Suppose you have any optimization problem that is symmetric. I somehow weaker question is: How much of the original symmetry carries over to the solutions? For the symmetric group $S_n$ if the degree of the polynomials that describe the problem is low (compared to the number of variables) the "degree and half  principle" says that one always finds minimizers  contained in the set of points invariant by a group $S_{l_1}\times\ldots\times S_{l_m}$ where $l_1+\ldots+l_m=n$