Compute $w^2 $ if $w = 782193456$, in cycle notation and compute $w^2$ using its expression in terms of generators $s_1...s_8$, simplifying it as much as possible.
my answer is $w^2 = (147)(256)(389)$ is this correct?
Can someone please help me expressing the answer in terms of generators. I have tried to do a wiring diagram, and from there do a diagram where I try to put the result in terms of generators,but I don't really know how. Thank you
I haven't checked your computation but let's assume it's correct. Get the result in one line notation. There's a simple algorithm for getting a minimal length expression in terms of generators. Find a right descent ($w^2(i)>w^2(i+1)$). The corresponding simple reflection can be used as the rightmost in some reduced expression. Act with the reflection and recurse until you get the identity. (This may be backward from the multiplication order you use, so make sure to check it works out.)
For a simple example, consider $a=4231$. $a(1)>a(2)$, so we can use $s_1$, then get $2431$. Position 2 is larger than position 3 in the new permutation, so so far we have $s_2s_1$ and are left with $2341$. Proceeding in this fashion we get $s_1s_2s_3s_2s_1$.