On Mar 19, 2007, at 9:58 AM, Mike Hansen wrote:
> I guess this is mostly directed toward David Joyner, but if anyone > else knows, feel free to chime in. I've been trying to figure out the > best way to do calculations in a group ring or group algebra. I've > checked around for GAP packages, but they seem to be pretty limited > and very awkward to use. Is there a nice way that I'm missing for > dealing with these in GAP? > > I ended up writing my own hack for a group algebra for some recent > work and would be interested in writing more complete code for it over > the summer. I would like to be able to do (almost) all of the > calculations below in SAGE: > http://magma.maths.usyd.edu.au/magma/htmlhelp/part10.htm Here's my long-term take on this question (anyone else please feel free to chime in and agree/disagree). There should really be a GroupAlgebra class, derived from Algebra (in algebra.py). Currently the functionality of Algebra, and its subclasses, is quite limited. There's some code for quarternion algebras, but it's not too efficient yet. I imagine that a GroupAlgebra would have a base ring R, and an associated group G. If the group G was finite, one could represent elements by vectors of elements of R; if G was infinite (or just really big), perhaps one would want a sparse representation. If G is abelian, and R commutative, special measures could be taken to speed up the arithmetic. (Does anyone do group algebras over non-commutative rings....? I don't know...). There would be coercions from R into R [G] and also from G into R[G] (assuming R is unital). There would be a specialisation for the case that R is a field. Then the harder stuff: decomposition into irreducibles, etc, which actually involves writing tricky code, or perhaps this has already been written.... I have no idea, I'm not a representation theory sort of guy. One could probably already define these things in SAGE as a quotient of a free algebra, but I bet it wouldn't be too efficient that way, and not very useful. So I guess we eventually want to see this: sage: G = SymmetricGroup(3) sage: ZZ[G] Group Algebra of Symmetric group of order 3! as a permutation group over Integer Ring David --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
