How to create a symmetry function

Questions
1

Hi,
I am new learner for Mathematica mainly following video and notebooks from Prof. Nikolay Gromov.
So I am trying to reproduce the Matrix calculation using orthogonal polynomial methods here http://msstp.org/sites/default/files/ProblemSet2_RandomMatricesOrthogonalPolynomials_Solutions_0.pdf . However, I am unable to reproduce the same result below. I try some changes but ultimately not the result wanted. Appreciate if anyone could give some hints? Thanks a lot.

test.png
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2

However, I am unable to reproduce the same result below.

There seems to be something badly wrong with the code given. But no wonder - even the plain text of this script if full of typos!

For that define a function Sym[expr_,n_] which will symmetrize expr by all permunations of x…

Is this:

1/5 f[x[1]] + 1/5 f[x[2]] + 1/5 f[x[3]] + 1/5 f[x[4]] + 1/5 f[x[5]]

then really the wanted result? Or should the code rather reads like so:

sym[expr_, n_] := Block[{prm},
  prm = Permutations[Table[x[i], {i, 1, n}]];
  Total[expr @@@ prm]/Length[prm] // Expand
  ]

Then:

In[]:= sym[f, 3]

Out[]= 
1/6 f[x[1], x[2], x[3]] + 1/6 f[x[1], x[3], x[2]] + 
 1/6 f[x[2], x[1], x[3]] + 1/6 f[x[2], x[3], x[1]] + 
 1/6 f[x[3], x[1], x[2]] + 1/6 f[x[3], x[2], x[1]]

Well - no idea! Maybe you should ask a more specific question.

3

Thanks so much for the reply. The purpose of constructing a symmetry function is to verify an equation for Vandermonde determinant. For example, n=2 case below, symmetrize a will be equal to b.

test2.png
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Delta[n_] = Product[x[i] - x[j], {i, n}, {j, i - 1}]

a = 2! Delta[2] Product[x[i]^(i - 1), {i, 2}] // Expand

b = Delta[2]^2 // Expand