Let $A=\mathbb{C}[X_1,X_2,X_3]/(X_1X_2X_3,X_1+X_2+X_3-1)$ and $B= \{ f(x_1,x_2,x_3) \in A \mid\forall \rho \in \mathfrak{S}_{3}, f(x_1,x_2,x_3)=f(x_{\rho(1)},x_{\rho(2)},x_{\rho(3)}) \}$ where $x_i$ denotes the image of $X_i$ in $A$.

Is it true that $B=\mathbb{C}[x_1+x_2+x_3,x_1x_2+x_1x_3+x_2x_3,x_1x_2x_3]=\mathbb{C}[x_1x_2+x_1x_3+x_2x_3]$?

I know that any symmetric polynomials is a polynomial in the elementary symmetric functions, but I don't know it is true for quotient of polynomial rings.

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    $\begingroup$ Yes, because any $S_3$-invariant element of the quotient ring $A $ is the projection of an $S_3$-invariant element of the polynomial ring $P:=\mathbb{C}\left [X_1,X_2,X_3\right] $. (To see this latter fact, write the former element as the projection of some polynomial $p\in P $, and then show that it is also the projection of the $S_3$-invariant polynomial $\dfrac {1}{3} \sum\limits_{w \in S_3} wp $.) This is actually an instance of a general argument that works for the invariants of any finite group over a field in characteristic $0$. $\endgroup$ – darij grinberg Jun 3 '17 at 15:35
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    $\begingroup$ More generally: If $f : U \to V$ is a surjective homomorphism between two representations of a group $G$, and if the representation $U $ is semisimple (aka completely reducible), then each $G $-invariant element of $V $ is an image of an $G $-invariant element of $U $. $\endgroup$ – darij grinberg Jun 3 '17 at 15:38
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    $\begingroup$ On the other hand, when do statements like this hold in positive characteristic? $\endgroup$ – darij grinberg Jun 3 '17 at 15:42
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    $\begingroup$ Correct $\dfrac13$ to $\dfrac16$ in my first comment. Sorry! $\endgroup$ – darij grinberg Jun 3 '17 at 15:44

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