> (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 -~----------~----~----~----~------~----~------~--~---
