Algebraic Invariants for Group of Permutations of Variables
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Theorem
Let $S = \set {x_1, x_2, \ldots,x_n}$ be a set of algebraic variables.
The algebraic invariants for the group of permutations of $S$ are those generated by the elementary symmetric polynomials:
\(\ds p_1\) | \(=\) | \(\ds x_1 + x_2 + \cdots + x_n\) | ||||||||||||
\(\ds p_2\) | \(=\) | \(\ds x_1 x_2 + x_1 x_3 + \cdots + x_{n - 1} x_n\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
\(\ds p_n\) | \(=\) | \(\ds x_1 x_2 \cdots x_n\) |
Proof
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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): invariant
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): invariant