Computer input/output: adams_7.17.part2.out

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This is 'atlas' (version 1.0.6, axis language version 0.9.5),
the Atlas of Lie Groups and Representations interpreter,
compiled on Jul 10 2017 at 11:13:00.   http://www.liegroups.org/
atlas> 
atlas> 
atlas> set G=F4_B4
Variable G: RealForm
atlas> G
Value: connected real group with Lie algebra 'f4(so(9))'
atlas> Cartan_classes (G)
Value: [Cartan class #0, occurring for 3 real forms and for 1 dual real form,Cartan class #1, occurring for 1 real form and for 1 dual real form,Cartan class #2, occurring for 2 real forms and for 1 dual real form,Cartan class #3, occurring for 1 real form and for 1 dual real form,Cartan class #4, occurring for 1 real form and for 1 dual real form,Cartan class #5, occurring for 1 real form and for 2 dual real forms,Cartan class #6, occurring for 1 real form and for 1 dual real form,Cartan class #7, occurring for 1 real form and for 3 dual real forms]
atlas> set x=KGB(G,0)
Variable x: KGBElt
atlas> set F4_
F4_B4                 F4_elliptic           F4_s
F4_c                  F4_ic                 F4_spherical_unitary
atlas> set E6_
E6_C4        E6_F4        E6_elliptic  E6_ic_e      E6_q
E6_D5T       E6_c         E6_h         E6_ic_s      E6_s
atlas> set C=Cartan_class(x)
Variable C: CartanClass (overriding previous instance, which had type string (constant))
atlas> print_Cartan_info (C)
compact: 4, complex: 0, split: 0
canonical twisted involution: e
twisted involution orbit size: 1; fiber size: 16; strong inv: 16
imaginary root system: F4
real root system: empty
complex factor: empty
atlas> 
atlas> 
atlas> 
atlas> 
atlas> 
atlas> G:=E6_F4 
Value: connected real group with Lie algebra 'e6(f4)'
atlas> set x=KGB(G,0)
Variable x: KGBElt (overriding previous instance, which had type KGBElt)
atlas> set K=K_0(x)
Variable K: RealForm
atlas> K
Value: compact connected real group with Lie algebra 'f4'
atlas> set C=Cartan_class(x)
Variable C: CartanClass (overriding previous instance, which had type CartanClass)
atlas> print_Cartan_info (C)
compact: 2, complex: 2, split: 0
canonical twisted involution: e
twisted involution orbit size: 45; fiber size: 4; strong inv: 180
imaginary root system: D4
real root system: empty
complex factor: A2
atlas> print_real_Weyl (C)
Error during analysis of expression at <standard input>:20:0-19
Type error:
  Subexpression C at <standard input>:20:17-18
  has wrong type: found CartanClass while (RealForm,CartanClass) was needed.
Expression analysis failed
atlas> print_real_Weyl (G,C)
real weyl group is W^C.((A.W_ic) x W^R), where:
W^C is isomorphic to a Weyl group of type A2
A is trivial
W_ic is a Weyl group of type D4
W^R is trivial

generators for W^C:
1,6
3,5
generators for W_ic:
2
4
3,4,5,4,3
1,3,4,5,6,5,4,3,1
atlas> whattype  dimension?
Overloaded instances of 'dimension'
  RootDatum->int
  (RootDatum,ratvec)->int
  Param->int
  (KGBElt,vec)->int
  (KGBElt,ratvec)->int
atlas> whattype  branch_irr ?
Overloaded instances of 'branch_irr'
  (Param,int)->ParamPol
  (ParamPol,int)->ParamPol
atlas> G:=SL(2,R)
Value: connected split real group with Lie algebra 'sl(2,R)'
atlas> 
atlas> 
atlas> 
atlas> set p=parameter(KGB(G,2),rho(G),[0])
Variable p: Param
atlas> p
Value: final parameter(x=2,lambda=[1]/1,nu=[0]/1)
atlas> LKT(p)
Value: (KGB element #2,[ 1 ]/1)
atlas> highest_LKT(p)
Error during analysis of expression at <standard input>:31:0-14
  Undefined identifier 'highest_LKT'
Expression analysis failed
atlas> highest_weight(LKT(p))
Value: (KGB element #0,[ 0 ])
atlas> branch_irr (p,4)
Value: 
1*parameter(x=2,lambda=[1]/1,nu=[0]/1) [0]
1*parameter(x=1,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=0,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=1,lambda=[3]/1,nu=[0]/1) [3]
1*parameter(x=0,lambda=[3]/1,nu=[0]/1) [3]
atlas> for @p in $ do prints(p, " ", highest_weight(LKT(p),KGB(G,0)) od
                                                                     ^^
syntax error, unexpected OD, expecting ')'
atlas> for @p in $ do prints(p, " ", highest_weight(LKT(p),KGB(G,0))) od
final parameter(x=2,lambda=[1]/1,nu=[0]/1) (KGB element #0,[ 0 ])
final parameter(x=1,lambda=[1]/1,nu=[0]/1) (KGB element #0,[ -2 ])
final parameter(x=0,lambda=[1]/1,nu=[0]/1) (KGB element #0,[ 2 ])
final parameter(x=1,lambda=[3]/1,nu=[0]/1) (KGB element #0,[ -4 ])
final parameter(x=0,lambda=[3]/1,nu=[0]/1) (KGB element #0,[ 4 ])
Value: [(),(),(),(),()]
atlas> branch_irr (8)
Error in expression branch_irr(8) at <standard input>:36:0-14
  Failed to match 'branch_irr' with argument type int
Expression analysis failed
atlas> branch_irr (p,8)
Value: 
1*parameter(x=2,lambda=[1]/1,nu=[0]/1) [0]
1*parameter(x=1,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=0,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=1,lambda=[3]/1,nu=[0]/1) [3]
1*parameter(x=0,lambda=[3]/1,nu=[0]/1) [3]
1*parameter(x=1,lambda=[5]/1,nu=[0]/1) [5]
1*parameter(x=0,lambda=[5]/1,nu=[0]/1) [5]
1*parameter(x=1,lambda=[7]/1,nu=[0]/1) [7]
1*parameter(x=0,lambda=[7]/1,nu=[0]/1) [7]
atlas> for @p in $ do prints(p, " ", highest_weight(LKT(p),KGB(G,0))) od
final parameter(x=2,lambda=[1]/1,nu=[0]/1) (KGB element #0,[ 0 ])
final parameter(x=1,lambda=[1]/1,nu=[0]/1) (KGB element #0,[ -2 ])
final parameter(x=0,lambda=[1]/1,nu=[0]/1) (KGB element #0,[ 2 ])
final parameter(x=1,lambda=[3]/1,nu=[0]/1) (KGB element #0,[ -4 ])
final parameter(x=0,lambda=[3]/1,nu=[0]/1) (KGB element #0,[ 4 ])
final parameter(x=1,lambda=[5]/1,nu=[0]/1) (KGB element #0,[ -6 ])
final parameter(x=0,lambda=[5]/1,nu=[0]/1) (KGB element #0,[ 6 ])
final parameter(x=1,lambda=[7]/1,nu=[0]/1) (KGB element #0,[ -8 ])
final parameter(x=0,lambda=[7]/1,nu=[0]/1) (KGB element #0,[ 8 ])
Value: [(),(),(),(),(),(),(),(),()]
atlas> set br=branch_irr (p,4)
Variable br: ParamPol
atlas> set dim(ParamPol P)=int: sum(for c@p in P do c*dimension(LKT(p)) od)
Error during analysis of expression at <standard input>:40:4-68
Type error:
  Subexpression sum( for c@p in P do *(c,dimension(LKT(p))) od ) at <standard input>:40:25-68
  has wrong type: found Split while int was needed.
Expression analysis failed
  Command 'set dim' not executed, nothing defined.
atlas> set dim(ParamPol P)=int: sum(for c@p in P do split_as_int(c*dimension(LKT(p))) od)
Defined dim: (ParamPol->int)
atlas> dim(br)
Value: 5
atlas> br
Value: 
1*parameter(x=2,lambda=[1]/1,nu=[0]/1) [0]
1*parameter(x=1,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=0,lambda=[1]/1,nu=[0]/1) [1]
1*parameter(x=1,lambda=[3]/1,nu=[0]/1) [3]
1*parameter(x=0,lambda=[3]/1,nu=[0]/1) [3]
atlas> dim(branch_irr (p,10))
Value: 11
atlas> for n:20 do prints(n, " ", dim(branch_irr(p,n))) od 
0 1
1 3
2 3
3 5
4 5
5 7
6 7
7 9
8 9
9 11
10 11
11 13
12 13
13 15
14 15
15 17
16 17
17 19
18 19
19 21
Value: [(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),(),()]
atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> 
atlas> 
atlas> 
atlas> set f(Param p, int height)=for n:height do dim(branch_irr(p,height)) od
Defined f: (Param,int->[int])
atlas> f(p,20)
Value: [21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21]
atlas> set f(Param p, int height)=for n:height do dim(branch_irr(p,n)) odRedefined f: (Param,int->[int])
atlas> f(p,20)
Value: [1,3,3,5,5,7,7,9,9,11,11,13,13,15,15,17,17,19,19,21]
atlas> G:=Sp(4,r)
Error during analysis of expression at <standard input>:54:0-10
  Undefined identifier 'r'
Expression analysis failed
atlas> G:=Sp(4,R)
Value: connected split real group with Lie algebra 'sp(4,R)'
atlas> set ds=all_discrete_series (G,rho(G))
Variable ds: [Param]
atlas> for p in ds do prints(p) od
final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=1,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=2,lambda=[2,1]/1,nu=[0,0]/1)
final parameter(x=3,lambda=[2,1]/1,nu=[0,0]/1)
Value: [(),(),(),()]
atlas> f(ds[2],10)
Value: [0,0,0,0,0,0,0,1,1,1]
atlas> f(ds[2],20)
Value: [0,0,0,0,0,0,0,1,1,1,1,1,1,4,4,5,5,5,5,10]
atlas> f(ds[2],40)
Value: [0,0,0,0,0,0,0,1,1,1,1,1,1,4,4,5,5,5,5,10,10,13,13,14,14,21,21,26,26,29,29,39,39,46,46,51,51,65,65,75]
atlas> f(ds[2],30)
Value: [0,0,0,0,0,0,0,1,1,1,1,1,1,4,4,5,5,5,5,10,10,13,13,14,14,21,21,26,26,29]
atlas> f(ds[0],30)
Value: [0,0,0,0,0,0,0,5,5,10,17,24,29,43,52,98,98,116,152,217,217,302,328,431,458,568,613,829,829,985]
atlas> set D ([int] L) = [int]: for i: #L-1 do L[i+1]-L[i] od 
Defined D: ([int]->[int])
atlas> set coset (int a, int d, [int] L) = [int] : while a<#L do L[a] next a+:=d od
Defined coset: (int,int,[int]->[int])
atlas> set L=$
Variable L: [int]
atlas> L
Value: [0,0,0,0,0,0,0,5,5,10,17,24,29,43,52,98,98,116,152,217,217,302,328,431,458,568,613,829,829,985]
atlas> coset(0,3,L)
Value: [0,0,0,10,29,98,152,302,458,829]
atlas> coset(1,3,L)
Value: [0,0,5,17,43,98,217,328,568,829]
atlas> let x=$ in for :#$-1 do x:=D(x) od
Value: [[0,5,12,26,55,119,111,240,261],[5,7,14,29,64,-8,129,21],[2,7,15,35,-72,137,-108],[5,8,20,-107,209,-245],[3,12,-127,316,-454],[9,-139,443,-770],[-148,582,-1213],[730,-1795],[-2525]]
atlas> let x=coset(0,2,L) in for :4 do x:=D(x) od
Value: [[0,0,0,5,12,12,23,46,54,65,111,130,155,216],[0,0,5,7,0,11,23,8,11,46,19,25,61],[0,5,2,-7,11,12,-15,3,35,-27,6,36],[5,-3,-9,18,1,-27,18,32,-62,33,30]]
atlas> let x=coset(1,2,L) in for :4 do x:=D(x) od
Value: [[0,0,5,5,14,19,55,18,101,85,129,137,261,156],[0,5,0,9,5,36,-37,83,-16,44,8,124,-105],[5,-5,9,-4,31,-73,120,-99,60,-36,116,-229],[-10,14,-13,35,-104,193,-219,159,-96,152,-345]]
atlas> let x=coset(0,4,L) in for :4 do x:=D(x) od
Value: [[0,5,24,69,119,241,371],[5,19,45,50,122,130],[14,26,5,72,8],[12,-21,67,-64]]
atlas> let x=coset(0,6,L) in for :4 do x:=D(x) od
Value: [[0,29,123,306],[29,94,183],[65,89],[24]]
atlas> let x=coset(0,8,L) in for :4 do x:=D(x) od
Value: [[5,93,360],[88,267],[179],[]]
atlas> let x=coset(0,7,L) in for :4 do x:=D(x) od
Value: [[5,47,250,527],[42,203,277],[161,74],[-87]]
atlas> let x=coset(3,7,L) in for :4 do x:=D(x) od
Value: [[17,99,342],[82,243],[161],[]]
atlas> L:=f(ds[2],30)
Value: [0,0,0,0,0,0,0,1,1,1,1,1,1,4,4,5,5,5,5,10,10,13,13,14,14,21,21,26,26,29]
atlas> f(ds[0],60)
Value: [0,0,0,0,0,0,0,5,5,10,17,24,29,43,52,98,98,116,152,217,217,302,328,431,458,568,613,829,829,985,1080,1371,1371,1686,1762,2121,2186,2588,2672,3300,3300,3780,3985,4730,4730,5547,5672,6573,6692,7640,7802,9138,9138,10258,10584,12145,12145,13786,14003,15778]
atlas> L:=$
Value: [0,0,0,0,0,0,0,5,5,10,17,24,29,43,52,98,98,116,152,217,217,302,328,431,458,568,613,829,829,985,1080,1371,1371,1686,1762,2121,2186,2588,2672,3300,3300,3780,3985,4730,4730,5547,5672,6573,6692,7640,7802,9138,9138,10258,10584,12145,12145,13786,14003,15778]
atlas> let x=coset(0,4,L) in for :4 do x:=D(x) od
Value: [[0,5,24,69,119,241,371,542,815,1114,1430,1962,2446,3007],[5,19,45,50,122,130,171,273,299,316,532,484,561],[14,26,5,72,8,41,102,26,17,216,-48,77],[12,-21,67,-64,33,61,-76,-9,199,-264,125]]
atlas> let x=coset(0,5,L) in for :4 do x:=D(x) od
Value: [[0,17,81,119,351,512,1041,1179,2247,2255,4343],[17,64,38,232,161,529,138,1068,8,2088],[47,-26,194,-71,368,-391,930,-1060,2080],[-73,220,-265,439,-759,1321,-1990,3140]]
atlas> for i:20 do let x=coset(0,i,L) in x.D.D.D.D od
^CUser interrupt
  [a=0, d=0, L=[0,0,0,0,0,0,0,5,5,10,17,24,29,43,52,98,98,116,152,217,217,302,328,431,458,568,613,829,829,985,1080,1371,1371,1686,1762,2121,2186,2588,2672,3300,3300,3780,3985,4730,4730,5547,5672,6573,6692,7640,7802,9138,9138,10258,10584,12145,12145,13786,14003,15778]]
(in call at <standard input>:82:18-30 of coset@(int,int,[int]), defined at <standard input>:64:4-76)
Evaluation aborted.
atlas> for i:4 do let x=coset(0,i,L) in x.D.D.D.D od
^CUser interrupt
  [a=0, d=0, L=[0,0,0,0,0,0,0,5,5,10,17,24,29,43,52,98,98,116,152,217,217,302,328,431,458,568,613,829,829,985,1080,1371,1371,1686,1762,2121,2186,2588,2672,3300,3300,3780,3985,4730,4730,5547,5672,6573,6692,7640,7802,9138,9138,10258,10584,12145,12145,13786,14003,15778]]
(in call at <standard input>:83:17-29 of coset@(int,int,[int]), defined at <standard input>:64:4-76)
Evaluation aborted.
atlas> for i:20 from 1 do let x=coset(0,i,L) in x.D.D.D.D od
Value: [[0,0,0,5,-15,20,-13,1,0,13,-25,56,-125,147,-64,11,-105,244,-294,280,-289,312,-307,384,-623,759,-589,474,-744,1093,-1160,1076,-1099,1208,-1286,1517,-2034,2280,-1863,1570,-2100,2847,-3071,2977,-3026,3169,-3226,3575,-4470,4966,-4370,3943,-4825,5998,-6267,6047],[5,-3,-9,18,1,-27,18,32,-62,33,30,-62,31,55,-127,96,51,-165,88,134,-256,131,124,-252,121,177],[-11,42,-106,176,-201,299,-544,810,-936,1181,-1763,2349,-2600,3012,-4019,5057],[12,-21,67,-64,33,61,-76,-9,199,-264,125],[-73,220,-265,439,-759,1321,-1990,3140],[24,44,35,41,6,62],[-87,414,-681,1412,-2160],[107,177,-16,151],[-258,1179,-1652],[214,231],[-142,2053],[580],[-556],[971],[],[],[],[],[],[]]
atlas>