# Description of certain invariant polynomials (not a group action)

Working on a recent question led me to the following invariant-computation problem : let

$$A=\bigg\lbrace P \in {\mathbb Q}[X_1,X_2,X_3,X_4] \ \bigg| \\ \quad\ P(X_1X_3+X_2X_4+X_1X_4,\ X_2X_3,\ X_1X_3+X_2X_4+X_2X_3,\ X_1X_4)=\\ =P(X_1,X_2,X_3,X_4) \bigg\rbrace$$

Then $A$ is a sub algebra of ${\mathbb Q}[X_1,X_2,X_3,X_4]$. Is it trivial ? Does anyone know how to find a finite generating set for it ?

• I can't tell what your question is. But let me remark that the ring of invariants $Q[X_1,X_2,X_3,X_4]^G$ is a fixed point set under a group action, since you suggested that it isn't in your question title. The usual action son a regular function $f \in Q[X_1,...X_4]$ is either given by $(g \cdot f )(x)= f{g^{-1} x)$ or $(g \cdot f)(x)=f(xg)$ both of which give group actions on $Q[X_1,...,X_4]$. – user062295 Apr 21 '16 at 4:31
• @user062295 1) Which part of my question is unclear to you ? 2) If you think that there is in fact a group action here, could you describe the group $G$ and the action ? Because I don't see any so far. – Ewan Delanoy Apr 21 '16 at 7:28