Atlas of Lie Groups and Representations

Documentation contents

atlas Library · Script File

group_operations.at

Source
atlas-scripts/group_operations.at (170 lines)
Definitions
26
Loads
Loaded by
finite_unipotents.at parabolics.at Weylgroup.at simple_factors.at representations.at twisted_root_datum.at affine.at conjugate.at W_order.at

Definitions in source order

some more general group constructions L5

real forms
L7
preferred terminology when argument is not an inner class: inner_forms
L9

inner_forms

L11inner_forms (RealForm G)
defining split groups
L21

split_form

L23split_form(InnerClass ic) = RealForm

also defined in basic.at

direct product of RootDatum, InnerClass and RealForm
L33

* 2 overloads

L35*(RootDatum R,RootDatum S) = RootDatum
L41*(InnerClass ic1,InnerClass ic2) = InnerClass

also in this file at line 49; also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at, stable.at, synthetic.at, Weylgroup.at, ratmat.at, tits.at, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

initial_torus_factor

L46initial_torus_factor(RealForm G) = ratvec

*

L49* (RealForm G,RealForm H) = RealForm

also in this file at line 35, line 41; also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at, stable.at, synthetic.at, Weylgroup.at, ratmat.at, tits.at, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

radical, i.e. maximal central torus L57

radical 2 overloads

L59radical (RootDatum rd) = RootDatum
central torus as separate root datum
L63radical (InnerClass ic) = RealForm
unique real form of central torus

maximal_central_torus 2 overloads

L68maximal_central_torus = (RootDatum->RootDatum)
L69maximal_central_torus = (InnerClass->RealForm)

derived, adjoint inner class and real form L71

derived

L73derived (InnerClass ic) = InnerClass

also in this file at line 96; also defined in basic.at

derived_info

L77derived_info(InnerClass ic) = (InnerClass,mat)

mod_central_torus

L81mod_central_torus (InnerClass ic) = InnerClass

also defined in basic.at

adjoint_transform

L86adjoint_transform (RootDatum rd, mat theta) = mat
transform involution theta to basis of simple roots

adjoint

L91adjoint (InnerClass ic) = InnerClass

also in this file at line 108; also defined in basic.at

derived

L96derived (RealForm G) = RealForm

also in this file at line 73; also defined in basic.at

adjoint

L108adjoint (RealForm G) = RealForm

also in this file at line 91; also defined in basic.at

tests for classes of groups L117

is_relatively_split

L119is_relatively_split (RealForm G) = bool
whether is_split(derived(G)))

is_abelian

L122is_abelian(RootDatum rd) = bool
converting between root data, and including a change of basis matrix
L124
Converting rd=(A,B)  (A=simple_roots, B=simple_coroots)
to  (A_new,B_new), also return  |ratmat| M (or |mat|)
M  = change of basis matrix from X*(T) to X^*(T_new)
^M = change of basis matrix from X_*(T_new) to X_*(T)
M satisfies:
M*A=A_new
^M*B_new=B
Note: ^A*B=C=^A_new*B_new (C is the Cartan matrix)
the built-in |derived_info| is of this form
with |mat| in place of |ratmat|
Similarly |mod_central_torus_info, except that in
M satisfies the transpose conditions:
^M*A=A_new
M*B_new=B
L126
derived_info and mod_central_torus_info are built-in
L142

adjoint_info

L145adjoint_info (RootDatum rd) = (RootDatum,mat)
adjoint root datum and change of basis matrix.
change weight basis to simple roots

simply_connected_cover_info

L152simply_connected_cover_info(RootDatum rd) = (RootDatum, ratmat)
simply connected cover of complex group, with change of basis matrix.

simply_connected_cover

L160simply_connected_cover(RootDatum rd) = RootDatum
simply connected cover of complex group

simply_connected_cover_derived_info

L164simply_connected_cover_derived_info(RootDatum rd) = (RootDatum,ratmat)
simply connected cover of derived group, with change of basis matrix.

simply_connected_cover_derived

L169simply_connected_cover_derived(RootDatum rd) = RootDatum
simply connected cover of complex group

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