Permutation on Polynomial/Examples/Polynomial on 3 Variables

Examples of Permutations on Polynomials

Consider the polynomial on $3$ variables:

$\map f {x_1, x_2, x_3} = {x_1}^2 + 2 x_1 x_2 = 4 x_1 x_2 {x_3}^2$

Let $\rho := \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ be a permutation on the Symmetric Group on 3 Letters $S_3$.

Then:

$\rho \circ f = {x_2}^2 + 2 x_2 x_3 = 4 x_2 x_3 {x_1}^2$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Example $9.12$