Script File

`restricted_roots.at`

compute the root system of A in G where
H=TA is the most split Cartan
this is possibly reduced, so restricted_roots_short
and restricted_roots_long give the two options
for a reduced root system
default: restricted_roots_short
for example restricted_roots_short(SU(n+1,n))=B_n

Source

atlas-scripts/restricted_roots.at (88 lines)

Definitions

Loads

<matrix.at

Loaded by

lusztig_cells.at

Definitions in source order

`basis_max_split_torus`

L13basis_max_split_torus(RealForm G) = mat

basis of X_*(H^{-theta_x}) where x <-> most split Cartan

id_mat in split case

L15

`remove_duplicates`

L18remove_duplicates([vec] list) = [vec]

eliminate duplicates from list

`remove_zeros`

L25remove_zeros([vec] list) = [vec]

remove zero vector from list

`prune`

L29prune(mat list) = mat

removed both duplicates and zeros

`restricted_roots_raw`

L37restricted_roots_raw(RealForm G) = (mat,mat)

return: (mat A,mat B) where
 A is the matrix of positive restricted roots
 B is the matrix of positive injected coroots
these are the roots of a possibly nonreduced root system

`keep_long`

L64keep_long([vec] list) = [vec]

if both v,2v occur keep only 2v

`keep_short`

L68keep_short([vec] list) = [vec]

if both v,2v occur keep only v

`keep_short_roots_long_coroots`

L71keep_short_roots_long_coroots(mat roots, mat coroots) = (mat,mat)

`keep_long_roots_short_coroots`

L74keep_long_roots_short_coroots(mat roots, mat coroots) = (mat,mat)

`restricted_roots_short`

L78restricted_roots_short(RealForm G) = RootDatum

reduced root system obtained by keep short roots/long coroots

`restricted_roots_long`

L83restricted_roots_long(RealForm G) = RootDatum

reduced root system obtained by keep long roots/short coroots

`restricted_roots`

L87restricted_roots(RealForm G) = RootDatum

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