On Saturday, February 27, 2016 at 3:59:21 PM UTC, Andrei Stoica wrote:
>
> Hello guys,
>
> I am trying to list the central orthogonal idempotents of a group algebra.
> I chose the dihedral group of order 6 over rational numbers.
>
> A4 = DihedralGroup(3).algebra(QQ)
> Z4 = A4.center()
> idempotents = Z4.central_orthogonal_idempotents()
> idempotents
>
> is the code I use.
>
> The results I get is:
> ︠[1/6*B[0] + 1/6*B[1] + 1/6*B[2], 2/3*B[0] - 1/3*B[2], 1/6*B[0] -
> 1/6*B[1] + 1/6*B[2]]
>
> Could someone help me with what elements B[0], B[1], B[2] are? Generally,
> how can I find those B[i]?
>
after running your code:
sage: Z4.basis()
Finite family {0: B[0], 1: B[1], 2: B[2]}
The B[i]'s you are interested in are the standard basis elements of Z4,
and they correspond to the linear combinations of the group elements by
sage: A4.center_basis()
[B[()], B[(2,3)] + B[(1,2)] + B[(1,3)], B[(1,2,3)] + B[(1,3,2)]]
(so, mathematically, B[i]'s are conjugacy class sums: B[0]=(),
B[1]=(12)+(13)+(12), B[2]=(123)+(132))
HTH,
Dima
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