good_W_representatives.at

EigenPair type

set_type EigenPair = (int order,[CyclotomicVec] vecs)

Fields: order, vecs

M: rational matrix of finite order
we want to record eigenvalues and eigenvectors for M
we record these as follows
an EigenPair is: (int order,[CyclotomicVec] vecs)
the vecs are all for a fixed cyclotomic field Q(\zeta_d)
these are a basis of the \zeta_d^order eigenspace of M

the underlying cyclotomic field is vecs[0].F
(vecs is necessarily non-empty, and all the vecs
are assumed to be for the same F

F

also defined in cyclotomic_Lie_algebra.at, cyclotomicMat.at

eigenvalues 2 overloads

compute all d^th roots of unity which are eigenvalues
of a matrix M (M is typically the matrix of a Weyl group
element of order d)
simply trial and error, computing in Q(zeta_d)

specify the field F=Q(zeta_d)
each integer k -> eigenvalue zeta_d^k

eigenvectors

assuming z is an eigenvalue of M

eigenvector

eigenpairs 2 overloads

F=Q(zeta_d)
first compute the eigenvalues (powers of zeta_d)
for each eigenvalue return basis of eigenspace

zero_coroots

coroots vanishing on a set of weights

levi

returns a root_datum for a *possibly non-standard* Levi

nested_Levis

nested_Levis_to_semisimple

map from a nested set of Levi subgroups, and list of orders,
to semisimple conjugacy classes
See Adams-He-Nie Section 2

centralizer

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

make_W_element

make an element of W(rd) from [W_Elt] for smaller root data

standardize

by induction on length of the sequence, conjugate a nested sequence
of Levi factors to one where each Levi is standard

nested levis are decreasing: [L_0=G,...,L_n=T]

also defined in induction.at

good_info 2 overloads

information about the good representative of a conjugacy classes

same as previous, except also use the affine Weyl group to
put v_0 in the fundamental alcove

good_report 2 overloads

information about good representatives of a list of elliptic elements

run_reports

show 2 overloads

also defined in modules.at, K_Nilpotent.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, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, exceptional.at, nilpotent_posets.at, lazy_lists.at