atlas Library · Script File

`twisted_conjugacy.at`

twisted semisimple conjugacy classes (including trivial case of trivial twist)

conjugacy of semisimple elements of G

test: s=exp(2\pi i v) is G-conjugate to t=exp(2\pi i w)

Source

atlas-scripts/twisted_conjugacy.at (104 lines)

Definitions

Loads

<twisted_root_datum.at

Loaded by

nilpotent_orbits.at

Definitions in source order

`is_semisimple_element_conjugate_in_semisimple`

L9is_semisimple_element_conjugate_in_semisimple(ratvec v,ratvec w,TwistedRootDatum trd)

case rd is semisimple

also in this file at line 89

Commented-out code, lines 16–32 (16 lines)

case rd is reductive
s is G-conjugate to t <=>
v -> v_dom via the affine Weyl group W_aff=W\rtimes R^vee
 v_dom is in the fundamental alcove for the action of W_aff
 i.e. <v_dom,alpha>\ge 0 for all simple roots and <v_dom,\beta><1
 for all highest roots beta
w -> w_dom also
set L=lattice spanned by simple coroots and radical_basis(rd)
 (the latter is a Z-basis of X_*(radical))
If G_der is simply connected the condition is
 v_dom-w_dom \in L
In general there is an additional term from
 the finite group X_*/L
So a set {v_i} in X_* representing X_*/L
Then the condition is
v_dom -w_dom + v_i \in L for some v_i

`is_semisimple_element_conjugate`

L34is_semisimple_element_conjugate(ratvec v,ratvec w,TwistedRootDatum trd)

also in this file at line 92

`is_semisimple_element_conjugate_mod_torus`

L56is_semisimple_element_conjugate_mod_torus(ratvec v,ratvec w,TwistedRootDatum trd,mat torus)

test if exp(2\pi iv)T is G-conjugate to exp(2\pi iw)T
where T is a torus

also in this file at line 95

`semisimple_conjugacy_class_reps`

L71semisimple_conjugacy_class_reps(TwistedRootDatum trd,[ratvec] S) = ([int],[ratvec])

the first value returned is the list of indices of chosen ratvecs

also in this file at line 99

`semisimple_conjugacy_class_mod_torus_reps`

L79semisimple_conjugacy_class_mod_torus_reps(TwistedRootDatum trd,[ratvec] S,mat torus) = ([int],[ratvec])

the first value returned is the list of indices of chosen ratvecs

also in this file at line 103

untwisted versions: rd -> trd=(rd,id_mat(rd.rank))

L87

`is_semisimple_element_conjugate_in_semisimple`

L89is_semisimple_element_conjugate_in_semisimple(ratvec v,ratvec w,RootDatum rd)

also in this file at line 9

`is_semisimple_element_conjugate`

L92is_semisimple_element_conjugate(ratvec v,ratvec w,RootDatum rd)

also in this file at line 34

`is_semisimple_element_conjugate_mod_torus`

L95is_semisimple_element_conjugate_mod_torus(ratvec v,ratvec w,RootDatum rd, mat torus)

also in this file at line 56

`semisimple_conjugacy_class_reps`

L99semisimple_conjugacy_class_reps(RootDatum rd,[ratvec] S)

also in this file at line 71

`semisimple_conjugacy_class_mod_torus_reps`

L103semisimple_conjugacy_class_mod_torus_reps(RootDatum rd,[ratvec] S,mat torus)

also in this file at line 79

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