Atlas of Lie Groups and Representations

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atlas Library · Script File

synthetic.at

Source
atlas-scripts/synthetic.at (192 lines)
Definitions
25
Loads
Loaded by
all_finite_order.at induction.at affine.at extended.at dual.at

Definitions in source order

KGBElt_gen type

L6
set_type
[ KGBElt_gen = (InnerClass inner_class, mat involution, ratvec torus_factor) ]

Fields: inner_class, involution, torus_factor

generalized KGB elements: x^2 not necessarily central
L8
KGBElt_gen:(ic,theta,torus_factor)
torus_factor must be theta_x-fixed (else error)
   could replace torus_factor with torus_factor*(1+theta)/2 but it seems better to flag this
[no: requirement: base_grading_vector - torus_factor\in Z^n don't have base_grading_vector]
xi=exp(2\pi i(torus_factor+rho^v)/2)\sigma_w\delta
L10

KGB_elt_gen 2 overloads

L17KGB_elt_gen(InnerClass ic,mat involution, ratvec v) = KGBElt_gen
L20KGB_elt_gen(KGBElt x)

KGB_elt

L21KGB_elt(KGBElt_gen x)

also defined in basic.at, parameters.at

grading_coweight

L23grading_coweight (KGBElt x) = ratvec

make_KGB

L27make_KGB (RealForm G, mat theta, ratvec grading_cowt) = KGBElt
alternative to KGB_elt that incorporates shift by rho_check_i
this is only used once, several lines down
cross actions
L30

cross

L32cross (mat w, KGBElt x) = KGBElt

also defined in basic.at, coherent_irreducible.at, extended_misc.at, coherent_characters.at

posroot_cross 3 overloads

L36posroot_cross (int i, KGBElt x) = KGBElt
L39posroot_cross (int i, Param p) = Param
L45posroot_cross (vec alpha, Param p) = Param
the following is what the now built-in function cross@(vec,Param) does
Cayley transforms
L51

posroot_Cayley 2 overloads

L53posroot_Cayley (int i, KGBElt x) = KGBElt
L70posroot_Cayley (int i, Param p) = Param

posroot_inv_Cayley

L85posroot_inv_Cayley (int i, Param p) = Param
some visualisation procedures
L105

find_posroot

L107find_posroot (RootDatum rd, vec alpha) = int

concat

L111concat ([string] l,string open, string sep, string close) = string

also defined in basic.at

posroot_status

L114posroot_status (int i, Param p) = int

new_block_line

L123new_block_line ([Param] B,int i) = string

is_conjugate

L149is_conjugate (ratvec v,ratvec w,mat theta) = bool
the following function supposes v,w are unnormalized torus factors
exp(2i\pi v)\xi is conjugate to exp(2i\pi w)\xi where \xi acts on H by theta
condition:             v-w\in (1-theta)X_Q+X,
which equivalent to:   (1+theta)(v-w)\in (1+theta)X
For normalized tourus factors, use (1+theta)*(v-w)/2 instead

also defined in all_finite_order.at, nilpotent_orbits.at

unnormalized_torus_factor

L153unnormalized_torus_factor (KGBElt_gen x) = ratvec

also defined in parameters.at

square

L156square (KGBElt_gen x) = ratvec

also defined in parameters.at

square_is_central

L158square_is_central(KGBElt_gen x) = bool

also defined in parameters.at, extended_misc.at

Note: square_is_central(KGB_elt y) gives the correct answer
=square_is_central(inner_class(y),involution(y), torus_factor(y))
square_is_central is defined in parameters.at
Note: generalized KGB elements x,y are conjugate
<=> is_conjugate(torus_factor(x)/2,torus_factor(y)/2,theta)=true
see discussion of unnormalized_torus_factor in parameters.at
L160

=

L168= (KGBElt_gen x1,KGBElt_gen x2) = bool

also defined in basic.at, combinatorics.at, modules.at, parabolics.at, hash.at, tits.at, K_highest_weights.at, nilpotent_orbits.at, cyclotomic.at, polynomial.at, W_reps_type_BC.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, extended.at, disconnected.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at

!=

L175!= ((KGBElt_gen,KGBElt_gen)p) = bool

also defined in basic.at, combinatorics.at, hash.at, tits.at, cyclotomic.at, polynomial.at, cyclotomicMat.at

Dangerous bend: this formula appears to be right, with the
unnormalized torus factors, no dividing by 2, and the transpose
I'm not 100% sure this is correct
L177

*

L183*(mat delta,KGBElt_gen(ic,theta,t)) = KGBElt_gen

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

is_fixed

L186is_fixed (mat delta,KGBElt_gen y) = bool

also defined in extended_misc.at

Commented-out code, lines 188–192 (3 lines)
set square_is_central(KGBElt_gen x)=bool:
let v=unnormalized_torus_factor(x) in
square_is_central(x.inner_class,x.involution,v)

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