Atlas of Lie Groups and Representations

Documentation contents

Script File

all_finite_order.at

The labels m_i on the Dynkin diagram representing the coefficient of the
simple roots in the highest root. The label of the lowest root is 1 (not
in the list of these labels).

simple_root_labels@RootDatum is now defined in basic.at
Source
atlas-scripts/all_finite_order.at (213 lines)
Definitions
31
Loads
Loaded by
none of the other all.at files

Definitions in source order

bounded_lists

L17bounded_lists ([int] limits,[int] labels, int order) = [ [int] ]
all nonnegative lists |L| of length |1+#limits| with |(limits#1)*L=0|, where
each entry (except final one) does not exceed the corresponding |label| entry

get_subsets_given_order

L26get_subsets_given_order(RootDatum rd, int order) = [ [int] ]

get_subsets

L29get_subsets(RootDatum rd, int max_order) = [[ [int] ]]

null

L33null(RootDatum rd) = ([int],RootDatum,int)

get_raw_data

L35get_raw_data(RootDatum rd,[[[int]]] S) = [[([int],RootDatum,int)]]

mysort

L48mysort = ([([int],RootDatum,int)] -> [([int],RootDatum,int)])

refine_raw_data

L51refine_raw_data([[([int],RootDatum,int)]] data) = [[([int],RootDatum,int)]]

get_data 2 overloads

L53get_data(RootDatum rd,[[[int]]] S) = [[([int],RootDatum,int)]]
L56get_data(RootDatum rd,int n) = [[([int],RootDatum,int)]]
main function

nice_classes

L58nice_classes(RootDatum rd, int n)

info

L66info([[([int],RootDatum,int)]] data, int j) = void

also in this file at line 76

info_reduced

L71info_reduced([[([int],RootDatum,int)]] data, int j) = void
don't include empty terms

info

L76info([[([int],RootDatum,int)]] data, int j, int bound) = void

also in this file at line 66

cox

L80cox(RootDatum rd,int n) = void

test

L84test([[([int],RootDatum,int)]] data, int n) = void

also defined in test_unitarity.at, nilpotent_centralizer.at, sommers.at

table 2 overloads

L97table([[([int],RootDatum,int)]] data) = void
set find([[([int],RootDatum,int)]] data, int n)=void:
let (,rd,)=data[0][0] in
let labels=for i:rank(rd) do rat_as_int(highest_root(rd)*fundamental_coweights(rd)[i]) od in
for i:#data do for b in data[i] do let (v,rdi,k)=b in if k=n then prints(i, " ", v, " ", Lie_type(rdi), " ", v*labels, " ", k) fi od od

set find_short([[([int],RootDatum,int)]] data, int n)=void:
let (,rd,)=data[0][0] in
let labels=for i:rank(rd) do rat_as_int(highest_root(rd)*fundamental_coweights(rd)[i]) od in
for i:#data do for j: min(10,#(data[i]))  do let (v,rdi,k)=data[i][j] in if k=n then prints(i, " ", v, " ", Lie_type(rdi), " ", v*labels, " ", k) fi od od
L125table([[([int],RootDatum,int)]] data,int bound) = void

table_reduced

L128table_reduced([[([int],RootDatum,int)]] data) = void

Kac_diags_given_order

L134Kac_diags_given_order(RootDatum rd, int order) = [[int]]
Given a positive integer m, (r+1)-tuples of integers a_i so that
sum_i(a_i m_i)=m, with a_i's relatively prime. The last entry
corresponds to the lowest root. These are essentially Kac diagrams.

Kac_diags_up_to_order

L139Kac_diags_up_to_order(RootDatum rd, int max_order) = [[[int]]]

Kac_x

L147Kac_x (RootDatum rd, vec v) = ratvec
Given a Kac diagram, compute the corresponding element of the Lie algebra.

order

L154order (RootDatum rd, vec v) = int
Given a Kac diagram, compute the order of the corresponding element
of G.

also defined in basic.at

Kac_diags_of_identity

L158Kac_diags_of_identity (RootDatum rd) = [[int]]
List the Kac diagrams for the identity element of a complex group G
(unique if G is simply connected).

identity_in_fund_domain

L164identity_in_fund_domain (RootDatum rd) = [ratvec]
List the elements in the fundamental domain for the affine Weyl Group
that exponentiate to the identity element.

is_conjugate

L169is_conjugate (ratvec v,ratvec w,RootDatum rd) = bool
Given two ratvecs representing elements of T, decide whether they are
conjugate in G.

is_conjugate_Kac

L172is_conjugate_Kac ([int] v,[int] w, RootDatum rd) = bool

Kac_classes_given_order_crude

L179Kac_classes_given_order_crude = Kac_diags_given_order@(RootDatum,int)
crude listing.

Kac_classes_given_order

L183Kac_classes_given_order (RootDatum rd,int order) = [[int]]
List all Kac elements of a given order, up to conjugacy of the
corresponding group element.

zero_roots

L193zero_roots(RootDatum rd,[int] Kac) = [int]
affine diagram roots labelled 0 in Kac diagram

complex_pseudo_Levi

L196complex_pseudo_Levi(RootDatum rd, [int] S) = RootDatum

centralizer_of_Kac

L210centralizer_of_Kac (RootDatum rd, [int] Kac) = RootDatum

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