Atlas of Lie Groups and Representations

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Script File

stable.at

stable virtual characters

some elementary operations needed, these should be moved elsewhere
Source
atlas-scripts/stable.at (329 lines)
Definitions
57
Loads
Loaded by
coherent_irreducible.at

Definitions in source order

evaluate_at_1 2 overloads

L14evaluate_at_1 ([i_poly] v) = vec
L15evaluate_at_1 (i_poly_mat M) = mat

left_kernel

L19left_kernel = cokernel@mat
defined as |cokernel| in basic.at

column

L21column (i_poly_mat M,int j) = [i_poly]

also defined in hodge_K_type_formula.at

null_poly_mat

L22null_poly_mat (int r,int c) = i_poly_mat

mat_as_poly_mat

L24mat_as_poly_mat (mat A) = i_poly_mat

*

L27* (mat A,i_poly_mat B) = i_poly_mat

also in this file at line 304, line 310; also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at

has_infinitesimal_character

L29has_infinitesimal_character ([Param] params) = bool

also defined in modules.at

make_regular

L33make_regular (Param p) = Param

make_param_pol

L40make_param_pol (RealForm G,[int] coefficients,[Param] params) = ParamPol
permutations
L44
S=[a_0,a_1,...] <-> permutation sigma(i)=a_i
L45

permutation_matrix_sort

L48permutation_matrix_sort([int] S) = mat

in_tau 2 overloads

L51in_tau (int s,Param p) = bool
whether s is in tau(p)
L53in_tau ([int] S,Param p) = bool
whether $S\subset\tau(p)$

also defined in modules.at, sub_cells.at

Commented-out code, lines 56–58 (2 lines)
This axis version does not allow declaring universal list type argument
set in_tau([*] S,Param p) = bool: true { whether $\emptyset\subset\tau(p)$ }

in_tau_complement 2 overloads

L60in_tau_complement (int s,Param p) = bool
whether s not in tau(p)
L62in_tau_complement ([int] S,Param p) = bool
whether $S\cap\tau(p)$ empty

also defined in modules.at

Psi_irr

L67Psi_irr ([Param] params,[int] S) = [Param]
translate irreducibles to singular infinitesimal character,
only keep those for which tau(p)\subset S-complement
from now on [int] S is a set of simple roots,
[Param] B is a block of parameters with regular infinitesimal character
  <not a Block in the sense of atlas, i.e. (x,y) pairs>
[Param] params is a list of parameters
L70

parameters_tau_containing

L77parameters_tau_containing ([int] S,[Param] params) = [int]
select indices of parameters whose |tau| contains |S|

parameters_tau_contained_in_complement

L81parameters_tau_contained_in_complement ([int] S,[Param] params) = [int]
select indices of parameters whose |tau| is disjoint from |S|

duality_permutation

L85duality_permutation ([Param] B) = [int]
recover permutation related to Vogan duality for given block

dual_parameters

L88dual_parameters ([int] S,[Param] B) = [int]
subset of parameters p^\vee on dual side satisfying: S^\vee\subset tau(p^\vee)

parameters

L97parameters ([int] S,[Param] B) = [int]
parameters dual to the previous ones, i.e. satisfying
tau(p)\subset S-complement in the order so that:
parameters=[a1,a2,...]
dual_parameters=[b1,b2,...]
then duality permutation takes a_i to b_j

parameters_singular

L103parameters_singular ([int] S,[Param] B) = [Param]

lengths_signs

L107lengths_signs ([Param] params) = [int]
vector of (-1)^{length(p)}, and corresponding diagonal matrix

also in this file at line 111

lengths_signs_matrix

L109lengths_signs_matrix ([Param] params) = mat

also in this file at line 113

lengths_signs

L111lengths_signs ([int] S,[Param] B) = [int]

also in this file at line 107

lengths_signs_matrix

L113lengths_signs_matrix ([int] S,[Param] B) = mat

also in this file at line 109

dual_parameters_matrix 3 overloads

L116dual_parameters_matrix ([int] S,[Param] B) = mat
L120dual_parameters_matrix ([Param] B) = mat
L123dual_parameters_matrix ([Param] B, [int] T) = mat
T is a subset of 1,...,n, keep only rows and columns in T

dual_parameters_standard_basis_poly_mat

L127dual_parameters_standard_basis_poly_mat ([Param] B) = i_poly_mat
nxn matrix, rows give formula for irreducible as sum of standards

dual_parameters_standard_basis 2 overloads

L129dual_parameters_standard_basis ([Param] B) = mat
L133dual_parameters_standard_basis ([int] S,[Param] B) = mat
r x n matrix, subset of rows of the previous, corresponding to rows in dual_parameter(S,B)

indices

L137indices ([Param] B,[Param] subset) = [int]

also defined in basic.at

subspace_injection_matrix

L139subspace_injection_matrix ([Param] B,[Param] subset) = mat

get_y

L144get_y ([Param] B) = [int]

stable_at_regular

L149stable_at_regular ([Param] B) = mat
B is a block

vanishing

L159vanishing ([int] S,[Param] B) = mat
r\times m matrix, r=#parameters, m=number of stable sums at regular
^(^dual_parameters_standard_basis(S,B)*^dual_parameters_matrix(B)*stable_at_regular(B))
L162

kernel_vanishing

L165kernel_vanishing ([int] S,[Param] B) = mat
r\times\ell  matrix, r=#parameters, \ell=dimension of (left) kernel of vanishing

stable_at_singular_unsorted

L170stable_at_singular_unsorted ([int] S,[Param] B) = (mat,[Param])
t\times r matrix, t=dimension of kernel of kernel_vanishing
unsorted version, shouldn't be used except for testing
B is a block (or union of blocks) of parameters at regular infinitesimal character
(^kernel(^kernel_vanishing(S,B))*lengths_signs_matrix(S,B),parameters_singular(S,B))
L174

stable_at_singular 2 overloads

L179stable_at_singular ([int] S,[Param] B) = (mat,[Param])
same as stable_at_singular_unsorted, except that it is sorted
B is a set of parameters at regular infinitesimal character
L190stable_at_singular ([int] S,[Param] B,[Param] subset_in) = (mat,[Param])
given block (or union of blocks) of parameters B at regular infinitesimal character,
and a subset\subset B
S=set of simple roots
find stable sums in subset pushed to the S-wall

also in this file at line 255

stable_sums

L206stable_sums([Param] singular_parameters) = [(mat,[Param])]

stable_sums_partial

L225stable_sums_partial([Param] singular_parameters)

stable_at_singular

L255stable_at_singular ([int] S,[Param] B,[Param] subset_in) = (mat,[Param])

also in this file at line 179, line 190

print_stable_at_singular

L271print_stable_at_singular ([int] S,[Param] B,[Param] subset) = void

also in this file at line 248

stable

L278stable ([Param] params) = (mat,[Param])

stable_test_Aq_packet 2 overloads

L297stable_test_Aq_packet (RealForm G,ComplexParabolic P) = void
L300stable_test_Aq_packet (RealForm G,[int] complex_parabolic) = void

* 2 overloads

L304*(vec v,[Param] params)
L310*(mat M,[Param] params)

also in this file at line 27; also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at

printParamPol

L312printParamPol(ParamPol P) = void

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