ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 25 Feb 2017 16:41:05 -0600how to sum up the function over all permutations of variables in associative non-commutative algebrahttp://ask.sagemath.org/question/34714/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 $i$.Sun, 04 Sep 2016 08:19:15 -0500http://ask.sagemath.org/question/34714/how-to-sum-up-the-function-over-all-permutations-of-variables-in-associative-non-commutative-algebra/Answer by dan_fulea for <p>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 $i$.</p>
http://ask.sagemath.org/question/34714/how-to-sum-up-the-function-over-all-permutations-of-variables-in-associative-non-commutative-algebra/?answer=36731#post-id-36731The 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.)Sat, 25 Feb 2017 16:41:05 -0600http://ask.sagemath.org/question/34714/how-to-sum-up-the-function-over-all-permutations-of-variables-in-associative-non-commutative-algebra/?answer=36731#post-id-36731