Atlas of Lie Groups and Representations

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Script File

projectors_using_character_tables.at

Source
atlas-scripts/projectors_using_character_tables.at (220 lines)
Definitions
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projectors_verbose

L9projectors_verbose = true

also in this file at line 96

projection on isotypic subspace L11

projector

L33projector (WeylClassTable tab, W_rep(dim,ops), [int] chi) = mat
scalar multiple of projection of representation space of |ops| on isotoypic
  subspace defined by character |chi| of an irreducible representation

  When |chi| is an irreducible character of $W$, and |tab| is the class table of
  $W$, the call |projector(tab,rep,chi)| for a $W$ representation given by |rep|
  computes a matrix describing a scalar multiple of the projection of the space
  $V$ on which |rep| acts to its isotypic subspace for |chi|, via the formula

		       $$ \sum_{w\in W}\chi(w)\rho(w), $$

  where $\rho(w)$ is the action in |rep| of |w|. The menioned scalar multiple
  is $|W|/\dim(\chi)$, as can be seen by taking $\rho$ to be the irredudible
  corresponding to $\chi$ itself, and computing the trace |W| of the above sum.

  If an irreducible character |chi| is not a constituent of the
  representation defined by |ops|, this gives the null matrix; then since
  the result depends linearly on |chi|, we can even use this when that
  character is reducible, but has a unique irreducible constituent in common
  with |ops|, giving (a multiple of) the projection for that isotypic subspace

also in this file at line 54, line 110; also defined in K_Nilpotent.at

this used to be:
for (w,action) in W_parabolic_with_action(rd,all_simples(rd),dim,ops)
do Q+:=character_value(tab,w,chi)*action
od; Q
L46

projector

L54projector (WeylClassTable tab, W_rep(dim,ops), [int] chi,(->)f) = mat
since the preceding function is very slow for large W because of
class_of(w), report on progress through W (if projectors_verbose=true

also in this file at line 33, line 110; also defined in K_Nilpotent.at

projector_verbose

L69projector_verbose ((WeylClassTable,W_rep,[int])(tab,rho,chi):args) = mat
since the preceding function is very slow for large W because of
class_of(w), report on progress through W (if projectors_verbose=true

projectors

L79projectors (WeylClassTable tab, W_rep (dim,ops), [[int]] chis,(->)f) = [mat]
since the preceding function is very slow for large W because of
class_of(w), combine several uses of it into one loop:
|chis| is a list of characters for which the isotypic
projections are to be computed

also in this file at line 113, line 210

projectors_verbose

L96projectors_verbose (WeylClassTable tab, W_rep rho, [[int]] chis) = [mat]

also in this file at line 9

restrict_representation_isotypic

L102restrict_representation_isotypic (WeylClassTable Wct, W_rep pi, vec chi) = W_rep

projector

L110projector (WeylClassTable Wct, WCell cell, vec chi) = mat
projector for a cell representation

also in this file at line 33, line 54; also defined in K_Nilpotent.at

projectors

L113projectors (WeylClassTable Wct, WCell cell, [vec] chis) = [mat]

also in this file at line 79, line 210

special_projector

L121special_projector (WeylClassTable Wct, WCell cell) = mat
given a cell, compute the cell representation pi, find the smallest k so that
<pi,sym^k(reflection)>!=0, then this inner product is 1, and the unique
irreducible in common is special; use the character of sym^k(reflection)
to compute the projection operator onto the special

also in this file at line 213

special_representation_of_cell

L129special_representation_of_cell (WeylClassTable Wct, WCell cell) = W_rep
given a cell, compute the special_projector, and use this to construct
the special representation itself

also in this file at line 216

special_projectors

L135special_projectors(WeylClassTable Wct, [WCell] cells) = [mat]
special_projectors for a list of cells

also in this file at line 219

is_rational_multiple

L139is_rational_multiple (vec v,vec w) = bool
test v, w same up to rational multiple

group_vectors_up_to_scalar

L145group_vectors_up_to_scalar ([vec] list) = [[int]]
given list of nonzero vectors return [[int]]
where each [int] is the indices of the vectors
which are the same up to multiple
example: [[1,2],[2,3],[2,4]] -> [[0,2],[1]]

group_vectors_up_to_scalar_alt

L159group_vectors_up_to_scalar_alt ([vec] list)
given list of nonzero vectors return [[int]]
where each [int] is the indices of the vectors
which are the same up to multiple
example: [[1,2],[2,3],[2,4]] -> [[0,2],[1]]
alt: no assert;
the result is [[parameters in kernel], [usual result]]

group_parameters_projection

L168group_parameters_projection (WCell cell,mat projector) = [[int]]

group_parameters_by_primitive_ideal

L179group_parameters_by_primitive_ideal (WCell cell,mat projector) = [[int]]
given a cell, compute the projector onto the special representation
Then cell vertices give the same primitive ideal iff they map to scalar
multiples of the same vector in the special representation, i.e. the two
corresponding columns of the matrix are the same up to rational multiple
Define some cases for |CharacterTable|, instead of |WeyClassTable|, as well
L208

projectors

L210projectors (CharacterTable ct, WCell cell, [vec] chis) = [mat]

also in this file at line 79, line 113

special_projector

L213special_projector (CharacterTable ct, WCell cell) = mat

also in this file at line 121

special_representation_of_cell

L216special_representation_of_cell (CharacterTable ct, WCell cell) = W_rep

also in this file at line 129

special_projectors

L219special_projectors(CharacterTable ct, [WCell] cells) = [mat]

also in this file at line 135

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