Atlas of Lie Groups and Representations

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atlas Library · Script File

twisted_root_datum.at

Source
atlas-scripts/twisted_root_datum.at (134 lines)
Definitions
16
Loads
Loaded by
weyltosemisimple.at twisted_conjugacy.at

Definitions in source order

is_distinguished

L6is_distinguished(RootDatum rd,mat delta) = bool

also defined in sommers.at, exceptional.at

TwistedRootDatum type

L9
set_type TwistedRootDatum= (RootDatum rd, mat delta)

Fields: rd, delta

*

L11*(TwistedRootDatum (rd1,delta1),TwistedRootDatum (rd2,delta2)) = TwistedRootDatum

also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at, stable.at, synthetic.at, Weylgroup.at, group_operations.at, ratmat.at, tits.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

order_twist

L16order_twist(TwistedRootDatum (rd,delta)) = int
order of twist: not just the order of delta
reason: given (rd,delta) we consider (factor,delta)
where factor is delta-invariant

pre_folded

L23pre_folded(TwistedRootDatum (rd,delta)) = (mat,mat)
rd is semsimple and delta is distinguished
returns roots and coroots matrices for the folded diagram

folded 2 overloads

L56folded(TwistedRootDatum(rd,delta):args) = (RootDatum,mat)
rd is semsimple and delta is distinguished
returns (RootDatum folded, mat T) where T is a matrix relating H and H^delta
L65folded (RootDatum rd,int n) = (RootDatum,mat)
make it easy to construct in simple case: search for a diagram automorphism
of order |n|
if you need to specify delta more precisely use folder(RootDatum,mat) above
to keep the type  checker happy
L72

inverse_image_simple_factor

L76inverse_image_simple_factor (TwistedRootDatum (rd,delta),RootDatum folded_factor, mat T) = RootDatum
all simple factors of rd mapping to given simple factor of folded root datum

affine_root_of_factor

L87affine_root_of_factor(TwistedRootDatum trd, RootDatum folded_factor,mat T)

affine_datum

L98affine_datum(TwistedRootDatum trd)

also in this file at line 125; also defined in affine.at

construct automorphism delta of (rd,rd,...,rd) (r copies)
(x_1,...,x_r) -> (tau(x_r),x_1,...,x_{r-1})
delta^r=tau (automorphism of first factor)
return ((rd,rd,...,rd),delta)
L104

cyclic_twist 2 overloads

L110cyclic_twist(RootDatum rd,mat tau,int r) = TwistedRootDatum
L120cyclic_twist(RootDatum rd,int r) = TwistedRootDatum
various constructions of AffineDatum
L123

affine_datum

L125affine_datum(LieType lt) = AffineDatum

also in this file at line 98; also defined in affine.at

affine_datum_long

L128affine_datum_long(LieType lt) = AffineDatum

affine_datum_short

L130affine_datum_short(LieType lt) = AffineDatum

dual

L134dual(AffineDatum (alpha_ch,rd,alpha)) = AffineDatum

also defined in hodge_K_type_formula.at, hodge_tensor.at, Vogan-dual.at

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