# Symmetric group action on polynomial ring

Let the symmetric group $$S_4$$ act on $$\mathbb R[x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]$$ by permuting the 1st $$4$$ variables and again permuting the last $$4$$ variables. We can restrict the action to the alternating group $$A_4$$. It seems there is a degree $$4$$ polynomial which is invariant under the $$A_4$$ action but not invariant under the $$S_4$$ action. But I can't figure out what is that polynomial. Any help in this direction will be helpful.

• Are you familiar with the proof that the parity of a permutation (defined as the parity of the number of transpositions in any expression of the permutation as a product of transpositions) is well defined, by acting on a discriminant? – Arturo Magidin Jan 30 at 17:02
• yes, I am familiar with even and odd permutations. But here the discriminant is of degree $36$. – jack Jan 30 at 17:06
• You only need to do it with some of the variables, since the action is independent. You only need to figure out how to drop it from 6 to 4, not from 36 to 4. – Arturo Magidin Jan 30 at 17:08
• Yes, I tried all possible degree 4 combinations like (x_i-y_j).... but it doesn't work. – jack Jan 30 at 17:11
• No, the point is that it acts on the $x$s separately from how it acts on the $y$s. You should look at polynomials that contain only $x$s, only $y$s, or have them in separate monomials that mirror each other. For example, if you just take the discriminant of the $x$s, $\prod_{i\lt j}(x_j-x_i)$, then this is invariant under the action of $A_4$ but not under the action of $S_4$. (And you could of course take the sum of the discriminant of the $x$s and the discriminant of the $y$s). I know this is degree 6, not 4, but you shouldn't be mixing your $x$s and $y$s, I think... – Arturo Magidin Jan 30 at 17:25