Atlas of Lie Groups and Representations

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atlas Library · Script File

kl.at

KL stuff, including character formulas and composition series
This implementation seeks to avoid using KL_block, and instead uses
KL_sum_at_s, as this function stores earlier results
Also (compared to old implementation) no regular/singular case distinction

Basic object: column of KL matrix (P-matrix), of polynomials evaluated as s,
which by setting s=1 gives formula for irreducible as sum of standards

main functions:

character_formula(p): irreducible(p) as sum of standards
composition_series(p): standard(p) as sum of irreducibles

signed_KL_mat_at_1(p): signed P-matrix evaluated at 1 (integral matrix)
KL_inverse_mat_at_1(p): Q-matrix evaluated at 1 (integral matrix)

also
  print_character_formula,print_composition_series
Source
atlas-scripts/kl.at (98 lines)
Definitions
15
Loads
Loaded by
modules.at induction.at coherent.at

Definitions in source order

character_formula 2 overloads

L27character_formula (Param p) = ParamPol
L28character_formula (ParamPol P) = ParamPol

also defined in extParamPol.at, modules.at

composition series
L32

composition_series 2 overloads

L34composition_series(ParamPol P) = ParamPol
L40composition_series (Param p) = ParamPol

also defined in modules.at

tabulate_at_1

L42tabulate_at_1 ([Param] block, (Param->ParamPol) f) = mat

signed_KL_mat_at_1

L47signed_KL_mat_at_1(Param p) = mat

KL_inverse_mat_at_1

L50KL_inverse_mat_at_1(Param p) = mat

full_signed_KL_mat_at_1

L54full_signed_KL_mat_at_1(Param p) = mat
versions that do the whole block rather than a partial block

full_KL_inverse_mat_at_1

L56full_KL_inverse_mat_at_1(Param p) = mat

composition_series_verbose

L73composition_series_verbose (ParamPol P) = ParamPol
print out high length terms as they become available

composition_series_trace

L91composition_series_trace (ParamPol P) = ParamPol
simpler and faster: trace progress while computing |composition_series|

Generated from atlas-scripts at commit 648506c (2026-10-13).