cyclotomic_Lie_algebra.at

CFLieAlgebraElement type

set_type CFLieAlgebraElement=(StructureConstantTable t,(CyclotomicVec,CyclotomicVec) X)

Fields: t, X

data type for an element of a reductive Lie algebra over CF(m)
N=number of roots
n=rank
X=CyclotomicVector of size N <-> \sum_{i=0}^{N-1} x_i X_i where X_i are the root vectors
H=CyclotomicVector of size n <-> X_* in atlas coordinates
arbitrary element of \g: (X,H)

F

also defined in good_W_representatives.at, cyclotomicMat.at

root_datum

also defined in basic.at, W_reps.at, K_Nilpotent.at, sommers.at, parabolics.at, K.at, nilpotent_orbits.at, W_classes.at, extended_misc.at, character_tables.at, springer_tables.at, disconnected.at, bracket.at, exceptional.at

root_part

also defined in bracket.at

semisimple_part

also defined in center.at, bracket.at

H

also in this file at line 59; also defined in groups.at, bracket.at, exceptional.at

=

also in this file at line 26; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.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, cyclotomicMat.at, bracket.at

null

also defined in all_finite_order.at, cyclotomicMat.at, bracket.at

=

also in this file at line 23; also defined in basic.at, combinatorics.at, modules.at, parabolics.at, synthetic.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, cyclotomicMat.at, bracket.at

+

also defined in basic.at, extParamPol.at, modules.at, ratmat.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomicMat.at, bracket.at, hodge_functions.at, lazy_lists.at

* 2 overloads

also in this file at line 39, line 134; 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, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

embed

also defined in cyclotomic.at, character_table_reps.at, cyclotomicMat.at, bracket.at

coordinates

(t,v,w)-> v##w

also defined in bracket.at

*

dot product of a cyclotomic vector and an integral vector

also in this file at line 31, line 33, line 134; 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, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

support

X=\sum a_\alpha X_\alpha
support: \alpha for which a_\alpha\ne 0

also defined in parabolics.at, nilpotent_orbits.at, W_reps_type_BC.at, bracket.at

lie_algebra_element

element of Lie algebra coming from ratvec of size n+N=dim(\g)

also in this file at line 64; also defined in bracket.at

lie_algebra_element_semisimple

also in this file at line 67; also defined in bracket.at

lie_algebra_element_root_vectors

also in this file at line 68; also defined in bracket.at

X_alpha 2 overloads

CF versions of constructors

also defined in bracket.at

H

also in this file at line 22; also defined in groups.at, bracket.at, exceptional.at

X 2 overloads

also defined in bracket.at, lazy_lists.at

lie_algebra_element

element of Lie algebra coming from ratvec of size n+N=dim(\g)

also in this file at line 50; also defined in bracket.at

lie_algebra_element_semisimple

also in this file at line 53; also defined in bracket.at

lie_algebra_element_root_vectors

also in this file at line 54; also defined in bracket.at

CFbasis

1\le i\le dim\g -> (t,(v,w)) where either v or w has exactly one entry 1

to_strings

nice string representation of LieAlgebraElement

root_part_array

also defined in bracket.at

show_long

also defined in character_tables.at, springer_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, weak_packets.at, bracket.at, exceptional.at

bracket_root_term

[X,Y] =[(X.root_term,X.H),(Y.root_term,Y.H)]=
[X.root_term,Y.root_term] + other terms
this function returns just the root term of this bracket
(not the torus part which may also be nonzero)
this is computed from the structure constant table

also defined in bracket.at

sum

also defined in basic.at, sommers.at, character_tables.at, lazy_lists.at

bracket_torus_term

torus term, coming from [X_alpha,X_-alpha]

also defined in bracket.at

*

also in this file at line 31, line 33, line 39; 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, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

bracket_off_diagonal_terms

also defined in bracket.at

bracket

put the three bracket calculations together

also defined in bracket.at

ad

also defined in bracket.at

any

also defined in basic.at, nilpotent_induction.at

a_solution

also defined in Gaussian_elim.at, cyclotomic_Gaussian_elim.at

jacobi_formula

general Jacobi formula [[X,Y],Z]+[[Z,X],Y]+[Y,X],X]

also defined in structure_constants.at, bracket.at

jacobi_identities

test all Jacobi identities for the Lie algebra
see jacobi_identity in structure_constants.at

also in this file at line 206; also defined in bracket.at

jacobi_identities_long

also in this file at line 207; also defined in bracket.at

jacobi_identities

also in this file at line 192; also defined in bracket.at

jacobi_identities_long

also in this file at line 198; also defined in bracket.at

centralizer

also defined in good_W_representatives.at, nilpotent_centralizer.at, K_highest_weights.at, subgroups.at, conjugate.at, bracket.at, weyl_character_formula.at

show

also defined in modules.at, K_Nilpotent.at, good_W_representatives.at, arthur_parameters.at, sub_cells.at, associated_variety_annihilator.at, geck_generic.at, L_packet.at, ratmat.at, Levi_subgroups.at, nilpotent_orbits.at, subgroups.at, poset.at, cyclotomic.at, polynomial.at, character_tables.at, springer_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, weak_packets.at, cyclotomicMat.at, bracket.at, exceptional.at, nilpotent_posets.at, lazy_lists.at