# how to sum up the function over all permutations of variables in associative non-commutative algebra

hello, i need to sum up $\lambda_{\sigma}(a_{\sigma(1)}a_{\sigma(2)}a_{\sigma(3)}-a_{\sigma(3)}a_{\sigma(4)}a_{\sigma(5)})$ over all $\sigma\in S_5$ where $a_i$ are elements of associative non-commutative algebra. the result should be $E_1a_1a_2a_3a_4a_5+\cdots+E_{120}a_5a_4a_3a_2a_1$ and i need to express $E_i$ in terms of $\lambda_{\sigma}$. actualy my tartget is to find non-zero solution of $E_i=0$ for all $i$.

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The following worked for me:

A.<a,b,c,d,e> = FreeAlgebra( QQ, 5 )
dic = { 1:a, 2:b, 3:c, 4:d, 5:e }
G = SymmetricGroup( 5 )

import random
scalars = dict( [ ( g, QQ( random.choice( [1..10] ) ) ) for g in G ] )

sum( [ scalars[g] * ( + prod( [ dic[g(k)] for k in (1,2,3) ] )
- prod( [ dic[g(k)] for k in (3,4,5) ] ) )
for g in G ] )


However, the result is a liniar combination of words of degree 3, not 5 as in the posted question. (The used scalars were randomly generated.)

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