Evaluating a function on permutations of its arguments

Question

Say I have a function "temp" of $n+1$ variables, $y,z1,z2,z3,...,zn$. I want to test if my function has certain symmetries like swapping $y$ with square of any $z$, swapping any two of the zs, replacing any z by its inverse, swapping any two zs by each other's inverse etc.

As of now I am having to type all of these cases explicitly by hand like for $n=3$ I am asking Mathematica to compute these differences for me (..if the symmetry exists then the answer would be 0..)

 Print[FullSimplify[temp[y, z1, z2, z3] - temp[z1^2, y^(1/2), z2, z3]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[z2^2, z1, y^(1/2), z3]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[z3^2, z1, z2, y^(1/2)]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, z2, z1, z3]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, z3, z2, z1]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, z1, z3, z2]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, 1/z2, 1/z1, z3]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, z1, 1/z3, 1/z2]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, 1/z3, z2, 1/z1]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, 1/z1, z2, z3]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, z1, 1/z2, z3]]]; 
 Print[FullSimplify[temp[y, z1, z2, z3] - temp[y, z1, z2, 1/z3]]]; 

Clearly doing the same with a large value of $n$ would get quite out of hand! I would like to know if there is a way to automate this.

• In general I would like to say have a $n \times n$ matrix acting on the zs (..thought of like a column n-vector..) and be able to test whether the function "temp" remains invariant under it.

  I wonder if Mathematica can generate representation matrices for groups. Like if I want to actually test that "temp" is invariant under a fundamental/vector action of the group $U(n)$ on the zs then I would like in the above case the $n\times n$ matrix to be that representation of the $U(n)$ group.