atlas> set G=GL(4,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> print_KGB(G) kgbsize: 10 Base grading: [111]. 0: 0 [C,n,C] 2 0 2 * 1 * (0,0,0,0)#0 e 1: 1 [C,r,C] 4 1 3 * * * (0,0,0,0) 1 2^e 2: 1 [C,C,C] 0 5 0 * * * (0,0,0,0) 0 1xe 3: 2 [C,C,C] 7 6 1 * * * (0,0,0,0) 1 3x2^e 4: 2 [C,C,C] 1 8 7 * * * (0,0,0,0) 1 1x2^e 5: 2 [n,C,n] 5 2 5 8 * 6 (0,0,0,0) 0 2x1xe 6: 3 [n,C,r] 6 3 6 9 * * (0,0,0,0) 1 2x3x2^e 7: 3 [C,n,C] 3 7 4 * 9 * (0,0,0,0)#1 1x3x2^e 8: 3 [r,C,n] 8 4 8 * * 9 (0,0,0,0) 1 1^2x1xe 9: 4 [r,r,r] 9 9 9 * * * (0,0,0,0)#2 1^2x3x2^e atlas> set S=[0,2] Variable S: [int] (overriding previous instance, which had type sparse_mat) atlas> {{0,2},{1,3,4,7},{5,6,8,9}} atlas> set kgps=KGP(G,S) Variable kgps: [KGPElt] atlas> kgps Value: [([0,2],KGB element #2),([0,2],KGB element #7),([0,2],KGB element #9)] atlas> equivalence_class_of(kgps[1]) Value: [KGB element #1,KGB element #3,KGB element #4,KGB element #7] atlas> has_theta_stable_Levi(kgps[1]) Value: false atlas> is_parabolic_theta_stable(kgps[0]) Value: true atlas> is_parabolic_theta_stable(kgps[2]) Value: false atlas> set P=Parabolic:([0,1,2],KGB(F4_s,20)) Variable P: KGPElt (overriding previous instance, which had type KGPElt) atlas> has_theta_stable_Levi(P) Value: true atlas> Levi(P) Value: connected quasisplit real group with Lie algebra 'so(4,3).u(1)' atlas> is_parabolic_theta_stable(P) Value: true atlas> set G=U(2,2) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> print_KGB(G) kgbsize: 21 Base grading: [111]. 0: 0 [n,n,n] 1 2 3 10 8 6 (0,0,0,0)#0 e 1: 0 [n,c,n] 0 1 4 10 * 7 (1,1,0,0)#0 e 2: 0 [c,n,c] 2 0 2 * 8 * (0,1,1,0)#0 e 3: 0 [n,c,n] 4 3 0 11 * 6 (0,0,1,1)#0 e 4: 0 [n,n,n] 3 5 1 11 9 7 (1,1,1,1)#0 e 5: 0 [c,n,c] 5 4 5 * 9 * (1,0,0,1)#0 e 6: 1 [n,C,r] 7 13 6 12 * * (0,0,0,0) 1 3^e 7: 1 [n,C,r] 6 14 7 12 * * (1,1,0,0) 1 3^e 8: 1 [C,r,C] 15 8 13 * * * (0,0,0,0) 1 2^e 9: 1 [C,r,C] 16 9 14 * * * (1,0,0,1) 1 2^e 10: 1 [r,C,n] 10 15 11 * * 12 (0,0,0,0) 1 1^e 11: 1 [r,C,n] 11 16 10 * * 12 (0,0,1,1) 1 1^e 12: 2 [r,C,r] 12 17 12 * * * (0,0,0,0) 2 1^3^e 13: 2 [C,C,C] 18 6 8 * * * (0,0,0,0) 1 2x3^e 14: 2 [C,C,C] 19 7 9 * * * (1,0,1,0) 1 2x3^e 15: 2 [C,C,C] 8 10 18 * * * (0,0,0,0) 1 1x2^e 16: 2 [C,C,C] 9 11 19 * * * (0,1,0,1) 1 1x2^e 17: 3 [C,C,C] 20 12 20 * * * (0,0,0,0)#2 2x1^3^e 18: 3 [C,n,C] 13 19 15 * 20 * (0,0,0,0)#1 1x2x3^e 19: 3 [C,n,C] 14 18 16 * 20 * (0,1,1,0)#1 1x2x3^e 20: 4 [C,r,C] 17 20 17 * * * (0,0,0,0) 2 1x2x1^3^e atlas> set x=KGB(G,2) Variable x: KGBElt (overriding previous instance, which had type KGBElt) atlas> set H=[1,1,1,0] Variable H: [int] (overriding previous instance, which had type string (constant)) atlas> set P=parabolic_by_cwt(H,x) Variable P: KGPElt (overriding previous instance, which had type KGPElt) atlas> Levi(P) Value: connected quasisplit real group with Lie algebra 'su(2,1).u(1).u(1)' atlas> set G=E7_s Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set S=[0,1,2,3,4,5] Variable S: [int] (overriding previous instance, which had type [int]) atlas> set list=theta_stable_parabolics_type(G,S) Variable list: [KGPElt] (overriding previous instance, which had type [RealForm]) atlas> list Value: [([0,1,2,3,4,5],KGB element #15865),([0,1,2,3,4,5],KGB element #15866)] atlas> set G=Sp(4,R) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set x=KGB(G,1) Variable x: KGBElt (overriding previous instance, which had type KGBElt) atlas> involution(x) Value: | 1, 0 | | 0, 1 | atlas> torus_factor(x) Value: [ -1, -1 ]/1 atlas> torus_factor(KGB(G,0)) Value: [ 0, 0 ]/1 atlas> torus_factor(KGB(G,2)) Value: [ 0, -1 ]/1 atlas> torus_factor(KGB(G,3)) Value: [ -1, 0 ]/1 atlas> set P=Parabolic:([0],KGB(G,10)) Variable P: KGPElt (overriding previous instance, which had type KGPElt) atlas> set L=Levi(P) Variable L: RealForm (overriding previous instance, which had type RealForm) atlas> L Value: disconnected split real group with Lie algebra 'sl(2,R).gl(1,R)' atlas> trivial(L) Value: final parameter(x=1,lambda=[1,-1]/2,nu=[1,-1]/2) atlas> real_induce_irreducible(trivial(L),G) Value: 1*parameter(x=10,lambda=[2,1]/1,nu=[1,1]/2) [0] atlas> set p=parameter(L,1,[3,1]/2,[1,-1]/2) Variable p: Param atlas> dimension(p) Value: 1 atlas> infinitesimal_character(p) Value: [ 1, -1 ]/2 atlas> real_induce_irreducible(p,G) Value: 1*parameter(x=10,lambda=[3,2]/1,nu=[1,1]/2) [0] atlas> set q=$ Variable q: ParamPol atlas> infinitesimal_character($) Value: [ 1, 1 ]/2 atlas> set G=U(2,2) Variable G: RealForm (overriding previous instance, which had type RealForm) atlas> set P=parabolic_by_cwt([1,1,1,0],KGB(G,2)) Variable P: KGPElt (overriding previous instance, which had type KGPElt) atlas> set L=Levi(P) Variable L: RealForm (overriding previous instance, which had type RealForm) atlas> set t=trivial(L) Variable t: Param atlas> t Value: final parameter(x=5,lambda=[1,0,-1,0]/1,nu=[1,0,-1,0]/1) atlas> set p=finite_dimensional(L,[2,1,1,1]) Variable p: Param (overriding previous instance, which had type Param) atlas> dimension(p) Value: 3 atlas> theta_induce_irreducible(p,G) Value: 1*parameter(x=15,lambda=[7,3,1,-1]/2,nu=[3,0,-3,0]/2) [8] atlas> goodness(p,G) Value: "Good" atlas> void: for i:9 from -4 do prints(theta_induce_irreducible(finite_dimensional(L,[2,1,1,i]),G)) od 1*parameter(x=15,lambda=[7,3,1,-11]/2,nu=[3,0,-3,0]/2) [23] 1*parameter(x=15,lambda=[7,3,1,-9]/2,nu=[3,0,-3,0]/2) [20] 1*parameter(x=15,lambda=[7,3,1,-7]/2,nu=[3,0,-3,0]/2) [17] 1*parameter(x=15,lambda=[7,3,1,-5]/2,nu=[3,0,-3,0]/2) [14] 1*parameter(x=15,lambda=[7,3,1,-3]/2,nu=[3,0,-3,0]/2) [11] 1*parameter(x=15,lambda=[7,3,1,-1]/2,nu=[3,0,-3,0]/2) [8] 1*parameter(x=15,lambda=[7,3,1,1]/2,nu=[3,0,-3,0]/2) [5] 1*parameter(x=18,lambda=[7,3,3,1]/2,nu=[3,0,0,-3]/2) [2] 1*parameter(x=20,lambda=[7,5,3,1]/2,nu=[3,1,-1,-3]/2) [0] 1*parameter(x=18,lambda=[7,5,3,1]/2,nu=[3,0,0,-3]/2) [3] 1*parameter(x=13,lambda=[7,5,3,1]/2,nu=[0,1,0,-1]/1) [6] atlas> void: for i:9 from -4 do prints(i," ",goodness(finite_dimensional(L,[2,1,1,i]),G)) od -4 Good -3 Good -2 Good -1 Good 0 Good 1 Good 2 Weakly good 3 Fair 4 None atlas> rho_u(P) Value: [ 1, 1, 1, -3 ]/2 atlas> theta_induce_standard(trivial(L),G) Value: 1*parameter(x=15,lambda=[3,1,-1,-3]/2,nu=[1,0,-1,0]/1) [6] atlas> trivial(L) Value: final parameter(x=5,lambda=[1,0,-1,0]/1,nu=[1,0,-1,0]/1) atlas> composition_series(trivial(L)) Value: 1*parameter(x=5,lambda=[1,0,-1,0]/1,nu=[1,0,-1,0]/1) [0] 1*parameter(x=4,lambda=[1,0,-1,0]/1,nu=[1,-1,0,0]/2) [3] 1*parameter(x=3,lambda=[1,0,-1,0]/1,nu=[0,1,-1,0]/2) [3] 1*parameter(x=0,lambda=[1,0,-1,0]/1,nu=[0,0,0,0]/1) [4] atlas> composition_series(theta_induce_standard(trivial(L),G)) Value: 1*parameter(x=15,lambda=[3,1,-1,-3]/2,nu=[1,0,-1,0]/1) [6] 1*parameter(x=10,lambda=[3,1,-1,-3]/2,nu=[1,-1,0,0]/2) [9] 1*parameter(x=8,lambda=[3,1,-1,-3]/2,nu=[0,1,-1,0]/2) [9] 1*parameter(x=0,lambda=[3,1,-1,-3]/2,nu=[0,0,0,0]/1) [10] atlas>