nLab elementary symmetric polynomial

Definition

The elementary symmetric polynomial on $n$ variables $\{X_i\}$ of degree $k \leq n$ is the polynomial

$$\sigma_k(X_1, \cdots, X_n) \coloneqq \sum_{1 \leq i_1 \lneq \cdots \lneq i_k \leq n} X_{i_1} \cdots X_{i_k} \,.$$

Equivalently these are the degree-$k$ summands in the polynomial

$$(1+X_1)(1+X_2) \cdots (1+X_n)$$

These polynomials form a basis for the Lambda-ring of symmetric functions.

Examples

• The Chern classes are, under the splitting principle, elementary symmetric polynomials of the first Chern classes.

References

• Wikipedia, Elementary symmetric polynomial