atlas Library · Script File

`finite_dimensional.at`

finite dimensional representations of G(C)

Source

atlas-scripts/finite_dimensional.at (140 lines)

Definitions

Loads

<Weylgroup.at { dominant }

Loaded by

finite_dimensional_signature.at adams_johnson.at induction.at weyl_character_formula.at

Definitions in source order

`finite_dimensional`

L7finite_dimensional(RealForm G, vec lambda) = Param

constructs parameter for the algebraic finite-dimensional of
extremal weight lambda.

also defined in representations.at

`is_finite_dimensional`

L12is_finite_dimensional (Param p) = bool

whether p is finite dimensional.

`fd_only`

L17fd_only (Param p) = Param

for protecting functions intended for finite dimensional parameters only.

`dimension`

L21dimension (RootDatum rd, ratvec lambda) = int

dimension of finite dimensional representation from its highest weight.

From the earlier documentation

Dimension of finite dimensional, valid up to 2^31-1

also in this file at line 57; also defined in basic.at, K_highest_weights.at, nilpotent_orbits.at, character_tables.at, W_reps_type_BC.at, tensor_product.at, exceptional.at

`Weyl_dim`

L32Weyl_dim (RootDatum rd, vec lambda) = int

`highest_weight_finite_dimensional`

L41highest_weight_finite_dimensional (Param p) = vec

If split torus in the center not necessarily a vec, this could fail.
Leave it for now because otherwise weyl_character_formula.at fails to load

highest weight of finite dimensional in X^* coordinates.

`highest_weight_finite_dimensional_ratvec`

L45highest_weight_finite_dimensional_ratvec (Param p) = ratvec

allow ratvec if G has split central torus

`on_fundamental_weights`

L48on_fundamental_weights (ratvec v,RootDatum rd) = vec

`fundamental_weight_coordinates`

L52fundamental_weight_coordinates (Param p) = vec

fundamental weight coordinates of finite dimensional parameter.

also defined in K_highest_weights.at

`dimension`

L57dimension(Param p) = int

use highest_weight_finite_dimensional_ratvec to all split central torus

dimension of finite dimensional parameter.

also in this file at line 21; also defined in basic.at, K_highest_weights.at, nilpotent_orbits.at, character_tables.at, W_reps_type_BC.at, tensor_product.at, exceptional.at

`complex_LKT`

L62complex_LKT (Param p) = vec

this function depends on a complex root datum, with half-swapping involution

first half

L64

some functions related to Frobenius-Schur indicator, real/quaternionic indicator,
and sign of Hermitian form on finite dimensional representation are
MOVED TO finite_dimensional_signature.at

L66

`ht`

L72ht(RealForm G, vec lambda) = int

`tensor_product`

L77tensor_product(RootDatum rd, vec gamma1, vec gamma2) = KTypePol

tensor product of algebraic reps of complex group of extremal wts gamma1
and gamma2, returned as KTypePol of corresponding GC K-types

also defined in character_tables.at, tensor_product.at

`tensor_product_weights`

L88tensor_product_weights(RootDatum rd, vec gamma1, vec gamma2) = [(int,vec)]

tensor product of algebraic reps of complex group of extremal wts gamma1
and gamma2, returned as list of (multiplicity,highest weight)

`complex_KT`

L97complex_KT (RealForm G, vec v) = KType

Finite dimensional algebraic representation with highest weight v as K-type
of the complexification of G

`Demazure`

L103Demazure (RootDatum rd, int i, KTypePol P) = KTypePol

in the next function |P| is for a pure torus sharing its $X^*$ with |rd|

one can get that torus as |compact_torus(rd.rank)|

`wts_of_fd`

L133wts_of_fd (RootDatum rd, vec hwt) = KTypePol

The weights of a finite dimensional algebraic representation with extremal
weight hwt

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