Hi Bruce,
On Tue, Sep 11, 2012 at 04:00:34AM -0700, Bruce wrote:
> I have just discovered that the character ring of a finite group is not
> recognised as a Ring.
>
> sage: G = SymmetricGroup(4)
> sage: R = G.character_ring(ZZ)
> sage: R is Ring
> False
I assume you mean:
sage: R in Rings()
False
> I came across this because I tried
>
> sage: SymmetricFunctions(R).schur()
>
> and got an AssertionError
>
> I have discovered that character rings are in fact written for finite
> semigroups.
Yes: this is experimental code that I wrote for my research on
representation theory of specific classes of semigroups (aperiodic,
...). I tried to write it as generically as possible, but most non
trivial features concern those specific class of semigroups. I am not
sure there currently is anything useful/non ridiculous for general
semigroups, and even more so for groups.
That being said, I'd be happy sharing work with others on that topic!
> I have looked at the class hierarchy for the character ring and
> found that it includes the category of commutative additive
> monoids with realizations.
>
> This means that all that is missing is the product of characters. It seems
> to me that this is defined for finite semigroups
Indeed!
> but I am not clear on whether there are issues involved in
> implementing this.
I guess all that needs to be implemented is the character table for
general semigroups, that is the change of basis from the C basis (the
analogue of the p basis of symmetric functions in the case of the
symmetric group; name to be discussed!) to the S basis (the analogue
of the s basis) in the character ring. Of course for groups this
should be just asking GAP.
Then, the product would be easily implemented: it's just the hadamard
product in the C basis, and can be obtained on the S basis by changing
back and forth.
Again: the current infrastructure is designed to be flexible, but is
experimental. In particular it's subject to some changes soon when
I'll be done with the functorial construction patch (which raised
interesting -but time consuming- theoretical discussions; but that's
another story).
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <[email protected]>
http://Nicolas.Thiery.name/
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