Atlas of Lie Groups and Representations

Documentation contents

atlas Library · Script File

groups.at

Source
atlas-scripts/groups.at (254 lines)
Definitions
97
Loads
Loaded by
combinatorics.at complex.at test_unitarity.at galois.at all_finite_order.at G2_unitary_dual.at speh.at isomorphism.at

Definitions in source order

trivial_datum

L3!trivial_datum = RootDatum

trivial_group

L4!trivial_group = compact_torus(0)
also split

complex

L7complex (RootDatum rd) = RealForm
complex group for |rd| (the group viewed as a real form of doubled type)

complexification

L15complexification = complex@RootDatum
complexification of a real group, another name for the same
some root systems
L17
type A root data
L19

GL_roots

L21GL_roots (int n) = mat

SL_coroots

L28SL_coroots (int n) = mat
equals |GL_roots(n)| without its final row

SL_roots

L35SL_roots (int n) = mat

GL

L42GL (int n) = RootDatum

also in this file at line 145

SL

L43SL (int n) = RootDatum

also in this file at line 144

PSL

L44PSL (int n) = RootDatum

also in this file at line 146

roots and coroots in nice coordinates for roots in types B,C,D
L46

type_B_roots

L48type_B_roots (int n) = mat
same as SL_coroots(n+1)

type_C_roots

L51type_C_roots (int n) = mat

type_D_roots

L55type_D_roots (int n) = mat
type C root data
L61

Sp

L63Sp (int n) = RootDatum

also in this file at line 158, line 172

PSp

L67PSp (int n) = RootDatum
same as adjoint(Sp(n))

also in this file at line 159, line 173

GSp

L71GSp (int n) = RootDatum

also in this file at line 174

type B/D root data
L77

SO

L80SO (int n) = RootDatum
use the standard basis for SO(n)

also in this file at line 197, line 208

(Spin, PSO)

L86(Spin,PSO) = …
two functions that share a local definition
real forms
L96
type A real forms
L98

SU 2 overloads

L101SU (int p,int q, RootDatum rd) = RealForm
here rd should be SL(p+q) or PSL(p+q)
L105SU (int p,int q) = RealForm

also in this file at line 107

PSU

L106PSU (int p,int q) = RealForm

also in this file at line 108

SU

L107SU (int n) = RealForm

also in this file at line 101, line 105

PSU

L108PSU (int n) = RealForm

also in this file at line 106

U

L111U (int p,int q) = RealForm
U(p,q), no isogenies in this case

SL_R

L114SL_R (int n) = RealForm

PSL_R

L115PSL_R (int n) = RealForm

PGL_R

L116PGL_R = (int->RealForm)

SL_C

L118SL_C (int n) = RealForm

PSL_C

L119PSL_C (int n) = RealForm

PGL_C

L120PGL_C = (int->RealForm)

SL_H

L122SL_H (int n) = RealForm

PSL_H

L123PSL_H (int n) = RealForm

PGL_H

L124PGL_H = (int->RealForm)

GL_R

L126GL_R (int n) = RealForm

GL_C

L127GL_C (int n) = RealForm

GL_H

L128GL_H (int n) = RealForm
complex and quaternionic groups of type A
L130

R

L133!R = "R"
to allow alternate syntax: for instance Sp(4,R) instead of Sp_R(4)

C

L133!C = "C"

H

L133!H = "H"
! means they are constants

also defined in cyclotomic_Lie_algebra.at, bracket.at, exceptional.at

by_field

L135by_field ((int->RealForm) f_R,(int->RealForm) f_C,(int->RealForm) f_H) = (string->(int->RealForm))

SL

L144SL (int n, string K)

also in this file at line 43

GL

L145GL (int n, string K)

also in this file at line 42

PSL

L146PSL (int n, string K)

also in this file at line 44

PGL

L147PGL (int n, string K)
type C real forms
L149

Sp_R

L152Sp_R (int n) = RealForm
split real forms; the calls to Sp(n), PSp(n), GSp(n) will catch odd n

PSp_R

L153PSp_R (int n) = RealForm

GSp_R

L154GSp_R (int n) = RealForm
Sp(p,q)
L156

Sp

L158Sp (int p,int q)

also in this file at line 63, line 172

PSp

L159PSp (int p,int q)

also in this file at line 67, line 173

complex and quaternionic real forms
L161

Sp_C

L163Sp_C (int n)
|Sp| will test |n| even

Sp_H

L164Sp_H (int n) = RealForm

PSp_C

L166PSp_C (int n)
|Sp| will test |n| even

PSp_H

L167PSp_H (int n) = RealForm

GSp_C

L169GSp_C (int n)
|Sp| will test |n| even

GSp_H

L170GSp_H (int n) = RealForm

Sp

L172Sp (int n, string K)

also in this file at line 63, line 158

PSp

L173PSp (int n, string K)

also in this file at line 67, line 159

GSp

L174GSp (int n, string K)

also in this file at line 71

type B/D real forms
L176

SO_inner_class

L178SO_inner_class (int p,int q) = string

SO_real_form_number

L187SO_real_form_number (int p,int q) = int

SO

L197SO (int p,int q,RootDatum rd) = RealForm

also in this file at line 80, line 208

SO_star

L200SO_star (int n, bool second, RootDatum rd) = RealForm

also in this file at line 212

Spin

L207Spin (int p,int q) = RealForm

SO

L208SO (int p,int q) = RealForm

also in this file at line 80, line 197

PSO

L209PSO (int p,int q) = RealForm

Spin_star

L211Spin_star (int n) = RealForm

SO_star

L212SO_star (int n) = RealForm

also in this file at line 200

PSO_star

L213PSO_star (int n) = RealForm
exceptional groups
L215

G2_ic

L217G2_ic = inner_class(simply_connected("G2"),"e")

G2_c

L219G2_c = quasicompact_form(G2_ic)

G2_s

L220G2_s = quasisplit_form(G2_ic)

F4_ic

L222F4_ic = inner_class(simply_connected("F4"),"e")

F4_c

L223F4_c = quasicompact_form(F4_ic)

F4_B4

L224F4_B4 = real_form(F4_ic,1)

F4_s

L225F4_s = quasisplit_form(F4_ic)

E6_ic_e

L227E6_ic_e = inner_class (simply_connected ("E6"),"e")

E6_c

L229E6_c = real_form(E6_ic_e,0)

E6_h

L230E6_h = real_form(E6_ic_e,1)

E6_D5T

L231E6_D5T = real_form(E6_ic_e,1)

E6_q

L232E6_q = real_form(E6_ic_e,2)

E6_ic_s

L234E6_ic_s = inner_class (simply_connected ("E6"),"s")

E6_F4

L236E6_F4 = real_form(E6_ic_s,0)

E6_s

L237E6_s = real_form(E6_ic_s,1)

E6_C4

L238E6_C4 = real_form(E6_ic_s,1)

E7_ic_e

L240E7_ic_e = inner_class (simply_connected ("E7"),"e")

E7_ic

L242E7_ic = inner_class (simply_connected ("E7"),"e")

E7_c

L243E7_c = real_form(E7_ic_e,0)

E7_h

L244E7_h = real_form(E7_ic_e,1)

E7_E6T

L245E7_E6T = real_form(E7_ic_e,1)

E7_q

L246E7_q = real_form(E7_ic_e,2)

E7_D6A1

L247E7_D6A1 = real_form(E7_ic_e,2)

E7_s

L248E7_s = real_form(E7_ic_e,3)

E7_A7

L249E7_A7 = real_form(E7_ic_e,3)

E8_ic

L251E8_ic = inner_class(simply_connected("E8"),"e")

E8_c

L252E8_c = real_form(E8_ic,0)

E8_q

L253E8_q = real_form(E8_ic,1)

E8_s

L254E8_s = real_form(E8_ic,2)

Generated from atlas-scripts at commit 648506c (2026-10-13).