What is this?
I don't understand how that can be interpreted to mean:
. . replace with every permutation of
I have this equation which has sigma notation. I need to expand the sigma into factors and solve for x.
my question is did I get the symmetry expansion right? I want to make sure before I try to solve for x. also, in general how do you think about this? I think you have 3 possible combinations in first factor. each of those has 3 combinations of a,b,c. 3+3+3=9 terms in total.
is this the total expansion?
this is notation for absolute-symmetric sum and cyclic-symmetric sum. where you have terms of the same type having the same coefficient. it shows the contraction of the sum of terms of the same type.
you only need to put abc under sigma if it is not obvious what variables to use.
a single term is understood to be absolute-symmetric. to denote cyclic use a letter like S. you can also use 'sym' and 'cyc'.
if there is more then one term in the group then cyclic symmetry is implied not absolute.
so the cyclic expansion of the expression is.
I notice that what I did was not correct in the first case. I did to many cycles, I did cycle combinations according to brackets. but the crazy part is that after you expand and collect like terms both versions are IDENTICAL??? I try to think about why this is the case. It has something to do with fact that the first factor has full symmetry and the last 3 have cyclic. so maybe that cancels out somehow through one complete cycle.
first case expansion:
second case expansion:
so I do a check using sigma again to lean to do calculations mentally. each term is absolute-symmetric and also heterogeneous so the product is also absolute-symmetric and heterogeneous. I realize that abc term has 3 sym and could have sigma also but that seems more confusing on paper.
so we get identity 0, answer is correct.