Script Reference
hermitian.at
Computes Hermitian (c-invariant) forms on standard and irreducible modules, and tests representations for unitarity. Central to the Atlas unitarity algorithm.
Mathematical background
For a real reductive group \(G(\mathbb{R})\), a representation \(\pi\) is unitary if it admits a positive-definite \(G(\mathbb{R})\)-invariant Hermitian form. The functions in this file compute the c-invariant form on a standard module \(I(x,\lambda,\nu)\) as a virtual \(K\)-type, which at \(\nu=0\) determines unitarity.
Definitions
| Name | Signature | Description |
|---|---|---|
| c_form_std | = (Param->KTypePol): { never involves |twisted_full_deform| } | never involves |twisted_full_deform| |
| twisted_c_form_std | = (Param->KTypePol): { c-form on extended group } | c-form on extended group |
| c_form_irreducible | (Param p) = KTypePol: | |
| twisted_c_form_irreducible_contributions | (Param p) = | |
| twisted_c_form_irreducible_unnormalized | (Param p) = KTypePol: | |
| twisted_c_form_irreducible | (Param p) = KTypePol: | |
| is_hermitian | (Param p) = bool: equivalent(twist(p),p) | |
| check_hermitian | (Param p, bool irreducible) = void: | |
| hermitian_form_irreducible | (Param p) = KTypePol: | |
| is_unitary | (Param p) = bool: | |
| map | ((Param->maybe_KTP)f, ParamPol P) = maybe_KTP: | |
| c_form_irreducible | (Param p,int time) = maybe_KTP: | |
| twisted_c_form_irreducible | (Param p,int time) = maybe_KTP: | |
| hermitian_form_irreducible | (Param p, int time) = maybe_KTP: | |
| is_unitary | (Param p, int time) = int: { -1: no, 0:timed out, 1: yes } | -1: no, 0:timed out, 1: yes |
| c_form_irreducible_long | (Param p) = | |
| twisted_c_form_irreducible_as_sum_of_standards | (Param p) = ParamPol: | |
| twisted_c_form_irreducible_long | (Param p) = | |
| twist_orbits | (ParamPol P) = ParamPol: | |
| print_twisted_c_form_irreducible_long | (Param p) = void: | |
| mixed | (Split w)= bool: not w.is_pure | |
| mixed_terms | (ParamPol P) = ParamPol: | |
| analyse | (ParamPol P) = void: | |
| hermitian_dual | (Param p) = Param: normal(twist(p)) | |
| hermitian_form_std | (Param p) = KTypePol: | |
| hermitian_form_irreducible | (Param p,KType t0) = KTypePol: | |
| hermitian_form_irreducible_long | (Param p) = | Hermitian form on an irreducible, with extra information |
| hermitian_form_irreducible_long | (Param p, KType t0) = | |
| print_hermitian_form_irreducible | (Param p) = void: | |
| print_hermitian_form_irreducible | ([Param] P) = void: | |
| print_hermitian_form_irreducible | (Param p,KType p0) = void: | |
| print_hermitian_form_irreducible | ([Param] P,KType p0) = void: | |
| print_hermitian_form_irreducible_long | (Param p) = void: | |
| print_hermitian_form_irreducible_long | (Param p,KType p0) = void: | |
| analyse_hermitian_form_irreducible | (Param p) = void: | |
| hermitian_form_is_pure | (Param p) = bool: | |
| print_is_unitary | (Param p) = void: | |
| is_weakly_unitary | (KTypePol P) = bool: let (,,mixed)=purity(P) in =mixed | |
| is_weakly_unitary | (Param p) = bool: | |
| test_line | (Param p) = void: | |
| weak_test | (Param p) = bool: | |
| strong_test | (Param p)=bool: | |
| branch_c_form_irreducible | p, int N) = KTypePol: |