That is *very* impressive! John
2008/6/1 Daniel Bump <[EMAIL PROTECTED]>: > > >> (hopefully with help from John Voight), and "Lie Algebras/Algebraic >> Groups" as a new package. For this last one I know that there are >> several freely available packages (e.g. LIE), but I'm not sure if they >> are actively maintained. > > Lie Algebras/Algebraic groups as a new package ... many of the things > LiE does can now be done natively in Sage. > > I'll take this as a cue to advertise the fact that Sage now (as of > 3.0.2) has nontrivial capability for Lie group/Lie algebra computations > including computation of Weyl characters, weight multiplicities, tensor > products and branching rules for characters, conjugation of roots and > weights by Weyl group elements. > > For example, we can create the spin representation of Spin(7): > > sage: B3=WeylCharacterRing(['B',3]) > sage: spin=B3(B3.lattice().fundamental_weights()[2]); spin > B3(1/2,1/2,1/2) > > Tensor it with itself and see how that decomposes into > irreducibles: > > sage: spin*spin > B3(0,0,0) + B3(1,0,0) + B3(1,1,0) + B3(1,1,1) > > Get the decomposition of that into weights: > > sage: b3=WeightRing(B3) > sage: b3(spin*spin) > b3(-1,-1,-1) + 2*b3(-1,-1,0) + b3(-1,-1,1) + 2*b3(-1,0,-1) + > 4*b3(-1,0,0) + 2*b3(-1,0,1) + b3(-1,1,-1) + 2*b3(-1,1,0) + b3(-1,1,1) + > 2*b3(0,-1,-1) + 4*b3(0,-1,0) + 2*b3(0,-1,1) + 4*b3(0,0,-1) + > 8*b3(0,0,0) + 4*b3(0,0,1) + 2*b3(0,1,-1) + 4*b3(0,1,0) + 2*b3(0,1,1) + > b3(1,-1,-1) + 2*b3(1,-1,0) + b3(1,-1,1) + 2*b3(1,0,-1) + 4*b3(1,0,0) + > 2*b3(1,0,1) + b3(1,1,-1) + 2*b3(1,1,0) + b3(1,1,1) > > Restrict it to GL(3) embedded as a Levi subgroup in Spin(7): > > sage: (spin*spin).branch(A2,rule="levi") > A2(-1,-1,-1) + 2*A2(0,-1,-1) + 3*A2(0,0,-1) + 4*A2(0,0,0) + A2(1,-1,-1) > + 2*A2(1,0,-1) + 3*A2(1,0,0) + A2(1,1,-1) + 2*A2(1,1,0) + A2(1,1,1) > > Conjugate weights around using the Weyl group: > > sage: W = B3.lattice().weyl_group() > sage: [s1,s2,s3]=W.simple_reflections() > sage: s1*s2*s3 > > [ 0 0 -1] > [ 1 0 0] > [ 0 1 0] > sage: b3(1/2,1/2,1/2).weyl_group_action(s1*s2*s3) > b3(-1/2,1/2,1/2) > > Dan > > > > --~--~---------~--~----~------------~-------~--~----~ 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://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
