subgroups.at

subgroup 2 overloads

subgroup(rd,roots):
roots are assumed to be the set of ALL (positive and negative) roots of a
reductive subgroup containing H
(otherwise an error)
let twochocheck=compute sum of positive coroots
simple roots are those with inner product 2 with this
then construct a RootDatum: root_datum(simple_roots,simple_coroots)

Function takes [vec] argument |roots|, which may be empty; always convert them
explicitly to a matrix |rd.rank#roots| (implicit fails when |roots| empty).

roots given by indices (in all roots) rather than vecs

subgroup_from_positive 2 overloads

only specify the positive roots

each root is given by index in positive roots, rather than a vec

subgroup_by_simple

specify a subgroup by a list of roots (not necessarily
simple or even positive), to serve as the simple roots of the subgroup

Levi_of_subspace

mat k: columns span a subspace U of X_*
construct RootDatum of Levi given by roots vanishing on U
this is the same as the Levi defined by the torus given
by exp(\C*v) for v the columns of the matrix

Levi_of_dual_subspace

matrix of elements of X^*

Levi_of_w

Levi subgroup defined by a group element: Levi of the dual subspace
w acts on X^*, U=kernel(w-1) \subset X^*, want roots alpha such that alpha^v
vanishes on U this is the Cartan if and only if w is elliptic

Levi_of_weight

Levi subgroup defined by weight in QX^*

Levi_of_coweight

Levi subgroup defined by weight in QX_*

normalizer 2 overloads

w in subgroup of W such that w permutes set_of_roots

w in W such that w permutes set_of_roots

also defined in conjugate.at

stabilizer_elementary

for testing

orthogonal_subgroup 2 overloads

subgroup orthogonal to a set of roots

subgroup orthogonal to a Levi

centralizer

root datum of coroots perpendicular to a set of elements of X_*

cwts is morally transposed

also in this file at line 148, line 170; also defined in good_W_representatives.at, nilpotent_centralizer.at, K_highest_weights.at, conjugate.at, cyclotomic_Lie_algebra.at, bracket.at, weyl_character_formula.at

centralizer_coweight

cowt is in X_*\otimes Q
t=exp(2\pi i cowt)\in T
compute (id component of) centralizer_G(t)
H=centralizer_coweight(G,v)
root datum of H: <alpha,v>\in Z
H is a subgroup of G

centralizer

centralizer of a set of torus elements, given as $\exp_1$ images of coweights

also in this file at line 134, line 170; also defined in good_W_representatives.at, nilpotent_centralizer.at, K_highest_weights.at, conjugate.at, cyclotomic_Lie_algebra.at, bracket.at, weyl_character_formula.at

centralizer_weight

wt is in X^*(T)_Q=X_*(T^vee)_Q
t=exp(2\pi i wt)\in T^vee
compute (id component of) centralizer_{G^vee}(t)
H=centralizer_weight(G,v)
root datum of H: <alpha^vee,v>\in Z
H^vee is a subgroup of G^vee

centralizer

default: v\in X_*(T)_Q, t=exp(2\pi iv)\in T
This is giving Cent_G(t), the most common case
unfortunately this terminology conflicts with the
terminology above for centralizer(RootDatum rd,[vec] wts)
should change this, but there are many cases in the scripts

Cent_G(exp(2\pi i v)

also in this file at line 134, line 148; also defined in good_W_representatives.at, nilpotent_centralizer.at, K_highest_weights.at, conjugate.at, cyclotomic_Lie_algebra.at, bracket.at, weyl_character_formula.at

find_conjugate

these functions are slow, by calling |classify_standard_Levis| each time

also defined in nilpotent_orbits.at

find_all_conjugate

group_by_conjugacy

also in this file at line 523

show 2 overloads

also in this file at line 579; 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, 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

can_locate

find a conjugate of sub_rd in rds, everything going on in rd_ambient

is_contained

test sub_rd_1 is rd-conjugate to a subgroup of sub_rd_2, where a list
of all subgroups of the latter is precomputed and passed as argument

also in this file at line 491

sort_by_dimension

also defined in nilpotent_orbits.at

sort_by_ss_rank

group_by_ss_rank

the next could be defined as |partition_by(ss_rank@RootDatum,<=@(int,int))|,
but that would not be equivalent, as it avoids empty |ss_rank| classes. MvL

group_by_number_simple_factors

group_by_ss_rank_and_number_simple_factors

find_weak

weak testing: only using rho and rho_check

filter_weak

keep one representative from each conjugacy class of subgroups

group_by_rho

filter

keep one representative from each conjugacy class of subgroups

also defined in basic.at, nilpotent_orbits.at

weakly_maximal_pseudo_Levis_simple

we call a pseudo Levi "weakly maximal" if it is obtained by
deleting a single node
[these aren't maximal, for that you should assume
the label is prime]
proper <=> label is not special (special: label=1)
we include the RootDatum rd itself for induction reasons
this does NOT include actual Levi subgroups (except rd)
assumption: rd has one simple factor, i.e. derived(rd) is simple

maximal_pseudo_Levis_simple

all_products

given nonsimple RootDatum rd=rd_1 * ... * rd_n
in each rd_i we have a list of c_i subgroups
construct the resulting list of
 c_1*...*c_n subgroups of rd
each subgroup RootDatum is made from roots in rd_ambient
construct Cartesian products (in rd_ambient) of simple factors
see maximal_pseudo_Levis below for the main application
and an example

maximal_pseudo_Levis

construct all maximal pseudo Levis of rd, not necessarily simple
no attempt to mod out by conjugacy
for example:
atlas> rd
Value: simply connected root datum of Lie type 'C2.C2'
atlas> for sub in maximal_pseudo_Levis(rd) do prints(sub) od
simply connected root datum of Lie type 'A1.A1.A1.A1'
simply connected root datum of Lie type 'C2.A1.A1'
simply connected root datum of Lie type 'A1.A1.C2'
simply connected root datum of Lie type 'C2.C2'

proper_maximal_pseudo_Levis

for the inductive step need to keep only proper ones

iterate_maximal_pseudo_Levis

iteratively compute all subgroups by repeatedly computing maximal pseudo Levis
ignore conjugacy

subgroups_raw

all equal rank reductive subgroups, ignoring conjugacy
algorithm:
1) iteratively compute all maximal pseudo Levis
2) for each sub sub_rd list representatives of all conjugacy classes of Levis

ignoring conjugacy

subgroups

cut down subgroups_raw to one representative of each conjugacy class
for efficiency group by ss_rank and number of simple factors before
testing for conjugacy
return two lists
first: [[RootDatum]] grouped by ss_rank
second:[RootDatum] ordered by dimension

Levi_subgroups

conjugacy classes of Levis as root data from W_classes.at

proper_Levi_subgroups

is_Levi

return true if rd_L is a (not necessarily standard) Levi subgroup

also defined in Levi_subgroups.at

regular_central_elements

semisimple element t so that L=Cent_G(t)^0

is_pseudo_Levi_in_simple_adjoint

adjoint_image

assuming |rd_L| is a Levi in |rd|, find corresponding Levin in |adjoint(rd)|

is_pseudo_Levi_in_simple

project_on_simple_factor

also defined in simple_factors.at

is_pseudo_Levi

regular_central_element

pseudo_Levi_subgroups 2 overloads

if you've computed subgroups=subgroups(rd) already, run through these
and select the pseudo_Levis. This works for any rd

this works for any rd: find all equal rank subgroups, and then
select the pseudo-Levis

strict_pseudo_Levi_subgroups

is_contained

test sub_rd_1 is W(rd)-conjugate to a subgroup of sub_rd_2

also in this file at line 223

proper_subgroups

subgroups_weak

cut down subgroups_raw to one representative of each conjugacy class
for efficiency group by ss_rank and number of simple factors before
testing for conjugacy

conjugacy_classes_from_subs

take each subgroup to its Coxeter element, embedded in W

group_by_conjugacy

also in this file at line 195

iterate_maximal_pseudo_Levis_long

iteratively compute all subgroups by repeatedly computing maximal pseudo Levis
ignore conjugacy
keep extra information: nested sequence of subgroups

NOT WORKING

transitive_containment

transitivity: for use in computing Hasse diagram,
avoid recomputing containments which arise from transitivity

subgroups_poset 2 overloads

assume sorted by dimension

show

also in this file at line 208, line 211; 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, 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

sub_graph 2 overloads

output the contents of a .dot file, suitable to be processed by graphviz
recommended method:
set g=sub_graph(...)
>file.dot prints(g)
command line:
%tred file.dot | dot -Tpdf -o file.pdf
(tred computes the reduced graph)
the ingredients are:
[[int]] data: hasse diagram (produced by hasse(rd,subs))
[RootDatum] subs: list of all subgroups, produced by sub_groups(rd)
[RootDatum] levis: list of levis, produced by Levi_subgroups(rd)
[RootDatum] pseudo_levis: list of pseudo_levis, produced by pseudo_Levi_subgroups(rd)
one node for each subgroup, arrow for containment (graph is not necessarily reduced)
node is a Levi: red; strict pseudo_Levi: green; neither: black

produce all of the ingredients needed from RootDatum rd

also defined in sub_cells.at

sub_graph_max_pseudo_levis_only

computing all pseudo Levis is slow, do just maximal ones

component_group_of_pseudo_Levi

centralizer_pseudo_Levi

normalizer_report

normalizer_reports

also in this file at line 743

default

also defined in extended.at

normalizer_reports

also in this file at line 730

filter_by_class

the following function could be done more efficiently using a |WeylClassTable|

reduce list of W elements to representatives of distinct conjugacy classes

embed_coxeter

embed Coxeter element of W(equal rank subgroup)

embed_coxeters 2 overloads

for debugging the previous ?