{getting started using the atlas software} jda@Leonidas:~/a$ ./atlas This is 'atlas' (version 0.6, axis language version 0.9), the Atlas of Lie Groups and Representations interpreter, compiled on Jan 19 2016 at 02:21:55. http://www.liegroups.org/ atlas> atlas> 1+1 Value: 2 atlas> {Ok, the software is running} atlas> {You need to be able to load the supplementary *.at files} atlas> {Give the command as follows:} atlas> quit Bye. jda@Leonidas:~/a$ jda@Leonidas:~/a$ ./atlas --path=atlas-scripts This is 'atlas' (version 0.6, axis language version 0.9), the Atlas of Lie Groups and Representations interpreter, compiled on Jan 19 2016 at 02:21:55. http://www.liegroups.org/ atlas> [int]) [lines deleted] Defined find_vec: ([vec],vec->int) Completely read file 'atlas-scripts/basic.at'. atlas> {OK, the .at files are loading correctly, let's load many of them} atlas> {define a root datum} atlas> set rd=simply_connected ("A1") Identifier rd: RootDatum atlas> rd Value: simply connected root datum of Lie type 'A1' atlas> simple_roots (rd) Value: | 2 | atlas> simple_coroots (rd) Value: | 1 | atlas> 2*1 Value: 2 atlas> [press the TAB key twice] Display all 903 possibilities? (y or n) ! != # [many lines deleted...] zero_poly_row zero_simple_coroots zero_simple_roots atlas> set rd=simply_connected ("E8") Identifier rd: RootDatum (hiding previous one of type RootDatum) atlas> simple_roots (rd) Value: | 2, 0, -1, 0, 0, 0, 0, 0 | | 0, 2, 0, -1, 0, 0, 0, 0 | | -1, 0, 2, -1, 0, 0, 0, 0 | | 0, -1, -1, 2, -1, 0, 0, 0 | | 0, 0, 0, -1, 2, -1, 0, 0 | | 0, 0, 0, 0, -1, 2, -1, 0 | | 0, 0, 0, 0, 0, -1, 2, -1 | | 0, 0, 0, 0, 0, 0, -1, 2 | atlas> simple_coroots (rd) Value: | 1, 0, 0, 0, 0, 0, 0, 0 | | 0, 1, 0, 0, 0, 0, 0, 0 | | 0, 0, 1, 0, 0, 0, 0, 0 | | 0, 0, 0, 1, 0, 0, 0, 0 | | 0, 0, 0, 0, 1, 0, 0, 0 | | 0, 0, 0, 0, 0, 1, 0, 0 | | 0, 0, 0, 0, 0, 0, 1, 0 | | 0, 0, 0, 0, 0, 0, 0, 1 | atlas> set rd1=adjoint("E8") Identifier rd1: RootDatum atlas> simple_roots (rd1) Value: | 1, 0, 0, 0, 0, 0, 0, 0 | | 0, 1, 0, 0, 0, 0, 0, 0 | | 0, 0, 1, 0, 0, 0, 0, 0 | | 0, 0, 0, 1, 0, 0, 0, 0 | | 0, 0, 0, 0, 1, 0, 0, 0 | | 0, 0, 0, 0, 0, 1, 0, 0 | | 0, 0, 0, 0, 0, 0, 1, 0 | | 0, 0, 0, 0, 0, 0, 0, 1 | atlas> simple_coroots (rd) Value: | 1, 0, 0, 0, 0, 0, 0, 0 | | 0, 1, 0, 0, 0, 0, 0, 0 | | 0, 0, 1, 0, 0, 0, 0, 0 | | 0, 0, 0, 1, 0, 0, 0, 0 | | 0, 0, 0, 0, 1, 0, 0, 0 | | 0, 0, 0, 0, 0, 1, 0, 0 | | 0, 0, 0, 0, 0, 0, 1, 0 | | 0, 0, 0, 0, 0, 0, 0, 1 | atlas> simple_coroots (rd1) Value: | 2, 0, -1, 0, 0, 0, 0, 0 | | 0, 2, 0, -1, 0, 0, 0, 0 | | -1, 0, 2, -1, 0, 0, 0, 0 | | 0, -1, -1, 2, -1, 0, 0, 0 | | 0, 0, 0, -1, 2, -1, 0, 0 | | 0, 0, 0, 0, -1, 2, -1, 0 | | 0, 0, 0, 0, 0, -1, 2, -1 | | 0, 0, 0, 0, 0, 0, -1, 2 | atlas> set pr=posroots (rd) Identifier pr: mat atlas> pr[0] Value: [ 2, 0, -1, 0, 0, 0, 0, 0 ] atlas> #pr Value: (8,120) atlas {the roots are the 120 columns of a matrix with 8 rows} atlas> pr[119] Value: [ 0, 0, 0, 0, 0, 0, 0, 1 ] atlas> {the highest root} atlas> set G=SL(2,R) Identifier G: RealForm atlas> G Value: connected split real group with Lie algebra 'sl(2,R)' atlas> simple_roots (G) Value: | 2 | atlas> set G=Sp(4,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> simple_roots (G) Value: | 1, 0 | | -1, 2 | atlas> {for classical groups the coordinates are the usual ones, roots are the columns} atlas> simple_coroots (G) Value: | 1, 0 | | -1, 1 | atlas> posroots (G) Value: | 1, 0, 1, 2 | | -1, 2, 1, 0 | atlas> for H in real_forms (G) do prints(H) od compact connected real group with Lie algebra 'sp(2)' connected real group with Lie algebra 'sp(1,1)' connected split real group with Lie algebra 'sp(4,R)' atlas> {the real forms of Sp(4,R)} atlas> set H=real_forms (G)[0] Identifier H: RealForm (hiding previous one of type string (constant)) atlas> H Value: compact connected real group with Lie algebra 'sp(2)' atlas> set G=PSp(4,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> G Value: disconnected split real group with Lie algebra 'sp(4,R)' atlas> simple_roots (G) Value: | 1, 0 | | 0, 1 | atlas> simple_coroots (G) Value: | 2, -1 | | -2, 2 | atlas> distinguished_involution (G) Value: | 1, 0 | | 0, 1 | atlas> set G=SL(4,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> G Value: connected split real group with Lie algebra 'sl(4,R)' atlas> set G=GL(4,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> simple_roots (G) Value: | 1, 0, 0 | | -1, 1, 0 | | 0, -1, 1 | | 0, 0, -1 | atlas> set A=simple_roots (G) Identifier A: mat atlas> set B=simple_coroots (G) Identifier B: mat atlas> B Value: | 1, 0, 0 | | -1, 1, 0 | | 0, -1, 1 | | 0, 0, -1 | atlas> A Value: | 1, 0, 0 | | -1, 1, 0 | | 0, -1, 1 | | 0, 0, -1 | atlas> ^A Value: | 1, -1, 0, 0 | | 0, 1, -1, 0 | | 0, 0, 1, -1 | atlas> ^A*B Value: | 2, -1, 0 | | -1, 2, -1 | | 0, -1, 2 | atlas> {^A*B is always the Cartan matrix} atlas> G Value: disconnected split real group with Lie algebra 'sl(4,R).gl(1,R)' atlas> set G=SL(4,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> for H in real_forms (G) do prints(H) od connected real group with Lie algebra 'sl(2,H)' connected split real group with Lie algebra 'sl(4,R)' atlas> nr_of_ nr_of_Cartan_classes nr_of_dual_real_forms nr_of_posroots nr_of_real_forms atlas> nr_of_Cartan_classes (G) Value: 3 atlas> Cartan_class(G,0) Value: Cartan class #0, occurring for 2 real forms and for 1 dual real form atlas> {this is the compact Cartan subgroup} atlas> set T=Cartan_class(G,0) Identifier T: CartanClass atlas> print_Cartan_info (T) compact: 1, complex: 1, split: 0 canonical twisted involution: e twisted involution orbit size: 3; fiber size: 2; strong inv: 6 imaginary root system: A1.A1 real root system: empty complex factor: A1 atlas> {the first line means: T(R)=S^1 x C^*} atlas> set T=Cartan_class(G,2) Identifier T: CartanClass (hiding previous one of type CartanClass) atlas> {This is the split Cartan T(R)=(R^*)^3} atlas> print_Cartan_info (T) compact: 0, complex: 0, split: 3 canonical twisted involution: 1,2,1,3,2,1 twisted involution orbit size: 1; fiber size: 1; strong inv: 1 imaginary root system: empty real root system: A3 complex factor: empty atlas> print_real_Weyl (G,T) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is a Weyl group of type A3 generators for W^R: 1 2 3 atlas> {This is the Weyl group W(G(R),T(R))} atlas> {See Vogan, Irreducible Characters IV, Duke, Proposition 4.16} atlas> set G=Sp(6,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> set T=Cartan_class(G,0) Identifier T: CartanClass (hiding previous one of type CartanClass) atlas> print_Cartan_info (T) compact: 3, complex: 0, split: 0 canonical twisted involution: e twisted involution orbit size: 1; fiber size: 8; strong inv: 8 imaginary root system: C3 real root system: empty complex factor: empty atlas> set G=SL(2,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> set T=Cartan_class(G,0) Identifier T: CartanClass (hiding previous one of type CartanClass) atlas> T Value: Cartan class #0, occurring for 2 real forms and for 1 dual real form atlas> print_Cartan_info (T) compact: 1, complex: 0, split: 0 canonical twisted involution: e twisted involution orbit size: 1; fiber size: 2; strong inv: 2 imaginary root system: A1 real root system: empty complex factor: empty atlas> print_real_Weyl (G,T) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is trivial atlas> {for SL(2,R), the Weyl group W(G(R),T(R)) of the compact Cartan is trivial} atlas> set G=PGL(2,R) Identifier G: RealForm (hiding previous one of type RealForm) atlas> set T=Cartan_class(G,0) Identifier T: CartanClass (hiding previous one of type CartanClass) atlas> print_real_Weyl (G,T) real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is an elementary abelian 2-group of rank 1 W_ic is trivial W^R is trivial generators for A 1 atlas> {for PGL(2,R), the Weyl group W(G(R),T(R)) of the compact Cartan is Z/2Z} atlas {the group A is sensitive to isogenies} atlas> set G=split_form(E6) Identifier G: RealForm (hiding previous one of type RealForm) atlas> for H in real_forms (G) do prints(H) od connected real group with Lie algebra 'e6(f4)' connected split real group with Lie algebra 'e6(R)' Value: [(),()] atlas> G Value: connected split real group with Lie algebra 'e6(R)' atlas> K_0(G) Value: compact connected real group with Lie algebra 'sp(4)' atlas> set H=real_forms (G)[0] Identifier H: RealForm (hiding previous one of type RealForm) atlas> H Value: connected real group with Lie algebra 'e6(f4)' atlas> K_0(H) Value: compact connected real group with Lie algebra 'f4' atlas> set T=Cartan_class(G,0) Identifier T: CartanClass (hiding previous one of type CartanClass) atlas> print_Cartan_info (T) 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 (G,T) 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 an elementary abelian 2-group of rank 2 W_ic is a Weyl group of type A1.A1.A1.A1 W^R is trivial generators for W^C: 1,6 3,5 generators for A 3,4,3,5,4,3 1,3,4,3,5,6,5,4,3,1 generators for W_ic: 2,4,2 2,3,4,5,4,2,3 1,2,3,4,5,6,5,4,2,3,1 4,3,1,5,4,2,3,4,5,6,5,4,2,3,1,4,3,5,4 atlas> print_real_Weyl (H,T) 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> {This says for E_6(K=F_4), the fundamental Cartan, W(G(R),H(R))=W(A_2)\ltimes W(D_4)} atlas> {Since W(A_2)=S_3, this is S_3\ltimes W(D_4) with S_3 acting by the outer automorphism group} atlas> {In fact this is isomorphic to W(F_4), the Weyl group of K} atlas> {This is because the quotient of the root system E_6 by the outer automorphism is F_4}