# Incorporating Signature with Permute for symmetric groups

Let $$f(x_1, x_2, x_3, x_4) := \frac{x_1^{-3} x_2^{-2} x_3^{-1}}{(1 - x_1^2) (1 -x_1^2 x_2^2) (1 - x_1^2 x_2^2 x_3^2) (1 - x_1^2 x_2^2 x_3^2 x_4^2)}.$$ I am trying to compute $$\sum_{w\in S_4} (-1)^{\overline{w}} w(f),$$ where $$\overline{w}$$ is the parity of the permutation element $$w$$ (it's the number of transpositions when $$w$$ is written as a product of such), and $$w$$ is acting $$f$$ via the indices of the variables $$x_i$$.

I have this code

f[x1_, x2_, x3_, x4_] := (x1^-3 x2^-2 x3^-1)/((1 - x1^2) (1 - x1^2 x2^2) (1 - x1^2 x2^2 x3^2)(1 - x1^2 x2^2 x3^2 x4^2));

Total[Permute[g @@ Array[Subscript[x, #] &, 4], #] & /@GroupElements[SymmetricGroup] /. g -> f]


but I am not sure how to incorporate the Signature[] of some permutation element so that it coincides with Permute in the code.

The problem is that Signature wants to operate on expressions and interprets each Cycles[...] object as a list of length 1. You can convert Cycles objects to list with PermutationsList and apply Signature afterwards.

With[{xx = Array[Subscript[x, #] &, 4]},

Sum[
Signature[PermutationList[σ, 4]] f @@ Permute[xx, σ],
{σ, GroupElements[SymmetricGroup]}
]

]


Alternatively, you may skip the group hokus-pokus and use Permutations:

With[{xx = Array[Subscript[x, #] &, 4]},

Sum[
Signature[p] f @@ xx[[p]],
{p, Permutations[Range]}
]

]

• Nice. Thank you. This is very helpful! – Mee Seong Im Mar 20 '19 at 18:24
• You're welcome. – Henrik Schumacher Mar 20 '19 at 18:27