[GAP Forum] Fwd: Recombining irreducible representations

[Jerry Swan]

A helpful further response from Marc Keilberg indicates that this is
possible for finite G.

Concretely, I'm working with $S_n$, which I should have clarified.

If anyone can spell out what I need to do to get back $g$ (as opposed to
just its class representative) that would be enormously helpful.

Best wishes,

Jerry.

On Sun, Jun 18, 2017 at 2:30 PM, Marc Keilberg <keilberg at usc.edu> wrote:

> Eh, jumped the gun.  So used to working with characters I failed to
> realize these are the full representations you're applying. In this case,
> at least for finite G: yes.  A faithful permutation representation of the
> group is determined by the representation theory (you can write it out in
> permutation matrices, in particular), and the group elements are uniquely
> determined by their representation as a permutation.
>
> On Sun, Jun 18, 2017 at 6:11 AM, Marc Keilberg <keilberg at usc.edu> wrote:
>
>> The irreducible characters are class functions, so you necessarily can't
>> get any more information than the conjugacy class.  And since the
>> irreducible characters are a basis for the class functions, that's
>> precisely what you can recover.  So you can only recover g itself if it's
>> in the center.
>>
>> On Sun, Jun 18, 2017 at 6:05 AM, Jerry Swan <dr.jerry.swan at gmail.com>
>> wrote:
>>
>>> Dear all,
>>>
>>> For some element g of a group G for which irr :=
>>> IrreducibleRepresentations(G) have been obtained, is it possible to
>>> recover
>>> g from images := List(irr,r->Image(r,g)) ?
>>>
>>> Best wishes,
>>>
>>> Jerry.
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>>
>>
>