I'm currently a freshman at Harvard College, probably concentrating in
mathematics / mathematics and CS. I've done a lot of mathematics in
high school (math Olympiads) and university (honors linear and
abstract algebra, real and complex analysis) so far, and I'm
interested in bringing mathematics and CS together.
I've been dealing with sympy for a couple of weeks now and was
wondering whether it'd be a good idea for GSoC to implement some more
complicated combinatorial functionality (e.g. graph algorithms,
generating functions, recurrence relations, operations on sets,...) ?
Also, I'm currently working on several little functions for
computation of the Galois group of quadratic/cubic/quartic
polynomials; I'll probably send the code in a couple of days. Maybe
I'll be able to develop some GSoC-like ideas in this direction
(abstract algebra) as well.
> I'm currently a freshman at Harvard College, probably concentrating in > mathematics / mathematics and CS. I've done a lot of mathematics in > high school (math Olympiads) and university (honors linear and > abstract algebra, real and complex analysis) so far, and I'm > interested in bringing mathematics and CS together.
> I've been dealing with sympy for a couple of weeks now and was > wondering whether it'd be a good idea for GSoC to implement some more > complicated combinatorial functionality (e.g. graph algorithms, > generating functions, recurrence relations, operations on sets,...) ?
networkx implements in python a lot of graph theory. I don't think sympy has an interface though. sympy has rsolve for sequences defined by a linear recurrence relation However, there isn't one for a vector-valued sequence (someone asked about that the other day on the sage-support list).
> Also, I'm currently working on several little functions for > computation of the Galois group of quadratic/cubic/quartic > polynomials; I'll probably send the code in a couple of days. Maybe > I'll be able to develop some GSoC-like ideas in this direction > (abstract algebra) as well.
Sympy doesn't have much of that, although Sage does. I think more abstract algebra in sympy would be very useful.
> -- > You received this message because you are subscribed to the Google Groups "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. > For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
> I've been dealing with sympy for a couple of weeks now and was > wondering whether it'd be a good idea for GSoC to implement some more > complicated combinatorial functionality (e.g. graph algorithms, > generating functions, recurrence relations, operations on sets,...) ?
Note, though, that graph theory is out of scope of SymPy, mainly because graphs are already implemented in a couple other Python libraries (e.g. NetworkX: http://networkx.lanl.gov/). Sorry, I cannot find the proper source to cite; but trust me, Aaron told me that about a month ago :-)
> Also, I'm currently working on several little functions for > computation of the Galois group of quadratic/cubic/quartic > polynomials; I'll probably send the code in a couple of days. Maybe > I'll be able to develop some GSoC-like ideas in this direction > (abstract algebra) as well.
I suppose that your contributions will be welcome :-)
As others have said, we will leave the straight graph theory to networkx and similar libraries. Those other things would be fitting, though. Take a look at what's already implemented in the combinatorics module, the sets module, and elsewhere.
And it would be awesome to have a group theory module. We presently only have a Permutation class in the combinatorics module, but other than that, we don't really have a good way to represent a group. Obviously, to compute the Galois group of a polynomial, you need a way to represent it, so for this idea, you would really need to implement a group theory framework that we can build upon.
What algorithm do you use to compute the Galois group?
Aaron Meurer
On Fri, Mar 16, 2012 at 6:06 AM, Aleksandar Makelov
> I'm currently a freshman at Harvard College, probably concentrating in > mathematics / mathematics and CS. I've done a lot of mathematics in > high school (math Olympiads) and university (honors linear and > abstract algebra, real and complex analysis) so far, and I'm > interested in bringing mathematics and CS together.
> I've been dealing with sympy for a couple of weeks now and was > wondering whether it'd be a good idea for GSoC to implement some more > complicated combinatorial functionality (e.g. graph algorithms, > generating functions, recurrence relations, operations on sets,...) ?
> Also, I'm currently working on several little functions for > computation of the Galois group of quadratic/cubic/quartic > polynomials; I'll probably send the code in a couple of days. Maybe > I'll be able to develop some GSoC-like ideas in this direction > (abstract algebra) as well.
> -- > You received this message because you are subscribed to the Google Groups "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. > For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
for the Galois group I'm using a rather naive approach for equations
of degree 4 or less - it's based on the theory given in a standard
textbook (Artin's Algebra, 2nd edition) - looking at the discriminant
and in the quartic case at the resolvent cubic. It's a solid algorithm
except for one of the cases where there is ambiguity between C_4 and
D_4, but that's OK for most of the polynomials you'd encounter in the
exercises... But now that I looked around I saw there might be some
more general algorithms out there (http://mathoverflow.net/questions/ 22923/computing-the-galois-group-of-a-polynomial/ for example), so
that's probably what I'm going to try next.
Respectively, representing groups will be just by name (not so many of
them in degrees less than 5...) but obviously in higher degrees it'd
be convenient to have a way of handling arbitrary groups. Galois
groups are naturally embedded in the symmetric groups S_n, so
permutations will probably pop up from the algorithms; as size gets
larger (approx. n!) it'd be better to have them in cycle notation, and
also try to extract a list of generators / generators with relations.
Other than that, I don't know... I'm currently doing representation
theory and it sounds tempting (eigenvals()? character tables?). But
that's just the mathematical side of it, and I'll have to think a lot
about implementation.
Aleksandar Makelov
On Mar 16, 4:32 pm, Aaron Meurer <asmeu...@gmail.com> wrote:
> As others have said, we will leave the straight graph theory to
> networkx and similar libraries. Those other things would be fitting,
> though. Take a look at what's already implemented in the
> combinatorics module, the sets module, and elsewhere.
> And it would be awesome to have a group theory module. We presently
> only have a Permutation class in the combinatorics module, but other
> than that, we don't really have a good way to represent a group.
> Obviously, to compute the Galois group of a polynomial, you need a way
> to represent it, so for this idea, you would really need to implement
> a group theory framework that we can build upon.
> What algorithm do you use to compute the Galois group?
> Aaron Meurer
> On Fri, Mar 16, 2012 at 6:06 AM, Aleksandar Makelov
> <amake...@college.harvard.edu> wrote:
> > Hi guys,
> > I'm currently a freshman at Harvard College, probably concentrating in
> > mathematics / mathematics and CS. I've done a lot of mathematics in
> > high school (math Olympiads) and university (honors linear and
> > abstract algebra, real and complex analysis) so far, and I'm
> > interested in bringing mathematics and CS together.
> > I've been dealing with sympy for a couple of weeks now and was
> > wondering whether it'd be a good idea for GSoC to implement some more
> > complicated combinatorial functionality (e.g. graph algorithms,
> > generating functions, recurrence relations, operations on sets,...) ?
> > Also, I'm currently working on several little functions for
> > computation of the Galois group of quadratic/cubic/quartic
> > polynomials; I'll probably send the code in a couple of days. Maybe
> > I'll be able to develop some GSoC-like ideas in this direction
> > (abstract algebra) as well.
> > --
> > You received this message because you are subscribed to the Google Groups "sympy" group.
> > To post to this group, send email to sympy@googlegroups.com.
> > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com.
> > For more options, visit this group athttp://groups.google.com/group/sympy?hl=en.
> for the Galois group I'm using a rather naive approach for equations > of degree 4 or less - it's based on the theory given in a standard > textbook (Artin's Algebra, 2nd edition) - looking at the discriminant > and in the quartic case at the resolvent cubic. It's a solid algorithm > except for one of the cases where there is ambiguity between C_4 and > D_4, but that's OK for most of the polynomials you'd encounter in the > exercises... But now that I looked around I saw there might be some > more general algorithms out there (http://mathoverflow.net/questions/ > 22923/computing-the-galois-group-of-a-polynomial/ for example), so > that's probably what I'm going to try next.
> Respectively, representing groups will be just by name (not so many of > them in degrees less than 5...) but obviously in higher degrees it'd > be convenient to have a way of handling arbitrary groups. Galois > groups are naturally embedded in the symmetric groups S_n, so > permutations will probably pop up from the algorithms; as size gets > larger (approx. n!) it'd be better to have them in cycle notation, and > also try to extract a list of generators / generators with relations. > Other than that, I don't know... I'm currently doing representation > theory and it sounds tempting (eigenvals()? character tables?). But > that's just the mathematical side of it, and I'll have to think a lot > about implementation.
> Aleksandar Makelov
Well, we want to be able to do more with a group than just have its name.
How you represent the group, and its elements, depends on what the group is (finite or infinite for one thing), and what you plan to do. It will be useful to be able to represent them in many different ways.
If you're interested in working on group theory, you might also look at the GAP project for some inspiration. They have the most powerful group theory software (as far as I know).
> On Mar 16, 4:32 pm, Aaron Meurer <asmeu...@gmail.com> wrote: >> As others have said, we will leave the straight graph theory to >> networkx and similar libraries. Those other things would be fitting, >> though. Take a look at what's already implemented in the >> combinatorics module, the sets module, and elsewhere.
>> And it would be awesome to have a group theory module. We presently >> only have a Permutation class in the combinatorics module, but other >> than that, we don't really have a good way to represent a group. >> Obviously, to compute the Galois group of a polynomial, you need a way >> to represent it, so for this idea, you would really need to implement >> a group theory framework that we can build upon.
>> What algorithm do you use to compute the Galois group?
>> Aaron Meurer
>> On Fri, Mar 16, 2012 at 6:06 AM, Aleksandar Makelov
>> <amake...@college.harvard.edu> wrote: >> > Hi guys,
>> > I'm currently a freshman at Harvard College, probably concentrating in >> > mathematics / mathematics and CS. I've done a lot of mathematics in >> > high school (math Olympiads) and university (honors linear and >> > abstract algebra, real and complex analysis) so far, and I'm >> > interested in bringing mathematics and CS together.
>> > I've been dealing with sympy for a couple of weeks now and was >> > wondering whether it'd be a good idea for GSoC to implement some more >> > complicated combinatorial functionality (e.g. graph algorithms, >> > generating functions, recurrence relations, operations on sets,...) ?
>> > Also, I'm currently working on several little functions for >> > computation of the Galois group of quadratic/cubic/quartic >> > polynomials; I'll probably send the code in a couple of days. Maybe >> > I'll be able to develop some GSoC-like ideas in this direction >> > (abstract algebra) as well.
>> > -- >> > You received this message because you are subscribed to the Google Groups "sympy" group. >> > To post to this group, send email to sympy@googlegroups.com. >> > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. >> > For more options, visit this group athttp://groups.google.com/group/sympy?hl=en.
> -- > You received this message because you are subscribed to the Google Groups "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. > For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
Yup I'm looking at the GAP website now and it seems like a lot of fun;
I'm also looking for some kind of algorithm reference for
computational group theory like the ones listed at GAP. I'll have a
lot of work to do in the next couple of days (break's over) but will
try to implement at least one way of representing groups and find some
algorithms that might help build a potential group theory module.
Aleksandar Makelov
On Mar 16, 6:39 pm, Aaron Meurer <asmeu...@gmail.com> wrote:
> > for the Galois group I'm using a rather naive approach for equations
> > of degree 4 or less - it's based on the theory given in a standard
> > textbook (Artin's Algebra, 2nd edition) - looking at the discriminant
> > and in the quartic case at the resolvent cubic. It's a solid algorithm
> > except for one of the cases where there is ambiguity between C_4 and
> > D_4, but that's OK for most of the polynomials you'd encounter in the
> > exercises... But now that I looked around I saw there might be some
> > more general algorithms out there (http://mathoverflow.net/questions/ > > 22923/computing-the-galois-group-of-a-polynomial/ for example), so
> > that's probably what I'm going to try next.
> > Respectively, representing groups will be just by name (not so many of
> > them in degrees less than 5...) but obviously in higher degrees it'd
> > be convenient to have a way of handling arbitrary groups. Galois
> > groups are naturally embedded in the symmetric groups S_n, so
> > permutations will probably pop up from the algorithms; as size gets
> > larger (approx. n!) it'd be better to have them in cycle notation, and
> > also try to extract a list of generators / generators with relations.
> > Other than that, I don't know... I'm currently doing representation
> > theory and it sounds tempting (eigenvals()? character tables?). But
> > that's just the mathematical side of it, and I'll have to think a lot
> > about implementation.
> > Aleksandar Makelov
> Well, we want to be able to do more with a group than just have its name.
> How you represent the group, and its elements, depends on what the
> group is (finite or infinite for one thing), and what you plan to do.
> It will be useful to be able to represent them in many different ways.
> If you're interested in working on group theory, you might also look
> at the GAP project for some inspiration. They have the most powerful
> group theory software (as far as I know).
> Aaron Meurer
> > On Mar 16, 4:32 pm, Aaron Meurer <asmeu...@gmail.com> wrote:
> >> As others have said, we will leave the straight graph theory to
> >> networkx and similar libraries. Those other things would be fitting,
> >> though. Take a look at what's already implemented in the
> >> combinatorics module, the sets module, and elsewhere.
> >> And it would be awesome to have a group theory module. We presently
> >> only have a Permutation class in the combinatorics module, but other
> >> than that, we don't really have a good way to represent a group.
> >> Obviously, to compute the Galois group of a polynomial, you need a way
> >> to represent it, so for this idea, you would really need to implement
> >> a group theory framework that we can build upon.
> >> What algorithm do you use to compute the Galois group?
> >> Aaron Meurer
> >> On Fri, Mar 16, 2012 at 6:06 AM, Aleksandar Makelov
> >> <amake...@college.harvard.edu> wrote:
> >> > Hi guys,
> >> > I'm currently a freshman at Harvard College, probably concentrating in
> >> > mathematics / mathematics and CS. I've done a lot of mathematics in
> >> > high school (math Olympiads) and university (honors linear and
> >> > abstract algebra, real and complex analysis) so far, and I'm
> >> > interested in bringing mathematics and CS together.
> >> > I've been dealing with sympy for a couple of weeks now and was
> >> > wondering whether it'd be a good idea for GSoC to implement some more
> >> > complicated combinatorial functionality (e.g. graph algorithms,
> >> > generating functions, recurrence relations, operations on sets,...) ?
> >> > Also, I'm currently working on several little functions for
> >> > computation of the Galois group of quadratic/cubic/quartic
> >> > polynomials; I'll probably send the code in a couple of days. Maybe
> >> > I'll be able to develop some GSoC-like ideas in this direction
> >> > (abstract algebra) as well.
> >> > --
> >> > You received this message because you are subscribed to the Google Groups "sympy" group.
> >> > To post to this group, send email to sympy@googlegroups.com.
> >> > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com.
> >> > For more options, visit this group athttp://groups.google.com/group/sympy?hl=en.
> > --
> > You received this message because you are subscribed to the Google Groups "sympy" group.
> > To post to this group, send email to sympy@googlegroups.com.
> > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com.
> > For more options, visit this group athttp://groups.google.com/group/sympy?hl=en.
> Yup I'm looking at the GAP website now and it seems like a lot of fun; > I'm also looking for some kind of algorithm reference for > computational group theory like the ones listed at GAP. I'll have a > lot of work to do in the next couple of days (break's over) but will > try to implement at least one way of representing groups and find some > algorithms that might help build a potential group theory module.
> Aleksandar Makelov > On Mar 16, 6:39 pm, Aaron Meurer<asmeu...@gmail.com> wrote: >> On Fri, Mar 16, 2012 at 4:12 PM, Aleksandar Makelov
>> <amake...@college.harvard.edu> wrote: >>> Hi, >>> for the Galois group I'm using a rather naive approach for equations >>> of degree 4 or less - it's based on the theory given in a standard >>> textbook (Artin's Algebra, 2nd edition) - looking at the discriminant >>> and in the quartic case at the resolvent cubic. It's a solid algorithm >>> except for one of the cases where there is ambiguity between C_4 and >>> D_4, but that's OK for most of the polynomials you'd encounter in the >>> exercises... But now that I looked around I saw there might be some >>> more general algorithms out there (http://mathoverflow.net/questions/ >>> 22923/computing-the-galois-group-of-a-polynomial/ for example), so >>> that's probably what I'm going to try next. >> http://www.maplesoft.com/support/help/Maple/view.aspx?path=galoismay >> also give you some inspiration.
>>> Respectively, representing groups will be just by name (not so many of >>> them in degrees less than 5...) but obviously in higher degrees it'd >>> be convenient to have a way of handling arbitrary groups. Galois >>> groups are naturally embedded in the symmetric groups S_n, so >>> permutations will probably pop up from the algorithms; as size gets >>> larger (approx. n!) it'd be better to have them in cycle notation, and >>> also try to extract a list of generators / generators with relations. >>> Other than that, I don't know... I'm currently doing representation >>> theory and it sounds tempting (eigenvals()? character tables?). But >>> that's just the mathematical side of it, and I'll have to think a lot >>> about implementation. >>> Aleksandar Makelov >> Well, we want to be able to do more with a group than just have its name.
>> How you represent the group, and its elements, depends on what the >> group is (finite or infinite for one thing), and what you plan to do. >> It will be useful to be able to represent them in many different ways.
>> If you're interested in working on group theory, you might also look >> at the GAP project for some inspiration. They have the most powerful >> group theory software (as far as I know).
>> Aaron Meurer
>>> On Mar 16, 4:32 pm, Aaron Meurer<asmeu...@gmail.com> wrote: >>>> As others have said, we will leave the straight graph theory to >>>> networkx and similar libraries. Those other things would be fitting, >>>> though. Take a look at what's already implemented in the >>>> combinatorics module, the sets module, and elsewhere. >>>> And it would be awesome to have a group theory module. We presently >>>> only have a Permutation class in the combinatorics module, but other >>>> than that, we don't really have a good way to represent a group. >>>> Obviously, to compute the Galois group of a polynomial, you need a way >>>> to represent it, so for this idea, you would really need to implement >>>> a group theory framework that we can build upon. >>>> What algorithm do you use to compute the Galois group? >>>> Aaron Meurer >>>> On Fri, Mar 16, 2012 at 6:06 AM, Aleksandar Makelov >>>> <amake...@college.harvard.edu> wrote: >>>>> Hi guys, >>>>> I'm currently a freshman at Harvard College, probably concentrating in >>>>> mathematics / mathematics and CS. I've done a lot of mathematics in >>>>> high school (math Olympiads) and university (honors linear and >>>>> abstract algebra, real and complex analysis) so far, and I'm >>>>> interested in bringing mathematics and CS together. >>>>> I've been dealing with sympy for a couple of weeks now and was >>>>> wondering whether it'd be a good idea for GSoC to implement some more >>>>> complicated combinatorial functionality (e.g. graph algorithms, >>>>> generating functions, recurrence relations, operations on sets,...) ? >>>>> Also, I'm currently working on several little functions for >>>>> computation of the Galois group of quadratic/cubic/quartic >>>>> polynomials; I'll probably send the code in a couple of days. Maybe >>>>> I'll be able to develop some GSoC-like ideas in this direction >>>>> (abstract algebra) as well. >>>>> -- >>>>> You received this message because you are subscribed to the Google Groups "sympy" group. >>>>> To post to this group, send email to sympy@googlegroups.com. >>>>> To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. >>>>> For more options, visit this group athttp://groups.google.com/group/sympy?hl=en. >>> -- >>> You received this message because you are subscribed to the Google Groups "sympy" group. >>> To post to this group, send email to sympy@googlegroups.com. >>> To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. >>> For more options, visit this group athttp://groups.google.com/group/sympy?hl=en.
I think a main reference is "Permutation Group Algorithms" by Akos Seress - Cambridge Tracts in Mathemathics 152 published 2003.
I worked as a student last year and may apply as mentor this year.
Please take a look at my branches in github. I was implementing the
Schreier Sims algorithm but I ran out of time unfortunately. You could
either help me merge my branches in or take off where I left.
> And it would be awesome to have a group theory module. We presently
> only have a Permutation class in the combinatorics module, but other
> than that, we don't really have a good way to represent a group.
Is this necessary? All groups are isomorphic to the permutation group
anyway. Groups for specific structures can make use of functionality
implemented for them (matrix group -> sympy matrices, galois -> polys)
for basic operations and can implement the mapping to the perm group
module for group theoretic operations.
> Obviously, to compute the Galois group of a polynomial, you need a way
> to represent it, so for this idea, you would really need to implement
> a group theory framework that we can build upon.
Again, most of the concrete algorithms for groups are for Permutation
groups only. I see that the book by Seress has already been
referenced.
I think a more fruitful venture would be to extend the perm groups
module and leverage that to implement matrix groups or galois groups
(but then again, I am probably biased ;)
> Is this necessary? All groups are isomorphic to the permutation group > anyway. Groups for specific structures can make use of functionality > implemented for them (matrix group -> sympy matrices, galois -> polys) > for basic operations and can implement the mapping to the perm group > module for group theoretic operations.
This seems incorrect. Zn is abelian for example and it is not isomorphic to any permutation group. Moreover, there are all the continuous groups.
Besides, it will be nicer to have some abstract object that is not tied to a concrete representation, even though it will probably just be a wrapper for all the representations supported by sympy.
>> And it would be awesome to have a group theory module. We presently >> only have a Permutation class in the combinatorics module, but other >> than that, we don't really have a good way to represent a group.
> Is this necessary? All groups are isomorphic to the permutation group > anyway. Groups for specific structures can make use of functionality > implemented for them (matrix group -> sympy matrices, galois -> polys) > for basic operations and can implement the mapping to the perm group > module for group theoretic operations.
As Stefan noted, this is only true for finite groups.
And anyway, the point I was trying to make was that a permutation represents an element of a group, whereas I was talking about a way to represent the whole group.
>> Obviously, to compute the Galois group of a polynomial, you need a way >> to represent it, so for this idea, you would really need to implement >> a group theory framework that we can build upon.
> Again, most of the concrete algorithms for groups are for Permutation > groups only. I see that the book by Seress has already been > referenced.
> I think a more fruitful venture would be to extend the perm groups > module and leverage that to implement matrix groups or galois groups > (but then again, I am probably biased ;)
> Cheers > Sapta
> -- > You received this message because you are subscribed to the Google Groups "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. > For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
>> Is this necessary? All groups are isomorphic to the permutation group >> anyway. Groups for specific structures can make use of functionality >> implemented for them (matrix group -> sympy matrices, galois -> polys) >> for basic operations and can implement the mapping to the perm group >> module for group theoretic operations.
> This seems incorrect. Zn is abelian for example and it is not > isomorphic to any permutation group. Moreover, there are all the > continuous groups.
It is not correct to say Zn (I assume you mean the ring of integers mod n) is not isomorphic to a permutation group. (Consider the cyclic group generated by the n-cycle (1,2,...,n) in disjoint cycle notation.)
Implementing Lie groups would be a relatively difficult undertaking I think...
> Besides, it will be nicer to have some abstract object that is not > tied to a concrete representation, even though it will probably just > be a wrapper for all the representations supported by sympy.
> -- > You received this message because you are subscribed to the Google Groups "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. > For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
On Sat, Mar 17, 2012 at 10:11 PM, David Joyner <wdjoy...@gmail.com> wrote: > On Sat, Mar 17, 2012 at 3:55 PM, krastanov.ste...@gmail.com > <krastanov.ste...@gmail.com> wrote:
>>> Is this necessary? All groups are isomorphic to the permutation group >>> anyway. Groups for specific structures can make use of functionality >>> implemented for them (matrix group -> sympy matrices, galois -> polys) >>> for basic operations and can implement the mapping to the perm group >>> module for group theoretic operations.
>> This seems incorrect. Zn is abelian for example and it is not >> isomorphic to any permutation group. Moreover, there are all the >> continuous groups.
> It is not correct to say Zn (I assume you mean the ring of integers > mod n) is not isomorphic to a permutation group. (Consider the > cyclic group generated by the n-cycle (1,2,...,n) in disjoint > cycle notation.)
I think difference should be made between
nZ = {n * k | k\in Z}
and
Z_n = {k mod n | k\in Z}.
Z_n is finite, while nZ is infinite. This might be the reason for the misunderstanding.
> On Sat, Mar 17, 2012 at 3:55 PM, krastanov.ste...@gmail.com > <krastanov.ste...@gmail.com> wrote: >>> Is this necessary? All groups are isomorphic to the permutation group >>> anyway. Groups for specific structures can make use of functionality >>> implemented for them (matrix group -> sympy matrices, galois -> polys) >>> for basic operations and can implement the mapping to the perm group >>> module for group theoretic operations.
>> This seems incorrect. Zn is abelian for example and it is not >> isomorphic to any permutation group. Moreover, there are all the >> continuous groups.
> It is not correct to say Zn (I assume you mean the ring of integers > mod n) is not isomorphic to a permutation group. (Consider the > cyclic group generated by the n-cycle (1,2,...,n) in disjoint > cycle notation.)
> Implementing Lie groups would be a relatively difficult undertaking > I think...
>> Besides, it will be nicer to have some abstract object that is not >> tied to a concrete representation, even though it will probably just >> be a wrapper for all the representations supported by sympy.
>> -- >> You received this message because you are subscribed to the Google Groups "sympy" group. >> To post to this group, send email to sympy@googlegroups.com. >> To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. >> For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
For Lie groups you might want to look at "Geometric Algebra for Physicists" by Doran and Lasenby section 11.3 - Lie Groups and section 11.4 - Complex Structures and Unitary Groups.
> I think a main reference is "Permutation Group Algorithms" by Akos
> Seress - Cambridge Tracts in Mathemathics 152 published 2003.
Thanks! The "Handbook of computational group theory" also looks like
serious business. Unfortunately, neither of these is a free resource;
I might end up buying one, I don't know.
> I worked as a student last year and may apply as mentor this year.
> Please take a look at my branches in github. I was implementing the
> Schreier Sims algorithm but I ran out of time unfortunately. You could
> either help me merge my branches in or take off where I left.
OK, I'll hopefully have the time to take a look this coming week.
> Is this necessary? All groups are isomorphic to the permutation group
> anyway. Groups for specific structures can make use of functionality
> implemented for them (matrix group -> sympy matrices, galois -> polys)
> for basic operations and can implement the mapping to the perm group
> module for group theoretic operations.
So I looked at the permutations module and it has a lot of nice group-
ish functions (like composing/inverting permutations, raising to
powers, conjugating permutations, getting the order (as an element of
the corresponding symmetric group) of a permutation, ...). These can
be incorporated in a representation of groups using permutation
groups; Galois groups would fit perfectly in this representation since
they naturally live inside the symmetric groups, and yes a lot of the
functions in the polys module will be helpful.
Also, there are generators for common groups like S_n, C_n, D_n, A_n
in the context of permutation representations. All this provides a
nice foundation for defining a Group class or something like that,
with one of the ways of representing it being the permutation
representation. Other ways (e.g., matrices, character tables, list of
generators and relations) could probably be added later, and
functionality to go from one to representation to another?
In other news, I found a bug inside the generators.py file in the
permutations module - the dihedral group D_2 of order 4 is given a
wrong permutation representation. I have a fix for this (well it's
quite straightforward, just manually considering the case n=2 and
outputting the right representation, because the general algorithm
fails there), what should I do about it?
>> I think a main reference is "Permutation Group Algorithms" by Akos >> Seress - Cambridge Tracts in Mathemathics 152 published 2003.
> Thanks! The "Handbook of computational group theory" also looks like > serious business. Unfortunately, neither of these is a free resource; > I might end up buying one, I don't know.
>> I worked as a student last year and may apply as mentor this year. >> Please take a look at my branches in github. I was implementing the >> Schreier Sims algorithm but I ran out of time unfortunately. You could >> either help me merge my branches in or take off where I left.
> OK, I'll hopefully have the time to take a look this coming week.
>> Is this necessary? All groups are isomorphic to the permutation group >> anyway. Groups for specific structures can make use of functionality >> implemented for them (matrix group -> sympy matrices, galois -> polys) >> for basic operations and can implement the mapping to the perm group >> module for group theoretic operations.
> So I looked at the permutations module and it has a lot of nice group- > ish functions (like composing/inverting permutations, raising to > powers, conjugating permutations, getting the order (as an element of > the corresponding symmetric group) of a permutation, ...). These can > be incorporated in a representation of groups using permutation > groups; Galois groups would fit perfectly in this representation since > they naturally live inside the symmetric groups, and yes a lot of the > functions in the polys module will be helpful.
> Also, there are generators for common groups like S_n, C_n, D_n, A_n > in the context of permutation representations. All this provides a > nice foundation for defining a Group class or something like that, > with one of the ways of representing it being the permutation > representation. Other ways (e.g., matrices, character tables, list of > generators and relations) could probably be added later, and > functionality to go from one to representation to another?
> In other news, I found a bug inside the generators.py file in the > permutations module - the dihedral group D_2 of order 4 is given a > wrong permutation representation. I have a fix for this (well it's > quite straightforward, just manually considering the case n=2 and > outputting the right representation, because the general algorithm > fails there), what should I do about it?
Submit a pull request! This can be your patch for the patch requirement.
> -- > You received this message because you are subscribed to the Google Groups "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. > For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
> On Sat, Mar 17, 2012 at 2:45 PM, Aleksandar Makelov > <amake...@college.harvard.edu> wrote: >>> I think a main reference is "Permutation Group Algorithms" by Akos >>> Seress - Cambridge Tracts in Mathemathics 152 published 2003. >> Thanks! The "Handbook of computational group theory" also looks like >> serious business. Unfortunately, neither of these is a free resource; >> I might end up buying one, I don't know.
>>> I worked as a student last year and may apply as mentor this year. >>> Please take a look at my branches in github. I was implementing the >>> Schreier Sims algorithm but I ran out of time unfortunately. You could >>> either help me merge my branches in or take off where I left. >> OK, I'll hopefully have the time to take a look this coming week.
>>> Is this necessary? All groups are isomorphic to the permutation group >>> anyway. Groups for specific structures can make use of functionality >>> implemented for them (matrix group -> sympy matrices, galois -> polys) >>> for basic operations and can implement the mapping to the perm group >>> module for group theoretic operations. >> So I looked at the permutations module and it has a lot of nice group- >> ish functions (like composing/inverting permutations, raising to >> powers, conjugating permutations, getting the order (as an element of >> the corresponding symmetric group) of a permutation, ...). These can >> be incorporated in a representation of groups using permutation >> groups; Galois groups would fit perfectly in this representation since >> they naturally live inside the symmetric groups, and yes a lot of the >> functions in the polys module will be helpful.
>> Also, there are generators for common groups like S_n, C_n, D_n, A_n >> in the context of permutation representations. All this provides a >> nice foundation for defining a Group class or something like that, >> with one of the ways of representing it being the permutation >> representation. Other ways (e.g., matrices, character tables, list of >> generators and relations) could probably be added later, and >> functionality to go from one to representation to another?
>> In other news, I found a bug inside the generators.py file in the >> permutations module - the dihedral group D_2 of order 4 is given a >> wrong permutation representation. I have a fix for this (well it's >> quite straightforward, just manually considering the case n=2 and >> outputting the right representation, because the general algorithm >> fails there), what should I do about it? > Submit a pull request! This can be your patch for the patch requirement.
> Aaron Meurer
>> Aleksandar Makelov
>> -- >> You received this message because you are subscribed to the Google Groups "sympy" group. >> To post to this group, send email to sympy@googlegroups.com. >> To unsubscribe from this group, send email to sympy+unsubscribe@googlegroups.com. >> For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
See attached!
survey of computational group theory.pdf
382K
Download
On Sat, Mar 17, 2012 at 8:27 PM, Alan Bromborsky <abro...@verizon.net> wrote: > On 03/17/2012 04:59 PM, Aaron Meurer wrote:
>> On Sat, Mar 17, 2012 at 2:45 PM, Aleksandar Makelov >> <amake...@college.harvard.edu> wrote:
>>>> I think a main reference is "Permutation Group Algorithms" by Akos >>>> Seress - Cambridge Tracts in Mathemathics 152 published 2003.
>>> Thanks! The "Handbook of computational group theory" also looks like >>> serious business. Unfortunately, neither of these is a free resource; >>> I might end up buying one, I don't know.
>>>> I worked as a student last year and may apply as mentor this year. >>>> Please take a look at my branches in github. I was implementing the >>>> Schreier Sims algorithm but I ran out of time unfortunately. You could >>>> either help me merge my branches in or take off where I left.
>>> OK, I'll hopefully have the time to take a look this coming week.
>>>> Is this necessary? All groups are isomorphic to the permutation group >>>> anyway. Groups for specific structures can make use of functionality >>>> implemented for them (matrix group -> sympy matrices, galois -> polys) >>>> for basic operations and can implement the mapping to the perm group >>>> module for group theoretic operations.
>>> So I looked at the permutations module and it has a lot of nice group- >>> ish functions (like composing/inverting permutations, raising to >>> powers, conjugating permutations, getting the order (as an element of >>> the corresponding symmetric group) of a permutation, ...). These can >>> be incorporated in a representation of groups using permutation >>> groups; Galois groups would fit perfectly in this representation since >>> they naturally live inside the symmetric groups, and yes a lot of the >>> functions in the polys module will be helpful.
>>> Also, there are generators for common groups like S_n, C_n, D_n, A_n >>> in the context of permutation representations. All this provides a >>> nice foundation for defining a Group class or something like that, >>> with one of the ways of representing it being the permutation >>> representation. Other ways (e.g., matrices, character tables, list of >>> generators and relations) could probably be added later, and >>> functionality to go from one to representation to another?
>>> In other news, I found a bug inside the generators.py file in the >>> permutations module - the dihedral group D_2 of order 4 is given a >>> wrong permutation representation. I have a fix for this (well it's >>> quite straightforward, just manually considering the case n=2 and >>> outputting the right representation, because the general algorithm >>> fails there), what should I do about it?
>> Submit a pull request! This can be your patch for the patch requirement.
>> Aaron Meurer
>>> Aleksandar Makelov
>>> -- >>> You received this message because you are subscribed to the Google Groups >>> "sympy" group. >>> To post to this group, send email to sympy@googlegroups.com. >>> To unsubscribe from this group, send email to >>> sympy+unsubscribe@googlegroups.com. >>> For more options, visit this group at >>> http://groups.google.com/group/sympy?hl=en.
> See attached!
> -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to > sympy+unsubscribe@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en.
On Sat, Mar 17, 2012 at 10:07 PM, Aaron Meurer <asmeu...@gmail.com> wrote: > Is that a preprint? Some of the sections seem unfinished (for > example, section 10).
> Aaron Meurer
> On Sat, Mar 17, 2012 at 8:27 PM, Alan Bromborsky <abro...@verizon.net> wrote: >> On 03/17/2012 04:59 PM, Aaron Meurer wrote:
>>> On Sat, Mar 17, 2012 at 2:45 PM, Aleksandar Makelov >>> <amake...@college.harvard.edu> wrote:
>>>>> I think a main reference is "Permutation Group Algorithms" by Akos >>>>> Seress - Cambridge Tracts in Mathemathics 152 published 2003.
>>>> Thanks! The "Handbook of computational group theory" also looks like >>>> serious business. Unfortunately, neither of these is a free resource; >>>> I might end up buying one, I don't know.
>>>>> I worked as a student last year and may apply as mentor this year. >>>>> Please take a look at my branches in github. I was implementing the >>>>> Schreier Sims algorithm but I ran out of time unfortunately. You could >>>>> either help me merge my branches in or take off where I left.
>>>> OK, I'll hopefully have the time to take a look this coming week.
>>>>> Is this necessary? All groups are isomorphic to the permutation group >>>>> anyway. Groups for specific structures can make use of functionality >>>>> implemented for them (matrix group -> sympy matrices, galois -> polys) >>>>> for basic operations and can implement the mapping to the perm group >>>>> module for group theoretic operations.
>>>> So I looked at the permutations module and it has a lot of nice group- >>>> ish functions (like composing/inverting permutations, raising to >>>> powers, conjugating permutations, getting the order (as an element of >>>> the corresponding symmetric group) of a permutation, ...). These can >>>> be incorporated in a representation of groups using permutation >>>> groups; Galois groups would fit perfectly in this representation since >>>> they naturally live inside the symmetric groups, and yes a lot of the >>>> functions in the polys module will be helpful.
>>>> Also, there are generators for common groups like S_n, C_n, D_n, A_n >>>> in the context of permutation representations. All this provides a >>>> nice foundation for defining a Group class or something like that, >>>> with one of the ways of representing it being the permutation >>>> representation. Other ways (e.g., matrices, character tables, list of >>>> generators and relations) could probably be added later, and >>>> functionality to go from one to representation to another?
>>>> In other news, I found a bug inside the generators.py file in the >>>> permutations module - the dihedral group D_2 of order 4 is given a >>>> wrong permutation representation. I have a fix for this (well it's >>>> quite straightforward, just manually considering the case n=2 and >>>> outputting the right representation, because the general algorithm >>>> fails there), what should I do about it?
>>> Submit a pull request! This can be your patch for the patch requirement.
>>> Aaron Meurer
>>>> Aleksandar Makelov
>>>> -- >>>> You received this message because you are subscribed to the Google Groups >>>> "sympy" group. >>>> To post to this group, send email to sympy@googlegroups.com. >>>> To unsubscribe from this group, send email to >>>> sympy+unsubscribe@googlegroups.com. >>>> For more options, visit this group at >>>> http://groups.google.com/group/sympy?hl=en.
>> See attached!
>> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To post to this group, send email to sympy@googlegroups.com. >> To unsubscribe from this group, send email to >> sympy+unsubscribe@googlegroups.com. >> For more options, visit this group at >> http://groups.google.com/group/sympy?hl=en.
> I wouldn't trust much from that section anyway, though, since the > paper is from 1998.
> Aaron Meurer
> On Sat, Mar 17, 2012 at 10:07 PM, Aaron Meurer<asmeu...@gmail.com> wrote: >> Is that a preprint? Some of the sections seem unfinished (for >> example, section 10).
>> Aaron Meurer
>> On Sat, Mar 17, 2012 at 8:27 PM, Alan Bromborsky<abro...@verizon.net> wrote: >>> On 03/17/2012 04:59 PM, Aaron Meurer wrote: >>>> On Sat, Mar 17, 2012 at 2:45 PM, Aleksandar Makelov >>>> <amake...@college.harvard.edu> wrote: >>>>>> I think a main reference is "Permutation Group Algorithms" by Akos >>>>>> Seress - Cambridge Tracts in Mathemathics 152 published 2003. >>>>> Thanks! The "Handbook of computational group theory" also looks like >>>>> serious business. Unfortunately, neither of these is a free resource; >>>>> I might end up buying one, I don't know.
>>>>>> I worked as a student last year and may apply as mentor this year. >>>>>> Please take a look at my branches in github. I was implementing the >>>>>> Schreier Sims algorithm but I ran out of time unfortunately. You could >>>>>> either help me merge my branches in or take off where I left. >>>>> OK, I'll hopefully have the time to take a look this coming week.
>>>>>> Is this necessary? All groups are isomorphic to the permutation group >>>>>> anyway. Groups for specific structures can make use of functionality >>>>>> implemented for them (matrix group -> sympy matrices, galois -> polys) >>>>>> for basic operations and can implement the mapping to the perm group >>>>>> module for group theoretic operations. >>>>> So I looked at the permutations module and it has a lot of nice group- >>>>> ish functions (like composing/inverting permutations, raising to >>>>> powers, conjugating permutations, getting the order (as an element of >>>>> the corresponding symmetric group) of a permutation, ...). These can >>>>> be incorporated in a representation of groups using permutation >>>>> groups; Galois groups would fit perfectly in this representation since >>>>> they naturally live inside the symmetric groups, and yes a lot of the >>>>> functions in the polys module will be helpful.
>>>>> Also, there are generators for common groups like S_n, C_n, D_n, A_n >>>>> in the context of permutation representations. All this provides a >>>>> nice foundation for defining a Group class or something like that, >>>>> with one of the ways of representing it being the permutation >>>>> representation. Other ways (e.g., matrices, character tables, list of >>>>> generators and relations) could probably be added later, and >>>>> functionality to go from one to representation to another?
>>>>> In other news, I found a bug inside the generators.py file in the >>>>> permutations module - the dihedral group D_2 of order 4 is given a >>>>> wrong permutation representation. I have a fix for this (well it's >>>>> quite straightforward, just manually considering the case n=2 and >>>>> outputting the right representation, because the general algorithm >>>>> fails there), what should I do about it? >>>> Submit a pull request! This can be your patch for the patch requirement.
>>>> Aaron Meurer
>>>>> Aleksandar Makelov
>>>>> -- >>>>> You received this message because you are subscribed to the Google Groups >>>>> "sympy" group. >>>>> To post to this group, send email to sympy@googlegroups.com. >>>>> To unsubscribe from this group, send email to >>>>> sympy+unsubscribe@googlegroups.com. >>>>> For more options, visit this group at >>>>> http://groups.google.com/group/sympy?hl=en.
>>> See attached!
>>> -- >>> You received this message because you are subscribed to the Google Groups >>> "sympy" group. >>> To post to this group, send email to sympy@googlegroups.com. >>> To unsubscribe from this group, send email to >>> sympy+unsubscribe@googlegroups.com. >>> For more options, visit this group at >>> http://groups.google.com/group/sympy?hl=en.
Link to download of - Handbook of Computational Group Theory
> > I wouldn't trust much from that section anyway, though, since the
> > paper is from 1998.
> > Aaron Meurer
> > On Sat, Mar 17, 2012 at 10:07 PM, Aaron Meurer<asmeu...@gmail.com> wrote:
> >> Is that a preprint? Some of the sections seem unfinished (for
> >> example, section 10).
> >> Aaron Meurer
> >> On Sat, Mar 17, 2012 at 8:27 PM, Alan Bromborsky<abro...@verizon.net> wrote:
> >>> On 03/17/2012 04:59 PM, Aaron Meurer wrote:
> >>>> On Sat, Mar 17, 2012 at 2:45 PM, Aleksandar Makelov
> >>>> <amake...@college.harvard.edu> wrote:
> >>>>>> I think a main reference is "Permutation Group Algorithms" by Akos
> >>>>>> Seress - Cambridge Tracts in Mathemathics 152 published 2003.
> >>>>> Thanks! The "Handbook of computational group theory" also looks like
> >>>>> serious business. Unfortunately, neither of these is a free resource;
> >>>>> I might end up buying one, I don't know.
> >>>>>> I worked as a student last year and may apply as mentor this year.
> >>>>>> Please take a look at my branches in github. I was implementing the
> >>>>>> Schreier Sims algorithm but I ran out of time unfortunately. You could
> >>>>>> either help me merge my branches in or take off where I left.
> >>>>> OK, I'll hopefully have the time to take a look this coming week.
> >>>>>> Is this necessary? All groups are isomorphic to the permutation group
> >>>>>> anyway. Groups for specific structures can make use of functionality
> >>>>>> implemented for them (matrix group -> sympy matrices, galois -> polys)
> >>>>>> for basic operations and can implement the mapping to the perm group
> >>>>>> module for group theoretic operations.
> >>>>> So I looked at the permutations module and it has a lot of nice group-
> >>>>> ish functions (like composing/inverting permutations, raising to
> >>>>> powers, conjugating permutations, getting the order (as an element of
> >>>>> the corresponding symmetric group) of a permutation, ...). These can
> >>>>> be incorporated in a representation of groups using permutation
> >>>>> groups; Galois groups would fit perfectly in this representation since
> >>>>> they naturally live inside the symmetric groups, and yes a lot of the
> >>>>> functions in the polys module will be helpful.
> >>>>> Also, there are generators for common groups like S_n, C_n, D_n, A_n
> >>>>> in the context of permutation representations. All this provides a
> >>>>> nice foundation for defining a Group class or something like that,
> >>>>> with one of the ways of representing it being the permutation
> >>>>> representation. Other ways (e.g., matrices, character tables, list of
> >>>>> generators and relations) could probably be added later, and
> >>>>> functionality to go from one to representation to another?
> >>>>> In other news, I found a bug inside the generators.py file in the
> >>>>> permutations module - the dihedral group D_2 of order 4 is given a
> >>>>> wrong permutation representation. I have a fix for this (well it's
> >>>>> quite straightforward, just manually considering the case n=2 and
> >>>>> outputting the right representation, because the general algorithm
> >>>>> fails there), what should I do about it?
> >>>> Submit a pull request! This can be your patch for the patch requirement.
> >>>> Aaron Meurer
> >>>>> Aleksandar Makelov
> >>>>> --
> >>>>> You received this message because you are subscribed to the Google Groups
> >>>>> "sympy" group.
> >>>>> To post to this group, send email to sympy@googlegroups.com.
> >>>>> To unsubscribe from this group, send email to
> >>>>> sympy+unsubscribe@googlegroups.com.
> >>>>> For more options, visit this group at
> >>>>>http://groups.google.com/group/sympy?hl=en.
> >>> See attached!
> >>> --
> >>> You received this message because you are subscribed to the Google Groups
> >>> "sympy" group.
> >>> To post to this group, send email to sympy@googlegroups.com.
> >>> To unsubscribe from this group, send email to
> >>> sympy+unsubscribe@googlegroups.com.
> >>> For more options, visit this group at
> >>>http://groups.google.com/group/sympy?hl=en.
> Link to download of - Handbook of Computational Group Theory
On Saturday, March 17, 2012 2:57:57 PM UTC-5, Aaron Meurer wrote:
> On Sat, Mar 17, 2012 at 1:42 PM, Saptarshi Mandal wrote:
> >> And it would be awesome to have a group theory module. We presently > >> only have a Permutation class in the combinatorics module, but other > >> than that, we don't really have a good way to represent a group.
> > Is this necessary? All groups are isomorphic to the permutation group > > anyway. Groups for specific structures can make use of functionality > > implemented for them (matrix group -> sympy matrices, galois -> polys) > > for basic operations and can implement the mapping to the perm group > > module for group theoretic operations.
> As Stefan noted, this is only true for finite groups.
> And anyway, the point I was trying to make was that a permutation > represents an element of a group, whereas I was talking about a way to > represent the whole group.
> Aaron Meurer
I was going to put group theory as a GSoC idea but you beat me to the punch :)
What I think would be cool is to have symbolic group elements. For example,
>>> x = GroupSymbol('x', order=3) >>> x**3 1 >>> x**-1
x**2
Then we can do neat things like create groups from arbitrary generators and assumptions!
> On Saturday, March 17, 2012 2:57:57 PM UTC-5, Aaron Meurer wrote:
> On Sat, Mar 17, 2012 at 1:42 PM, Saptarshi Mandal wrote:
> >> And it would be awesome to have a group theory module. We > presently > >> only have a Permutation class in the combinatorics module, but > other > >> than that, we don't really have a good way to represent a group.
> > Is this necessary? All groups are isomorphic to the permutation > group > > anyway. Groups for specific structures can make use of functionality > > implemented for them (matrix group -> sympy matrices, galois -> > polys) > > for basic operations and can implement the mapping to the perm group > > module for group theoretic operations.
> As Stefan noted, this is only true for finite groups.
> And anyway, the point I was trying to make was that a permutation > represents an element of a group, whereas I was talking about a way to > represent the whole group.
> Aaron Meurer
> I was going to put group theory as a GSoC idea but you beat me to the > punch :)
> What I think would be cool is to have symbolic group elements. For > example, > >>> x = GroupSymbol('x', order=3) > >>> x**3 > 1 > >>> x**-1 > x**2
> Then we can do neat things like create groups from arbitrary > generators and assumptions! > -- > You received this message because you are subscribed to the Google > Groups "sympy" group. > To view this discussion on the web visit > https://groups.google.com/d/msg/sympy/-/8BKF_m6cyR4J. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to > sympy+unsubscribe@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en.
If you are interested in Lie Groups you might find the attached paper interesting.
Lie groups as spin groups - Doran, Hestenes, Sommen, Van Acker - 1993.pdf
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