Detect a Symmetric polynomial
Mathematica, 43 bytes

Last@SymmetricReduction[#,Variables@#]===0&amp;


Unnamed function taking as input a polynomial in the given format (except that juxtaposed variables must be separated by a space) and returning True or False. Variables@# detects the variables appearing in the input (and thus the input can contain all kinds of weird variable names, not just single letters). SymmetricReduction returns an ordered pair of polynomials, where the first one is symmetric and the two sum to the original polynomial; therefore we can detect whether the input is symmetric by seeing whether the second polynomial is identically 0.