Atlas of Lie Groups and Representations

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Example \(G=SO(3,2)\)

Let’s study the minimal principal series for this group

atlas> G:SO(3,2)
Variable G: RealForm (overriding previous instance, which had type RealForm)
atlas> set parameters=all_parameters_gamma (G, rho(G))
Variable parameters: [Param] (overriding previous instance, which had type [Param])
atlas> rho(G)
Value: [ 3, 1 ]/2
atlas> #parameters
Value: 12
atlas> void: for p in parameters do prints(p) od
final parameter (x=0,lambda=[3,1]/2,nu=[0,0]/1)
final parameter (x=1,lambda=[3,1]/2,nu=[0,0]/1)
final parameter (x=2,lambda=[3,1]/2,nu=[1,-1]/2)
final parameter (x=3,lambda=[3,1]/2,nu=[0,1]/2)
final parameter (x=3,lambda=[3,3]/2,nu=[0,1]/2)
final parameter (x=4,lambda=[3,1]/2,nu=[3,0]/2)
final parameter (x=4,lambda=[5,1]/2,nu=[3,0]/2)
final parameter (x=5,lambda=[3,1]/2,nu=[1,1]/1)
final parameter (x=6,lambda=[3,1]/2,nu=[3,1]/2)
final parameter (x=6,lambda=[5,1]/2,nu=[3,1]/2)
final parameter (x=6,lambda=[3,3]/2,nu=[3,1]/2)
final parameter (x=6,lambda=[5,3]/2,nu=[3,1]/2)
atlas>

We are looking only at the minimal principal series. So we are for the moment only interested in the last four representations corresponding to the KGB element x=6.

Note that here we can also just use the command all_minimal_principal_series:

atlas> ps:=all_minimal_principal_series (G,rho(G))
Value: [final parameter(x=6,lambda=[3,1]/2,nu=[3,1]/2),final parameter(x=6,lambda=[5,1]/2,nu=[3,1]/2),final parameter(x=6,lambda=[3,3]/2,nu=[3,1]/2),final parameter(x=6,lambda=[5,3]/2,nu=[3,1]/2)]
atlas>

And to write them one line at a time we do:

atlas> void: for p in ps do prints(p) od
final parameter(x=6,lambda=[3,1]/2,nu=[3,1]/2)
final parameter(x=6,lambda=[5,1]/2,nu=[3,1]/2)
final parameter(x=6,lambda=[3,3]/2,nu=[3,1]/2)
final parameter(x=6,lambda=[5,3]/2,nu=[3,1]/2)
atlas>

Let us look at the tau invariants for these standard representations:

atlas> void: for p in ps do prints(p," ",tau(p)) od
final parameter(x=6,lambda=[3,1]/2,nu=[3,1]/2) [0,1]
final parameter(x=6,lambda=[5,1]/2,nu=[3,1]/2) [1]
final parameter(x=6,lambda=[3,3]/2,nu=[3,1]/2) [1]
final parameter(x=6,lambda=[5,3]/2,nu=[3,1]/2) [0,1]
atlas>

Now, we see that two of them have tau invariant [0,1]. This is because they are both one-dimensional representations. The group is disconnected and has two one-dimensional representations. Each is equivalent to the other one tensor the sign representation. This interchanges the two representations. And likewise, the two representations labeled with the tau invariant [1] get interchanged.

Now let us look at composition series for one of those pairs of representations

atlas> p:ps[3]
Variable p: Param (overriding previous instance, which had type Param)
atlas> p
Value: final parameter(x=6,lambda=[5,3]/2,nu=[3,1]/2)
atlas>
atlas> show(composition_series(I(p)))
1*J(x=6,lambda=[5/2,3/2],nu=[3/2,1/2])
1*J(x=4,lambda=[5/2,1/2],nu=[3/2,0/1])
1*J(x=5,lambda=[3/2,1/2],nu=[1/1,1/1])
1*J(x=3,lambda=[3/2,1/2],nu=[0/1,1/2])
1*J(x=3,lambda=[3/2,3/2],nu=[0/1,1/2])
1*J(x=2,lambda=[3/2,1/2],nu=[1/2,-1/2])
1*J(x=0,lambda=[3/2,1/2],nu=[0/1,0/1])
atlas>


atlas> p:ps[0]
Variable p: Param (overriding previous instance, which had type Param)
atlas> show(composition_series(I(p)))
1*J(x=6,lambda=[3/2,1/2],nu=[3/2,1/2])
1*J(x=4,lambda=[3/2,1/2],nu=[3/2,0/1])
1*J(x=5,lambda=[3/2,1/2],nu=[1/1,1/1])
1*J(x=3,lambda=[3/2,1/2],nu=[0/1,1/2])
1*J(x=3,lambda=[3/2,3/2],nu=[0/1,1/2])
1*J(x=2,lambda=[3/2,1/2],nu=[1/2,-1/2])
1*J(x=0,lambda=[3/2,1/2],nu=[0/1,0/1])
atlas>

These are almost identical but not quite. For example, the lambdas are different in lines 1 and 2.

Similarly if we look at parameters ps[1] and ps[2] we have

atlas> p:ps[1]
Variable p: Param (overriding previous instance, which had type Param)
atlas> show(composition_series(I(p)))
1*J(x=6,lambda=[5/2,1/2],nu=[3/2,1/2])
1*J(x=4,lambda=[5/2,1/2],nu=[3/2,0/1])
1*J(x=3,lambda=[3/2,3/2],nu=[0/1,1/2])
1*J(x=2,lambda=[3/2,1/2],nu=[1/2,-1/2])
1*J(x=1,lambda=[3/2,1/2],nu=[0/1,0/1])
1*J(x=0,lambda=[3/2,1/2],nu=[0/1,0/1])
atlas>

atlas> p:ps[2]
Variable p: Param (overriding previous instance, which had type Param)
atlas> show(composition_series(I(p)))
1*J(x=6,lambda=[3/2,3/2],nu=[3/2,1/2])
1*J(x=4,lambda=[3/2,1/2],nu=[3/2,0/1])
1*J(x=3,lambda=[3/2,1/2],nu=[0/1,1/2])
1*J(x=2,lambda=[3/2,1/2],nu=[1/2,-1/2])
1*J(x=1,lambda=[3/2,1/2],nu=[0/1,0/1])
1*J(x=0,lambda=[3/2,1/2],nu=[0/1,0/1])
atlas>

These are smaller standard representations, have less complicated and also very similar composition series.