I'm referring to the dual group in the context of the Petery-Weyl theorem from harmonic analysis, which applies to any compact group. The dual group consists of irreducible homomorphisms from the group to the space of unitary operators. And by matrix representations, I mean defining the elements of a group as matricies in such a way that group multiplication can be computed as matrix multiplication. In SymPy and GAP groups are represented in the form of a set of permutations, with group multiplication being permutation multiplication, I think using matrix representations might be more useful, as many Lie groups (like the Heisenberg group) are defined in this way.
On Tuesday, April 23, 2013 6:47:40 PM UTC-4, David Joyner wrote: > > It's not clear from your email if you have looked at what SymPy has > already, > or if you have looked at the archives of the sympy list to see what > previous replies > on this topic are. It is also not clear what exactly the procedure for > implementing > character theory would be. (GAP for example uses a lot of predefined > tables.) > By dual group, I assume you men the group of multiplicative characters of > an > abelian group? Finally, you say "matrix representations: either over the > integers or > Z/(2^n Z)." Do you mean matrix representation over a finite ring? I don't > think matrices > over finite fields are implemented in sympy, much less over finite rings. > I'm not trying to be discouraging, but in short, more details would be > appreciated. > > > > On Tue, Apr 23, 2013 at 6:32 PM, Tyler Hannan > <tha...@u.rochester.edu<javascript:> > > wrote: > >> Hello, >> I thought that working with Group Theory for my GSoC 2013 project would >> be ideal. >> >> I have familiarized myself with the GAP library and their approach to >> group theory >> >> problems, and have a fairly extensive knowledge of group theory (at the >> undergraduate >> >> level, at any rate). >> >> The standard approach focuses heavily on representations in the form of >> >> permutations, but does not touch on matrix representations: either over >> the integers or >> >> Z/(2^n Z). This representation is more natural for defining, for example, >> Lie groups and >> >> reduces group multiplication to matrix multiplication. Group >> multiplication as matrix >> >> multiplication has the potential to be more efficient in terms of space >> and time than a >> >> permutation if certain precautions are taken. The other key advantage to >> a matrix >> >> representation is in regard to Lie Groups. The exponential map which >> defines the >> >> relationship between a Lie Group and its associated Lie Algebra can be >> computed using the >> >> power series expansion of a matrix in the Lie Group. >> >> Another area to focus on is the notion of algorithmic efficiency for >> tests for >> >> subgroups, normal supgroups, and the use of Sylow Theory. As defined in >> the GAP library, >> >> every group object is annotated with flags describing the group's >> properties, which are >> >> lazily evaluated. Many of these properties can be filled in purely by the >> group's >> >> construction. Using these properties, and weighing the relative >> algorithmic costs of >> >> testing them, could create an algorithm which adapts to take advantage of >> group structure. >> >> For instance, when testing whether a given subgroup N is normal to G, we >> can use either a >> >> straight test to insure gN = Ng for all g in G, or we can use Sylow's >> theorems. Sylow >> >> theory provides a far quicker test, assuming we already know N to be a >> Sylow p-subgroup of >> >> G. So, to choose which method to use, we must consider the amount of time >> it would take to >> >> confirm N is a Sylow p-subgroup. >> >> Finally, if this works out, I would like to implement support for >> character theory >> >> and the computation of dual groups, at least in the case of finite order. >> >> Any thoughts or criticisms would be much appreciated. >> >> -Tyler >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to sympy+un...@googlegroups.com <javascript:>. >> To post to this group, send email to sy...@googlegroups.com <javascript:> >> . >> Visit this group at http://groups.google.com/group/sympy?hl=en-US. >> For more options, visit https://groups.google.com/groups/opt_out. >> >> >> > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy?hl=en-US. For more options, visit https://groups.google.com/groups/opt_out.
