jda@Leonidas:~$ cd atlasSoftware/master/atlasofliegroups/atlas-scripts/ jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ ../atlas all This is 'atlas' (version 1.0.7, axis language version 0.9.6), the Atlas of Lie Groups and Representations interpreter, compiled on Jul 17 2017 at 20:01:50. http://www.liegroups.org/ atlas> [Param],[int]) Added definition [4] of dual_block: (Param,mat->[Param],[int]) Added definition [2] of KL_Q_polynomials_via_dual: ([Param],mat->[[vec]]) Defined signs: (vec->[[vec]]) Redefined KL_P_polynomials: ([Param],mat->[[vec]]) Completely read file 'twisted_dual.at'. atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> atlas> quit Bye. jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ evince ~/Desktop/hermitianFormsSMFrefB.pdf jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ jda@Leonidas:~/atlasSoftware/master/atlasofliegroups/atlas-scripts$ ../atlas all This is 'atlas' (version 1.0.7, axis language version 0.9.6), the Atlas of Lie Groups and Representations interpreter, compiled on Jul 17 2017 at 20:01:50. http://www.liegroups.org/ atlas> atlas> atlas> atlas> atlas> atlas> [Param],[int]) Added definition [4] of dual_block: (Param,mat->[Param],[int]) Added definition [2] of KL_Q_polynomials_via_dual: ([Param],mat->[[vec]]) Defined signs: (vec->[[vec]]) Redefined KL_P_polynomials: ([Param],mat->[[vec]]) Completely read file 'twisted_dual.at'. atlas> set G=SL(3,R) Variable G: RealForm atlas> atlas> set delta=distinguished_involution (G) Variable delta: mat atlas> delta Value: | 1, 0 | | 1, -1 | atlas> simple_roots (G) Value: | 1, 1 | | -1, 2 | atlas> delta*simple_roots (G) Value: | 1, 1 | | 2, -1 | atlas> atlas> atlas> set p=trivial(G) Variable p: Param atlas> print_block(p) Parameter defines element 3 of the following block: 0: 0 [C+,C+] 2 1 (*,*) (*,*) *(x=0,lam_rho= [0,0], nu= [0,0]/1) e 1: 1 [i2,C-] 1 0 (3,4) (*,*) *(x=1,lam_rho= [0,0], nu= [3,3]/2) 2xe 2: 1 [C-,i2] 0 2 (*,*) (3,5) *(x=2,lam_rho= [0,0], nu= [3,0]/2) 1xe 3: 2 [r2,r2] 4 5 (1,*) (2,*) *(x=3,lam_rho= [0,0], nu= [2,1]/1) 1^2xe 4: 2 [r2,rn] 3 4 (1,*) (*,*) *(x=3,lam_rho= [1,1], nu= [2,1]/1) 1^2xe 5: 2 [rn,r2] 5 3 (*,*) (2,*) *(x=3,lam_rho= [1,0], nu= [2,1]/1) 1^2xe atlas> set Bfixed=fixed_block_of (delta,p) Variable Bfixed: [Param] atlas> #Bfixed Value: 2 atlas> for p in Bfixed do prints(p) G > G > od final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1) final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1) Value: [(),()] atlas> atlas> atlas> set B=block_of (trivial(G)) Variable B: [Param] atlas> set P=KL_P_signed_polynomials(B) Variable P: [[vec]] atlas> P Value: [[[ 1 ],[ -1 ],[ -1 ],[ 1 ],[ ],[ ]],[[ ],[ 1 ],[ ],[ -1 ],[ -1 ],[ ]],[[ ],[ ],[ 1 ],[ -1 ],[ ],[ -1 ]],[[ ],[ ],[ ],[ 1 ],[ ],[ ]],[[ ],[ ],[ ],[ ],[ 1 ],[ ]],[[ ],[ ],[ ],[ ],[ ],[ 1 ]]] atlas> printPolyMatrix (P) +1 -1 -1 +1 0 0 0 +1 0 -1 -1 0 0 0 +1 -1 0 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> atlas> atlas> atlas> set Q=KL_Q_polynomials(B) Variable Q: [[vec]] atlas> printPolyMatrix (Q) +1 +1 +1 +1 +1 +1 0 +1 0 +1 +1 0 0 0 +1 +1 0 +1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> printPolyMatrix (P*Q) +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> atlas> atlas> atlas> print_extended_ print_extended_block print_extended_composition_series print_extended_character_formula print_extended_indices atlas> print_extended_indices (B,delta) |B|=6 delta-fixed parameters:=[0,3] complete indices=[(0,1),(0,-1),(1,0),(3,1),(3,-1),(4,0)] delta_action=[0,2,1,3,5,4] atlas> atlas> atlas> atlas> atlas> atlas> set Pbig=big_KL_P_signed_polynomials(B,delta) Variable Pbig: [[vec]] atlas> printPolyMatrix (Pbig) +1 0 -1 +1 0 0 0 +1 -1 0 +1 0 0 0 +1 -1 -1 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> printPolyMatrix (P) +1 -1 -1 +1 0 0 0 +1 0 -1 -1 0 0 0 +1 -1 0 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> printPolyMatrix (Pbig) +1 0 -1 +1 0 0 0 +1 -1 0 +1 0 0 0 +1 -1 -1 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> atlas> Bf Error during analysis of expression at :48:0-2 Undefined identifier 'Bf' Expression analysis failed atlas> Bfixed Value: [final parameter(x=0,lambda=[2,1]/1,nu=[0,0]/1),final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1)] atlas> set Pdelta=KL_P_signed_polynomials(delta,Bfixed ) Error in expression KL_P_signed_polynomials(delta,Bfixed) at :50:11-49 Failed to match 'KL_P_signed_polynomials' with argument type (mat,[Param]) Expression analysis failed Command 'set Pdelta' not executed, nothing defined. atlas> set Pdelta=KL_P_signed_polynomials(delta,p) Error in expression KL_P_signed_polynomials(delta,p) at :51:11-43 Failed to match 'KL_P_signed_polynomials' with argument type (mat,Param) Expression analysis failed Command 'set Pdelta' not executed, nothing defined. atlas> whattype KL_P_signed_polynomials? Overloaded instances of 'KL_P_signed_polynomials' Param->[[vec]] [Param]->[[vec]] (Param,mat)->[[vec]] ([Param],mat)->[[vec]] atlas> set Pdelta=KL_P_signed_polynomials(p,delta) Variable Pdelta: [[vec]] atlas> printPolyMatrix (Pdelta) +1 +1 0 +1 atlas> atlas> atlas> atlas> atlas> atlas> printPolyMatrix (P) +1 -1 -1 +1 0 0 0 +1 0 -1 -1 0 0 0 +1 -1 0 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> atlas> printPolyMatrix (Pdelta ) +1 +1 0 +1 atlas> atlas> atlas> atlas> atlas> printPolyMatrix (P) +1 -1 -1 +1 0 0 0 +1 0 -1 -1 0 0 0 +1 -1 0 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> atlas> atlas> printPolyMatrix (Pbig ) +1 0 -1 +1 0 0 0 +1 -1 0 +1 0 0 0 +1 -1 -1 -1 0 0 0 +1 0 0 0 0 0 0 +1 0 0 0 0 0 0 +1 atlas> atlas> atlas> printPolyMatrix (Pdelta ) +1 +1 0 +1 atlas> atlas> atlas> atlas> print_block(B) Error in expression print_block(B) at :77:0-14 Failed to match 'print_block' with argument type [Param] Expression analysis failed atlas> print_block(p) Parameter defines element 3 of the following block: 0: 0 [C+,C+] 2 1 (*,*) (*,*) *(x=0,lam_rho= [0,0], nu= [0,0]/1) e 1: 1 [i2,C-] 1 0 (3,4) (*,*) *(x=1,lam_rho= [0,0], nu= [3,3]/2) 2xe 2: 1 [C-,i2] 0 2 (*,*) (3,5) *(x=2,lam_rho= [0,0], nu= [3,0]/2) 1xe 3: 2 [r2,r2] 4 5 (1,*) (2,*) *(x=3,lam_rho= [0,0], nu= [2,1]/1) 1^2xe 4: 2 [r2,rn] 3 4 (1,*) (*,*) *(x=3,lam_rho= [1,1], nu= [2,1]/1) 1^2xe 5: 2 [rn,r2] 5 3 (*,*) (2,*) *(x=3,lam_rho= [1,0], nu= [2,1]/1) 1^2xe atlas> atlas> atlas> atlas> print_extended_block (p,delta) 0: [3Ci ] +0; +3 x=0, [ 0, 0 ] 3: [3r ] +3; +0 x=3, [ 0, 0 ] atlas> atlas> atlas> atlas> atlas> set E=E(delta,p) Variable E: (InnerClass,mat,ratvec,vec,mat,ratvec,vec,mat,vec,vec) atlas> display(E) x=KGB element #3 gamma=[ 2, 1 ]/1 g=[ 1, 0 ]/1 lambda_rho=[ 0, 0 ] tau=[ 0, 0 ] l=[ 0, 0 ] t=[ 0, 0 ] atlas> atlas> atlas> atlas> atlas> parameter(E) Value: final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1) atlas> p Value: final parameter(x=3,lambda=[2,1]/1,nu=[2,1]/1) atlas> G:=SL(4,R) Value: connected split real group with Lie algebra 'sl(4,R)' atlas> print_ext_block (trivial(G),d) Error during analysis of expression at :96:0-30 Undefined identifier 'd' Expression analysis failed atlas> set delta=distinguished_involution (G) Variable delta: mat (overriding previous instance, which had type mat) atlas> print_ext_block (trivial(G),delta) Error in expression print_ext_block(trivial(G),delta) at :98:0-34 Failed to match 'print_ext_block' with argument type (Param,mat) Expression analysis failed atlas> print_ext_block (delta,trivial(G)) 0 2Ci 1i1 2Ci: final parameter(x=0,lambda=[3,2,1]/1,nu=[0,0,0]/1) 1 2Ci 1i1 2Ci: final parameter(x=1,lambda=[3,2,1]/1,nu=[0,0,0]/1) 2 2C+ 1r1f 2C+: final parameter(x=2,lambda=[3,2,1]/1,nu=[0,1,-1]/2) 3 2Cr 1C+ 2Cr: final parameter(x=3,lambda=[3,2,1]/1,nu=[1,0,1]/1) 4 2Cr 1C+ 2Cr: final parameter(x=4,lambda=[3,2,1]/1,nu=[1,0,1]/1) 7 2i12 1C- 2i12: final parameter(x=7,lambda=[3,2,1]/1,nu=[2,2,0]/1) 8 2i12 1C- 2i12: final parameter(x=8,lambda=[3,2,1]/1,nu=[2,2,0]/1) 10 2C- 1i2f 2C-: final parameter(x=10,lambda=[3,2,1]/1,nu=[6,3,3]/2) 12 2r21 1r2 2r21: final parameter(x=12,lambda=[3,2,1]/1,nu=[3,2,1]/1) 13 2r21 1rn 2r21: final parameter(x=12,lambda=[4,3,1]/1,nu=[3,2,1]/1) 14 2rn 1r2 2rn: final parameter(x=12,lambda=[3,3,2]/1,nu=[3,2,1]/1) atlas> print_extended_block (delta,trivial(G)) Error during analysis of expression at :100:0-39 Type error: Subexpression (delta,trivial(G)) at :100:21-39 has wrong type: found (mat,Param) while (Param,mat) was needed. Expression analysis failed atlas> print_extended_block (trivial(G),delta) 0: [2Ci ,1i1 ] +0 +1; +3 +2 x= 0, [ 0, 0, 0 ] 1: [2Ci ,1i1 ] +1 +0; +4 +2 x= 1, [ 0, 0, 0 ] 2: [2C+ ,1r1f ] +10 +2; . +0,+1 x= 2, [ 0, 0, 0 ] 3: [2Cr ,1C+ ] +3 +7; +0 . x= 3, [ 0, 0, 0 ] 4: [2Cr ,1C+ ] +4 +8; +1 . x= 4, [ 0, 0, 0 ] 7: [2i12f,1C- ] +7 +3; +12,+13 . x= 7, [ 0, 0, 0 ] 8: [2i12f,1C- ] +8 +4; +12,-13 . x= 8, [ 0, 0, 0 ] 10: [2C- ,1i2f ] +2 +10; . +12,+14 x=10, [ 0, 0, 0 ] 12: [2r21f,1r2 ] +12 +14; +7,+8 +10 x=12, [ 0, 0, 0 ] 13: [2r21f,1rn ] +13 +13; +7,-8 . x=12, [ 1, 1, 0 ] 14: [2rn ,1r2 ] +14 +12; . +10 x=12, [ 0, 1, 1 ] atlas> atlas> atlas>