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My setup is the following: Fix some field $k$, set $M=k(X_1,\dots,X_d,Y_1,\dots,Y_e)$ with the obvious $S_{d+e}$-action and let $L=M^{S_d\times S_e}$ and $K=M^{S_{d+e}}$. Is there a way to let Sage compute the minimal polynomial of $X_1+\dots+X_d\in L$ over $K$?