nLab
elementary symmetric polynomial

Contents

Definition

The elementary symmetric polynomial on nn variables {X i}\{X_i\} of degree knk \leq n is the polynomial

σ k(X 1,,X n) 1i 1i knX i 1X i k. \sigma_k(X_1, \cdots, X_n) \coloneqq \sum_{1 \leq i_1 \leq \cdots \leq i_k \leq n} X_{i_1} \cdots X_{i_k} \,.

Equivalently these are the degree-kk summands in the polynomial

(1+X 1)(1+X 2)(1+X n) (1+X_1)(1+X_2) \cdots (1+X_n)

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

Examples

References

Created on March 29, 2014 at 03:32:54. See the history of this page for a list of all contributions to it.