On Tue, Apr 23, 2013 at 7:27 PM, Tyler Hannan <than...@u.rochester.edu>wrote:
> 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 Peter-Weyl Theorem does not refer to a "dual group". Maybe it is a minor terminology issue, but the algebraic dual (which is the dual referred to in PW) of a compact, or even reductive, Lie group is in general not a group. In fact, the only time the algebraic dual of a group is itself a group is when the group is abelian. If you want to construct matrix representations of finite groups then you might consider looking at the GAP package repsn http://www.gap-system.org/Packages/repsn.html and decide if you think you can implement any of it is sympy. If you want to work with abelian groups (which is much, much harder than it sounds), that would be great too. > 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>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. >>> To post to this group, send email to sy...@googlegroups.com. >>> >>> Visit this group at >>> http://groups.google.com/**group/sympy?hl=en-US<http://groups.google.com/group/sympy?hl=en-US> >>> . >>> For more options, visit >>> https://groups.google.com/**groups/opt_out<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. > > > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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