Tutorial

atlas choice of coordinates

How do we interpret the way the software writes the simple roots for \(SL(n,\mathbb R)\)? Let us look at an example:

atlas> set G= SL(5,R)
Identifier G: RealForm (hiding previous one of type RealForm)
atlas> simple_roots(G)
Value:
| 1, 0, 0, 1 |
| -1, 1, 0, 1 |
| 0, -1, 1, 1 |
| 0, 0, -1, 2 |
atlas> simple_coroots(G)
Value:
| 1, 0, 0, 0 |
| -1, 1, 0, 0 |
| 0, -1, 1, 0 |
| 0, 0, -1, 1 |
atlas> rho(G)
Value: [ 4, 3, 2, 1 ]/1
atlas>sum(simple_roots(G))
Value: [ 2, 1, 1, 1 ]
atlas>

So, atlas chooses a set of coordinates to work with. They were chosen in the roots.at file so that the matrix of the simple coroots for the simply connected group is the identity matrix:

atlas> set g=LieType: A4
Identifier g: LieType (hiding previous one of type LieType)
atlas> set rd=simply_connected(A4)
Identifier rd: RootDatum (hiding previous one of type RootDatum)
atlas> rd
Value: simply connected root datum of Lie type 'A4'

atlas> simple_coroots (rd)
Value:
| 1, 0, 0, 0 |
| 0, 1, 0, 0 |
| 0, 0, 1, 0 |
| 0, 0, 0, 1 |

atlas>

But look what happens when we type:

atlas> G:=SU(5)
Value: compact connected real group with Lie algebra 'su(5)'
atlas> simple_coroots (G)
Value:
|  1,  0,  0, 0 |
| -1,  1,  0, 0 |
|  0, -1,  1, 0 |
|  0,  0, -1, 1 |

atlas>

If you want to ask atlas about a vector in the Cartan subgroupor say an infinitesimal character, you need to write it in terms of the simple roots. Then the software will give you the vector in terms of the atlas coordinates. You then have to understand which coordinates the software is using in order to both, ask the question and interpret the answer. For example, it is useful to know what rho is in the coordinates the software is using:

atlas> rho(rd)
Value: [ 1, 1, 1, 1 ]/1
atlas> rho(G)
Value: [ 4, 3, 2, 1 ]/1

atlas> sum(simple_roots (G))
Value: [ 2, 1, 1, 1 ]
atlas>sum(simple_roots (rd))
Value: [ 1, 0, 0, 1 ]
atlas>

Note the expression in the case when \(G=SL(5,\mathbb R)\). atlas is using only four coordinates when it is costumary to use five coordinates to describe the weights in \(G\).

Alternatively, you can try to phrase the question in a way that atlas will use coordinates you are familiar with:

atlas> set G= GL(5,R)
Identifier G: RealForm (hiding previous one of type RealForm)
atlas> simple_coroots(G)
Value:
| 1, 0, 0, 0 |
| -1, 1, 0, 0 |
| 0, -1, 1, 0 |
| 0, 0, -1, 1 |
| 0, 0, 0, -1 |

atlas> rho(G)
Value: [ 2, 1, 0, -1, -2 ]/1

Remark: Once you defined a root datum or group, atlas fixes some coordinates. However, as we have seen, it is often possible to redefine the group in a different way so that the coordinates are easier to work with:

atlas> set rd=root_datum ([[6,2]],[[1,-2]])
Identifier rd: RootDatum (hiding previous one of type RootDatum)
atlas> simple_roots(rd)
Value:
| 6 |
| 2 |
atlas> simple_coroots(rd)
Value:
| 1 |
| -2 |
atlas> rho (rd)
Value: [ 3, 1 ]/1
atlas> ^simple_roots(rd)*simple_coroots(rd)
Value:
| 2 |
atlas> rd
Value: simply connected root datum of Lie type 'A1.T1'
atlas>

So this is a root datum of the above Lie type and with some ridiculous coordinates. It is not clear which group it is. There are three isomorphism classes of root data of this type. But once we recognize the Lie Type we can redefine it so that it looks nicer:

atlas> set rd_new=root_datum ([[1,-1]],[[1,-1]])
Identifier rd_new: RootDatum
atlas> rd_new
Value: simply connected adjoint root datum of Lie type 'A1.T1'
atlas>

Notice this is a version of the root_datum command that we had not seen. It asks for the root datum for the given set of simple roots and coroots. That is what a root datum is in atlas. So you can define the root datum by giving the matrices you want for the simple roots and coroots and atlas will accept them as a root datum.

Now lets look at a another example:

 atlas> set rd=simply_connected (C4)
 Identifier rd: RootDatum (hiding previous one of type RootDatum)
 atlas> simple_roots (rd)
 Value:
 |  2, -1,  0,  0 |
 | -1,  2, -1,  0 |
 |  0, -1,  2, -2 |
|  0,  0, -1,  2 |

 atlas> simple_coroots (rd)
 Value:
 | 1, 0, 0, 0 |
 | 0, 1, 0, 0 |
 | 0, 0, 1, 0 |
 | 0, 0, 0, 1 |

 atlas> ^simple_roots (rd)*simple_coroots (rd)
 Value:
 |  2, -1,  0,  0 |
 | -1,  2, -1,  0 |
 |  0, -1,  2, -1 |
 |  0,  0, -2,  2 |

 atlas>

Again these are not the usual simple roots and corroots. But as you can see we get the Cartan matrix with the above product. These are the fundamental weight coordinates. Observe also that the simple coroots (resp. simple roots) give the identity matrix (resp. the Cartan matrix), which you would expect for the simply connected group of type C4.

In these corrdinates rho is:

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

So, in fundamental weight coordinates, the coordinates of rho are all 1. You can also check that if you use the adjoint root datum for C4, the simple roots matrix will be the identity etc.

But now, if we use the defined real form \(Sp(8,\mathbb R)\), we get root data in the usual coordinates:

atlas> G:=Sp(8,R)
Value: connected split real group with Lie algebra 'sp(8,R)'
atlas> simple_roots (G)
Value:
|  1,  0,  0, 0 |
| -1,  1,  0, 0 |
|  0, -1,  1, 0 |
|  0,  0, -1, 2 |

atlas> rho(G)
Value: [ 4, 3, 2, 1 ]/1
atlas>

These are isomorphic root data. They are equal up to a change of coordinates. We just need to be aware of which coordinates atlas is using.