Tutorial

Change of Coordinates

For this and following sessions you want to download the files basic.at, groups.at and lietypes.at. Or, if you prefer, the file all.at to have most of the commands and scripts you will need.

To change atlas coordinates for roots and other weights to your prefered coordinate system you can also download the file coordinates.at (included in the all.at file) and do the following:

atlas> set rd=simply_connected(C2)
Identifier rd: RootDatum
atlas> simple_roots(rd)
Value:
| 2, -2 |
| -1, 2 |

atlas> set A=mat:[[1,-1],[0,2]]
Identifier A: mat atlas> A
Value:
| 1, 0 |
| -1, 2 |

atlas> set C=change_basis_integral (rd,A)
  Identifier C: mat (hiding previous one of type string (constant))
atlas> C
Value:
| 1, 1 |
| 0, 1 |

atlas> set sr=simple_roots (rd)
Identifier sr: mat
atlas> sr
Value:
| 2, -2 |
|-1, 2 |

atlas> C*sr
Value:
| 1, 0 |
| -1, 2 |

atlas> C*sr=A
Value:true
atlas> rho(rd)
Value:
[ 1, 1 ]/1

atlas> C*rho(rd)
Value:
[ 2, 1]/1

atlas>

So, C is the change of basis matrix and you can change, in the usual way, any vector like rho written in the original basis, into one expressed in terms of the new basis using the matrix C.

Now, for this example this was not so necessary since we can use the real form expression of the group:

atlas> simple_roots(Sp(4,R))
Value:
| 1, 0 |
| -1, 2 |
atlas>

However, the change of basis matrix is needed for example for \(SL(3,\mathbb R)\). Moreover, in this case we do not get integral matrices. So we need a more general command:

 atlas> set G=SL(3,R)
 Identifier G: RealForm (hiding previous one of type RealForm)
 atlas> sr:=simple_roots (G)
 Value:
 | 1, 1 |
 | -1, 2 |

 atlas> A:=[[1,-1,0],[0,1,-1]]
 Value:
 | 1, 0 |
 | -1, 1 |
 | 0, -1 |

 atlas> set C=change_basis (G,A)
 Identifier C: [ratvec] (hiding previous one of type mat)
 atlas> C
 Value:
 [[ 2, -1, -1 ]/3,[ -1, 2, -1 ]/3]

atlas> C*sr
 Value:
 [[ 1, -1, 0 ]/1,[ 0, 1, -1 ]/1]
 atlas> make_integral(C*sr)
 Value:
 | 1, 0 |
 | -1, 1 |
 | 0, -1 |

 atlas> C*rho(G)
 Value:
 [ 1, 0, -1]/1

 atlas>

Note the use of the new command:make_integral. Since the end result is an integral matrix, atlas can re-write it as such.

Now to translate things back to atlas coordinates we use the inverse change of coordinates matrix:

atlas> set D=inverse_change_basis(SL(3,R),A)
Identifier D:[ratvec]
atlas> D
Value:
[[ 1, 0 ]/1,[ 0, 1 ]/1,[ -1, -1 ]/1]

atlas> D*C
Value:
[[ 1, 0 ]/1,[ 0, 1 ]/1]

atlas> make_integral(D*C)
Value:
| 1, 0 |
| 0, 1 |

atlas>

For more information about this look at the coordinates.at file and watch the supplemental video on coordinates on the atlas website.