On 2012-08-20, bsmile <[email protected]> wrote: > ------=_Part_1_15301451.1345493498755 > Content-Type: text/plain; charset=ISO-8859-1 > > > > On Monday, August 20, 2012 1:41:42 PM UTC-5, John H Palmieri wrote: >> >> >> >> On Monday, August 20, 2012 11:21:04 AM UTC-7, bsmile wrote: >>> >>> >>> >>> On Monday, August 20, 2012 12:56:07 PM UTC-5, John H Palmieri wrote: >>>> >>>> >>>> >>>> I don't know much about this aspect of Sage, but you can also try this: >>>> >>>> sage: rep = SymmetricGroupRepresentation([3,1]) >>>> >>>> Now rep is the representation of S_4 (since 4=3+1) corresponding to the >>>> partition [3,1]. We can compute find the matrix associated to each element >>>> of S_4: >>>> >>>> sage: rep([1,4,3,2]) >>>> [ 1 0 0] >>>> [ 1 -1 1] >>>> [ 0 0 1] >>>> sage: REPS = [(x,rep(x)) for x in S4.list()] >>>> >>>> This defines REPS to be a list of pairs, (elt of S_4, corresponding >>>> matrix). >>>> >>>> Is that the sort of thing you want? >>>> >>> >>> Thanks for your response. It's close, but not quite. It seems the way you >>> pointed out does give a 3D irreducible matrix representation for S4. But >>> how can I get the 2D irreducible matrix representation for S4? >>> >> >> Use a different partition. Every partition of 4 will give an irreducible >> representation. You can get each of the representations like this: >> >> sage: [SymmetricGroupRepresentation(p) for p in Partitions(4).list()] >> > > Thanks, this exactly gives what I am trying to calculate. Summarizing your > input and I learned to write the following, > > sage: S4=SymmetricGroup(4) > sage: S4.character_table() > sage: rep=SymmetricGroupRepresentation([2,2]) > sage: REPS=[(p,rep(p)) for p in S4] > sage: REPS > > It seems REPS now saves an irreducible 2D matrix representation for S4. > > Can I ask further, > > (1) what's the advantages towards each of the three options, "specht", > "orthogonal", "seminormal" for SymmetricGroupRepresentation? It seems > "orthogonal" gives something matching the physically meaningful Td group > result. well, I suppose people here don't know about Td group... The "orthogonal" gives you, as the name suggests, orthogonal matrices. (which have a disadvantage that in some cases you need square roots of rational numbers, whereas for "seminormal" rational numbers suffice)
"specht" is important in representation theory of S_n for some other reasons. > (2) How can I output to file so that fortran can directly read in the > matrices for subsequent calculation? Sage is just a (huge :-)) Python library. You can write Python code to do whatever you need. You can even call Fortran code directly, without leaving Sage. > (3) Is there a nice way to print the output on the screen so that the > matrix is clearly seen? It seems REPS=[rep(p) for p in S4] would do pretty > good, but I will lose track of which element corresponds to which matrix. > (4) Is there command in SAGE so that I can use to check that the REPS > really gives a representation for the S4 group? sage: rep.verify_representation() will do this. You can see what it does by doing sage: rep.verify_representation?? HTH, Dmitrii > > Thanks a lot! > > Sincerely, > Jon > >> >> -- >> John >> >> > > -- -- -- To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
