How to write an involution in Weyl group as a product of $2$-cycles?
Is there some method in Sage to decompose an involution in Weyl group as a product of $2$-cycles (transpositions)?
For example, I define
W=WeylGroup(['A', 3], prefix='s')
t1=[1,2,3,1,2,1]
t2=W.from_reduced_word(t1)
How to write t2 as a product of $2$-cycles? The result should be $(1,4)(2,3)$. Thank you very much.
Something along these lines should do the job:
@max, thank you very much! But the result I got is [(1, 12), (2, 11), (3, 10), (4, 9), (5, 8), (6, 7)] not (1,4)(2,3). Do you know what is the problem? Thank you very much.
It depends on how you interpret
t2
as a permutation. In the code above I consider it as a permutation of the 12 roots. If you have something else in mind, modify the code accordingly.E.g. try to change
.roots()
to.simple_roots()
.@max, thank you very much. I tried simple_roots() but it has errors. The t2 is the product of simple reflections $s_1s_2s_3s_1s_2s_1$.
And how this representation translates into a permutation (and permutatio of what)? How do you see that the expected result is (1,4) (2,3) ?
@max, thank you very much. It is a permutation on the set ${1,2,3,4}$. $s_i$ interchange $i,i+1$. So under t2, 1,2,3,4 is sent to 4,3,2,1, respectively. Writing t2 as a permutation, it is $\begin{array}{cccc} 1&2&3&4\ 4&3&2&1\end{array}$. So writing t2 as product of transpositions, it is (1,4)(2,3), meaning that 1,4 are interchanged and 2,3 are interchanged.