Symmetric function

Remarks

The ring $\Lambda$ is graded, $\Lambda = \bigoplus_{k \ge 0} \Lambda^k$, where $\Lambda^k$ consists of the homogeneous symmetric functions of degree $k$. For a partition $\lambda = (\lambda_1, \ldots, \lambda_r)$, the monomial symmetric function $m_\lambda$ is the sum of all distinct monomials $x_{i_1}^{\lambda_1} \cdots x_{i_r}^{\lambda_r}$ with distinct indices. The $m_\lambda$ with $\lambda$ a partition of $k$ form a $\Z$-basis of $\Lambda^k$. Setting $x_{n+1} = x_{n+2} = \cdots = 0$ maps $\Lambda$ onto the ring of symmetric polynomials in $x_1, \ldots, x_n$. Equivalently, $\Lambda^k$ is the inverse limit of the degree-$k$ symmetric polynomials in $n$ variables, as $n \to \infty$.

Example: the power sum $p_2 = x_1^2 + x_2^2 + x_3^2 + \cdots$ is a symmetric function of degree $2$. Both clauses hold: all monomials have degree $2$, and a permutation of the variables only permutes the terms.

Non-example: the product $\prod_{i \ge 1} (1 + x_i) = 1 + e_1 + e_2 + \cdots$ is invariant under all permutations of the variables, but it fails clause 1. It contains the monomial $x_1 x_2 \cdots x_k$ of degree $k$ for every $k$.

The elementary symmetric functions $e_k = \sum_{i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}$ are algebraically independent, and $\Lambda = \Z[e_1, e_2, \ldots]$. The same holds for the complete homogeneous symmetric functions $h_k$, the sum of all monomials of degree $k$.