---
tag: DA03KQ311J
kind: definition
name: Schur polynomial
msc: ["05E05", "05E10"]
defines: 43VFMJCDYJ
uses: [0JV30XN73Z, 0C3CFHDN6R, 0VHSHB9B6J, 0P84B554AS]
sources:
  - url: https://en.wikipedia.org/wiki/Schur_polynomial
---

## Definition

Let $n \ge 1$ and let $\lambda = (\lambda_1, \ldots, \lambda_n)$ be an [[integer partition|0JV30XN73Z]] with at most $n$ nonzero parts, padded with zeros, so $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$.
The *Schur polynomial* of $\lambda$ in the variables $x_1, \ldots, x_n$ is
$$s_\lambda(x_1, \ldots, x_n) = \frac{\det\big(x_i^{\lambda_j + n - j}\big)_{1 \le i, j \le n}}{\det\big(x_i^{n - j}\big)_{1 \le i, j \le n}}.$$
Here $\det$ is the [[determinant|0C3CFHDN6R]], and the denominator is the [[Vandermonde determinant|0P84B554AS]] $\prod_{i < j} (x_i - x_j)$.

## Remarks

The numerator is an alternating polynomial, so it is divisible by the Vandermonde determinant, and the quotient is a [[symmetric polynomial|0VHSHB9B6J]] 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|Q6TH9PH2R7]] $T$ 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|0X5QVCF06N]] for $k$ ones.
Letting $n \to \infty$ gives the Schur functions in the ring of [[symmetric functions|PAJEHWXW6M]].