Script File

`finite_dimensional_signature.at`

some functions related to Frobenius-Schur indicator,
real/quaternionic indicator, and sign of Hermitian form
on finite dimensional representation

Source

atlas-scripts/finite_dimensional_signature.at (203 lines)

Definitions

Loads

<finite_dimensional.at { for |highest_weight_finite_dimensional| }
<representations.at { for |finite_dimensional@(RealForm,vec)| }
<K_types.at { for |K_signature_irr| }
<weyl_character_formula.at { for |trace_strong_real@(Param,ratvec)| }
<K_highest_weights.at

Loaded by

none of the other all.at files

Definitions in source order

`verbose_fdsig`

L11verbose_fdsig = true

`long_rat_as_integer_string`

L15long_rat_as_integer_string (rat r)

assuming a rational is an integer, replace it with a string,
to allow output of integers up to 2^63-1

cut off /1

`is_self_dual`

L18is_self_dual ( Param p) = bool

finite dimensional

`Frobenius_Schur`

L22Frobenius_Schur ( Param p) = int

finite dimensional

`fs`

L28fs = Frobenius_Schur@ Param

finite dimensional

`is_self_conjugate`

L30is_self_conjugate ( Param p) = bool

finite dimensional

`real_quaternionic`

L34real_quaternionic ( Param p) = int

finite dimensional

`rq`

L40rq = real_quaternionic@ Param

finite dimensional

`signature`

L43signature ( Param p) = (int,int)

signature (p,q) of Hermitian form on a finite dimensional representation.

finite dimensional

`signature_string`

L53signature_string ( Param p) = string

signature as a string.

finite dimensional

`index`

L58index ( Param p) = int

index min(p,q) of Hermitian form on a finite dimensional representation.

finite dimensional

`index_string`

L60index_string ( Param p) = string

finite dimensional

`anisotropic_part`

L63anisotropic_part ( Param p) = int

finite dimensional

`anisotropic_part_string`

L66anisotropic_part_string ( Param p) = string

finite dimensional

`fd_report`

L70fd_report (RealForm G, int k) = void

report using direct computation of Hermitian form

Now all that same stuff using an alternative (faster) computation of
signature and index, in equal rank case, using Weyl character formula,
evaluated at strong real form (equal rank case only)

L102

`signature_trace`

L107signature_trace ( Param p) = (int,int)

equal rank finite dimensional

`signature_trace_string`

L115signature_trace_string ( Param p) = string

signature as a string.

equal rank finite dimensional

`index_trace`

L119index_trace ( Param p) = int

equal rank finite dimensional

`index_trace_string`

L122index_trace_string ( Param p) = string

equal rank finite dimensional

`anisotropic_part_trace`

L125anisotropic_part_trace ( Param p) = int

equal rank finite dimensional

`anisotropic_part_trace_string`

L128anisotropic_part_trace_string( Param p) = string

equal rank finite dimensional

`fd_report_trace`

L133fd_report_trace (RealForm G, int k) = void

report using computation of Hermitian form via the Weyl character formula

`trace_report`

L165trace_report (RootDatum rd, ratvec v, int k) = void

detailed information about K-signature of finite dimensional
also see print_K_types:print_K_signature_irr (defined for any p)

L179

`show_signature`

L182show_signature ( Param p) = void

finite dimensional

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