Example :math:`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.