# Revision history [back]

Yes, this looks strange indeed. You can proceed like that instead:

sage: R = RootSystem(['A', 2])
sage: WL = R.weight_lattice()
sage: FW = R.weight_lattice().basis()
sage: SR = WL.simple_roots()
sage: FW
Finite family {1: Lambda, 2: Lambda}
sage: SR
Finite family {1: 2*Lambda - Lambda, 2: -Lambda + 2*Lambda}


Yes, this looks strange indeed. You can proceed like that instead:

sage: R = RootSystem(['A', 2])
sage: WL = R.weight_lattice()
sage: FW = R.weight_lattice().basis()
sage: SR = WL.simple_roots()
sage: FW
Finite family {1: Lambda, 2: Lambda}
sage: SR
Finite family {1: 2*Lambda - Lambda, 2: -Lambda + 2*Lambda}


EDIT: here is how to get the Weyl group acting on the weight lattice

sage: W = WL.weyl_group()
sage: s = W.gens()
sage: s.action(FW)
-Lambda + Lambda


Yes, this looks strange indeed. You can proceed like that instead:

sage: R = RootSystem(['A', 2])
sage: WL = R.weight_lattice()
sage: FW = R.weight_lattice().basis()
sage: SR = WL.simple_roots()
sage: FW
Finite family {1: Lambda, 2: Lambda}
sage: SR
Finite family {1: 2*Lambda - Lambda, 2: -Lambda + 2*Lambda}


EDIT: here is how to get the Weyl group acting on the weight lattice

sage: W = WL.weyl_group()
sage: s = W.gens()
sage: s.action(FW)
-Lambda + Lambda


EDIT: with the same notations

sage: v=s.action(SR)
sage: v.to_ambient().is_positive_root()
False
sage: v=s.action(SR)
sage: v.to_ambient().is_positive_root()
True
sage: v=s.action(SR)
sage: v.to_ambient().is_positive_root()
False
sage: v=s.action(SR)
sage: v.to_ambient().is_positive_root()
True