Elementary symmetric polynomial

2020 Mathematics Subject Classification: Primary: 12E05 [MSN][ZBL]

elementary symmetric function

The elementary symmetric polynomials of $n$ indeterminates $x_1, \ldots, x_n$ are the polynomials $\sigma_k(x_1,\ldots,x_n)$ for $k=0,\ldots,n$ where the $k$-th polynomial is obtained by summing all distinct monomials which are products of $k$ distinct $x_i$: we write $\sigma_0 = 1$ and $\sigma_k = 0$ for $k > n$. Thus $\sigma_k$ has degree $k$ and contains $\binom{n}{k}$ terms.

One can write formally $$ \prod_{i=1}^n (T - x_i) = \sum_{i=0}^n (-1)^i \sigma_{n-i} T^i \ . $$

The $\sigma_k$ are symmetric polynomials, in that each is invariant under any permutation of the indeterminates, and form a complete system of invariants for the symmetric group $S_n$, so that any symmetric polynomial in the $x_i$ can be written as a polynomial in the $\sigma_k$.

References

• Olver, Peter J. "Classical invariant theory" London Mathematical Society Student Texts. Cambridge University Press (1999) Zbl 0971.13004