\lambda = (2, 0)$, the exponents in the numerator are $\lambda_1 + 1 = 3$ and $\lambda_2 + 0 = 0$. So $s_{(2)} = \frac{x_1^3 - x_2^3}{x_1 - x_2} = x_1^2 + x_1 x_2 + x_2^2$. For $\lambda = (1, 1)$, the same computation gives $s_{(1,1)} = \frac{x_1^2 x_2 - x_1 x_2^2}{x_1 - x_2} = x_1 x_2$.

Non-example: the sequence $(1, 2)$ is not weakly decreasing, so it is not a partition, and $s_{(1,2)}$ is not a Schur polynomial. The formula would give a numerator with two equal columns, hence $0$.

Combinatorial form: $s_\lambda(x_1, \ldots, x_n) = \sum_T x^T$, a sum over the semistandard Young tableaux $T$ of shape $\lambda$ with entries in $\set{1, \ldots, n}$. Here $x^T = \prod_i x_i^{t_i}$, where $t_i$ is the number of entries of $T$ equal to $i$. Special cases are $s_{(k)} = h_k$, the complete homogeneous symmetric polynomial, and $s_{(1, \ldots, 1)} = e_k$, the elementary symmetric polynomial for $k$ ones. Letting $n \to \infty$ gives the Schur functions in the ring of symmetric functions.