Atlas software: user guide

Jeffrey Adams

July 11, 2006

1 Introduction

This is intended as an quick introduction to the Atlas software. It should give prospective users enough information to start using the software, and lots of examples and practical tips. For more on the mathematical background see [?]. The online documentation is also quite helpful, and is included in the appendix.

This describes the current state of the software. This will be substantially changed by the time of Version 1.0.

2 Obtaining and installing the software

…

3 Defining an Inner class of real forms

The basic object we can define is a complex reductive group, together with a choice of inner class. The latter is an element γ ∈ Out(G) of order 2. This is accomplished by the type command.

This proceeds in three steps:

(1) Define the complex Lie algebra of G, or equivalently a group G^{∗} which is a product of a torus and a simply connected semisimple group;

(2) Define a finite subgroup of Z(G^{∗})

(3) Choose γ

In response to the type command, the software asks for the Lie type:. A basic entry is of the form Xn where X=A,B,…, E,T and n is a positive integer. A general entry is a sequence of basic entries, separated by “.”. Examples:

• A1

• A1.B2.C3.D4.E6.F4.G2

• T1.A1.A1.G2.T1.E6.T3

Order does not matter, except that pairs corresponding to a complex group must be consecutive. Note that T2 is the same as T1.T1, except that you must use the latter to make a complex torus.

This completes step 1. The software then asks for a finite subgroup of the center. (If the center

is trivial this step is skipped.) The finite cyclic group of order n is denoted Z/n, and is viewed as the group

\frac{1}{n}ℤ/ℤ = \{\frac{0}{n}, \frac{1}{n}, …, \frac{n − 1}{n}\}

The elements of finite order in S^{1} are isomorphic to ℚ/ℤ (via the map a/b → e^{2πia/b}). An element of this group is given by an element of ℚ.

To specify a finite subgroup of Z the user gives a list of elements of Z of finite order. Each such element is given as a list of fractions, separated by commas. See the preceding examples.

This completes step 2. Finally the software asks the user for an inner class (even if there is a

unique such class). For each simple or torus factor the user can specify:

• c: compact

• s: split

• e: equal rank

• u: unequal rank

For each consecutive pair of identical entries the user can also specify:

• C: complex

Type u is only allowed in type A_{n} (n > 1), D_{n} and E_{6}.

3.1 Examples

Define SL(2) together with its (unique) inner class:

empty: type
Lie type: A1
elements of finite order in the center of the simply connected group:
Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
sc
enter inner class(es): s

This is the simply connected group of type A_{1}, and the inner class of the split real form (there is only one inner class). Here’s how to make the adjoint group:

empty: type
Lie type: A1
elements of finite order in the center of the simply connected group:
Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
ad
enter inner class(es): s

Instead we could specify the generator 1/2 of ℤ/2ℤ = {0/1, 1/2}:

main: type
Lie type: A1
elements of finite order in the center of the simply connected group:
Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2
enter inner class(es): s
main:

Now let’s make the inner class of ℝ^{×} × SL(2, ℝ):

main: type
Lie type: T1.A1
elements of finite order in the center of the simply connected group:
Q/Z.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
enter inner class(es): ss

Now let’s make the inner class of GL(2, ℝ). Note that

(3.1)(a)  GL(2, ℂ) = ℂ^{×}SL(2, ℂ)

where ℂ^{×} is on the diagonal. The intersection of ℂ^{×} and SL(2, ℂ) is ±1. The elements of finite order in ℂ^{×} are isomorphic to ℚ/ℤ, and −1 corresponds to the element 1/2. The center of SL(2, ℂ) is ℤ/2ℤ = \{0/2, 1/2\}, so −1 = \frac{1}{2}. The kernel of the map ℂ^{×} ×SL(2, ℂ) ↠ GL(2, ℂ) is the cyclic group of order 2 generated by (1/2, 1/2). So:

main: type
Lie type: T1.A1
elements of finite order in the center of the simply connected group:
Q/Z.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2,1/2
enter inner class(es): ss
main:

The construction of the inner class of U(1, 1) is similar:

main: type
Lie type: T1.A1
elements of finite order in the center of the simply connected group:
Q/Z.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2,1/2
enter inner class(es): cs

We could use cc also for the last entry.

Now we’ll define the inner class of the real group SL(2, ℂ). The type is A1.A1 and the inner class is C for Complex:

empty: tpe
tpe : not found
empty: type
Lie type: A1.A1
elements of finite order in the center of the simply connected group:
Z/2.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
enter inner class(es): C

If the center is trivial the software will not ask about it:

main: type
Lie type: E8
enter inner class(es): s

There is substantial overlap between the types s,c,e and u. In particular u is only needed to define the inner class of SO(n + 1, n − 1) with n even. For example:

main: type
Lie type: D4
elements of finite order in the center of the simply connected group:
Z/2.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
enter inner class(es): u
main: realform
(weak) real forms are:
0: so(7,1)
1: so(5,3)

Note that the quasisplit group SO(5, 3) is not split, nor is it inner to a split group.

Here is SO(2n):

main: type
Lie type: D4
elements of finite order in the center of the simply connected group:
Z/2.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2,1/2
enter inner class(es): c

Tip: You can specify options sc and ad, and the inner class, on a single line:

real: type
Lie type: A1 sc s
main: type
Lie type: A1 ad c
main:

4 Defining a real group

Suppose we have defined a complex group G, together with an inner class of real forms. Defining a real form of G is not simply a question of choosing from a list.

Lie type: A1
elements of finite order in the center of the simply connected group:
Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
sc
enter inner class(es): s
main: realform
(weak) real forms are:
0: su(2)
1: sl(2,R)
enter your choice: 1
real:

The list is a list of the real forms of the complex Lie algebra. For example in type B_{n} the choices are 𝔰𝔬(p, q) with p + q = n. This is independent of isogeny, so will serve to choose Spin(p, q) or SO(p, q).

real: type
Lie type: B3 sc s
main: realform
(weak) real forms are:
0: so(7)
1: so(6,1)
2: so(5,2)
3: so(4,3)
enter your choice: 3
real:

The command showrealforms will give you a list:

main: type
Lie type: E8 sc s
main: showrealforms
(weak) real forms are:
0: e8
1: e8(e7.su(2))
2: e8(R)

You can also obtain a list of the real forms of the dual group:

main: type
Lie type: A2 sc s
main: showrealforms
(weak) real forms are:
0: sl(3,R)
main: showdualforms
(weak) dual real forms are:
0: su(3)
1: su(2,1)

Note that specifying an inner class of real forms of G specifies an inner class of real forms of G^{∨}; even if the root system is self dual these might not be the same.

Once we have specified a real group, there is a lot more structure theory we can ask about, including Cartan subgroups and Weyl groups. See the next section. One small bit of structure theory which is quite useful is the component group of G.

For example the split real form SL(2, ℝ) of SL(2, ℂ) is connected, while the split real form of the adjoint group P SL(2, ℂ), i.e. P SL(2, ℝ) ≃ SO(2, 1), is not:

Lie type: A1 sc s
main: realform
(weak) real forms are:
0: su(2)
1: sl(2,R)
enter your choice: 1
real: components
group is connected
real: type
Lie type: A1 ad s
main: realform
(weak) real forms are:
0: su(2)
1: sl(2,R)
enter your choice: 1
real: components
component group is (Z/2)^1

This is helpful for determining when groups are not isomorphic, as in the next example.

There is one subtle point about the definition of real forms, which arises only in type D_{2n}. Our notion of equivalence of real forms only allows action of Int(G) not all of Out(G). There are two real forms of SO(2n, ℂ) which are locally isomorphic to SO^{∗}(2n). Their behavior under isogenies is very subtle. See [?].

To distinguish these two groups they are denoted so*(2n)[1,0] and

so*(2n)[0,1] by the realforms command.

For example, here is an interesting example of a funny quotient of Spin(2n). There are two groups locally isomorphic to SO^{∗}(2n), which are not isomorphic.

enter inner class(es): c
main: realform
(weak) real forms are:
0: so(12)empty: type
1: so(10,2)
2: so*(12)[1,0]
3: so*(12)[0,1]
4: so(8,4)
5: so(6,6)
enter your choice: 2
real: components
component group is (Z/2)^1

This group is not connected. However

real: realform
(weak) real forms are:
0: so(12)
1: so(10,2)
2: so*(12)[1,0]
3: so*(12)[0,1]
4: so(8,4)
5: so(6,6)
enter your choice: 3
real: components
group is connected
real:

The situation is even more confusing in D_{4}. There are three real forms of the complex Lie algebra, labelled SO(6, 2), SO^{∗}(8)[1, 0] and SO^{∗}(8)[0, 1]. In Spin(8) these are all interchanged by graph automorphisms, and are isomorphic. However in SO(8), SO(6, 2) is not isomorphic to SO^{∗}(8):

real: type
Lie type: D4
elements of finite order in the center of the simply connected group:

Z/2.Z/2
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/2,1/2
enter inner class(es): c
main: realform
(weak) real forms are:
0: so(8)
1: so(6,2)
2: so*(8)[0,1]
3: so*(8)[1,0]
4: so(4,4)
enter your choice: 1
real: components
component group is (Z/2)^1
real: realform
(weak) real forms are:
0: so(8)
1: so(6,2)
2: so*(8)[0,1]
3: so*(8)[1,0]
4: so(4,4)
enter your choice: 2
real: components
group is connected
real: realform
(weak) real forms are:
0: so(8)
1: so(6,2)
2: so*(8)[0,1]
3: so*(8)[1,0]
4: so(4,4)
enter your choice: 3
real: components
group is connected

5 Structure Theory of G: Cartan subgroups

and Weyl Groups 6 K orbits on the flag variety and the space

X 7 Representation Theory: Blocks 8 Kazhdan-Lusztig Polynomials 9 Appendix: Online documentation References

[1] J. Adams and F. du Cloux. Algorithms for representation theory of real reductive groups I. preprint.