Atlas of Lie Groups and Representations

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good_W_representatives.at

good conjugacy class representatives in W,
constructed using the algorithm of He-Nie,
in terms of eigenvalues
Source
atlas-scripts/good_W_representatives.at (180 lines)
Definitions
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Definitions in source order

EigenPair type

L23
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

L24F(EigenPair ep) = CyclotomicField

eigenvalues 2 overloads

L31eigenvalues(mat M,int d) = (CyclotomicField,[int])
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)
L40eigenvalues(WeylElt w) = (CyclotomicField,[int])
specify the field F=Q(zeta_d)
each integer k -> eigenvalue zeta_d^k

eigenvectors

L43eigenvectors(mat M,CyclotomicFieldElement z) = [CyclotomicVec]
assuming z is an eigenvalue of M

eigenvector

L46eigenvector(mat M,CyclotomicFieldElement z) = CyclotomicVec

eigenpairs 2 overloads

L53eigenpairs(mat M,int d) = [EigenPair]
F=Q(zeta_d)
first compute the eigenvalues (powers of zeta_d)
for each eigenvalue return basis of eigenspace
L60eigenpairs(WeylElt w) = [EigenPair]

zero_coroots

L64zero_coroots(RootDatum rd,[CyclotomicVec] wts) = [int]
coroots vanishing on a set of weights

levi

L72levi(RootDatum rd,[CyclotomicVec] vecs) = RootDatum
returns a root_datum for a *possibly non-standard* Levi

nested_Levis

L77nested_Levis (RootDatum rd, [EigenPair] pairs) = ([int],[RootDatum])

nested_Levis_to_semisimple

L95nested_Levis_to_semisimple(RootDatum rd,[int] orders,[RootDatum] levis, int d)
map from a nested set of Levi subgroups, and list of orders,
to semisimple conjugacy classes
See Adams-He-Nie Section 2

centralizer

L102centralizer(RootDatum rd,RootDatum rd_L) = RootDatum

also defined in nilpotent_centralizer.at

make_W_element

L106make_W_element ([WeylElt] S,RootDatum rd) = WeylElt
make an element of W(rd) from [W_Elt] for smaller root data

standardize

L112standardize (RootDatum rd_orig,[RootDatum] nested_levis) = (WeylElt,[RootDatum])
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]

good_info 2 overloads

L130good_info (WeylElt w) = ([(int,[[CyclotomicFieldElement]])] ,[int] ,[RootDatum] ,[RootDatum] ,WeylElt ,ratvec)
information about the good representative of a conjugacy classes
L146good_info(WeylElt w,AffineDatum ad)
same as previous, except also use the affine Weyl group to
put v_0 in the fundamental alcove

good_report 2 overloads

L152good_report([WeylElt] ell,AffineDatum ad) = void
information about good representatives of a list of elliptic elements
L160good_report(RootDatum rd) = void

run_reports

L163run_reports() = void

show 2 overloads

L169show(EigenPair ep) = void
L174show([EigenPair] eigenpairs) = void

also defined in modules.at, K_Nilpotent.at, arthur_parameters.at, sub_cells.at, associated_variety_annihilator.at, geck_generic.at, L_packet.at

Generated from atlas-scripts at commit 7e1b958 (2026-09-17).