Hello,

On Thu, Mar 26, 2009 at 12:09 PM, David Joyner <[email protected]> wrote:
> On Thu, Mar 26, 2009 at 12:41 AM, [email protected]
> <[email protected]> wrote:
>>
>> Hello,
>>
>> I've been trying to use Sage to play around with representation
>> theory, but I'm running into trouble when dealing with characters. In
>> particular, I can't seem to build any character with complex values.
>> So, for example, I'm trying to build the irreducible characters for a
>> cyclic subgroup of order 3 in A_4.
>>
>> H = AlternatingGroup(4)
>> g = H.list()[1]
>> K = H.subgroup([g])
>>
>> # All integers works great
>> c = K.character([1,1,1])
>>
>> # It doesn't seem to work with elements coming from anywhere else
>> # for example
>> zeta2 = e^((I*pi*2)/3)
>> c = K.character([1,zeta2,zeta2**2])
>>
>> # Or from an other source;
>> k.<z> = NumberField(x^2+x+1)
>> zeta2 = k.roots_of_unity()[3]
>> c = K.character([1,zeta2,zeta2**2])
>>
>> Both of these will both produce errors.
>>
>> I'm using Sage 3.4. Any ideas?

> It seems to be an issue with the conversion between
> Sage's elements of CyclotomicField(3) and GAP's version
> of that field.
>
> Franco, do you know?

Yes, it must be exactly that. If you start with the gap version of
that field element, then one can create the character without any
problems:

sage: H = AlternatingGroup(4)
sage: g = H.list()[1]
sage: K = H.subgroup([g])
sage: z = gap.E(3)
sage: c = K.character([1, z, z**2])
sage: c
Character of Subgroup of AlternatingGroup(4) generated by [(2,3,4)]
sage: [c(k) for k in K]
[1, zeta3, -zeta3 - 1]

Take care,
Franco

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