1 | initial version |

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[1], 2: Lambda[2]}
sage: SR
Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2]}
```

2 | No.2 Revision |

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[1], 2: Lambda[2]}
sage: SR
Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2]}
```

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[0].action(FW[1])
-Lambda[1] + Lambda[2]
```

3 | No.3 Revision |

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[1], 2: Lambda[2]}
sage: SR
Finite family {1: 2*Lambda[1] - Lambda[2], 2: -Lambda[1] + 2*Lambda[2]}
```

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[0].action(FW[1])
-Lambda[1] + Lambda[2]
```

EDIT: with the same notations

```
sage: v=s[0].action(SR[1])
sage: v.to_ambient().is_positive_root()
False
sage: v=s[0].action(SR[2])
sage: v.to_ambient().is_positive_root()
True
sage: v=s[1].action(SR[2])
sage: v.to_ambient().is_positive_root()
False
sage: v=s[1].action(SR[1])
sage: v.to_ambient().is_positive_root()
True
```

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