hermitian.at

c_form_std

write c-invariant form on standard module as a KTypePol

never involves |twisted_full_deform|

built-in

twisted_c_form_std

c-form on extended group

built-in

c_form_irreducible

untwisted |c_form_irreducible|, for efficiency; only used in equal rank case

c-invariant form on irreducible module of p as KTypePol

also in this file at line 97

twisted_c_form_irreducible_contributions

utility: compute fixed and unfixed contributions to c-form on irreducible(p)
no deformation is done, but orientation numbers are included;
fixed contributions are already halved; for unfixed ones this needs to wait

(fixed part, 2*unfixed part)

twisted_c_form_irreducible_unnormalized

The following function implements algorithm from Vogan in email 9/22/16:
* for each delta-fixed term q add c_form_std(q) (fully twisted-deformed)
* for each pair {q,delta(q)} add q*0 (deformation to nu=0 of q).
For second kind it it convenient to add q*0 and delta(q)*0
separately and at the end divide the sum of such contributions by 2.

twisted c-invariant form on an irreducible in terms of standardrepKs.

twisted_c_form_irreducible

normalization here means making the leading term 1 (rather than |s|)

also in this file at line 105

is_hermitian

whether J(p) admits an invariant Hermitian form.

check_hermitian

hermitian_form_irreducible

Hermitian form on a irreducible module, normalization from its initial term.

also in this file at line 118, line 253

is_unitary

compute Hermitian form on p, and report if it is unitary

also in this file at line 127

maybe_KTP type

set_type maybe_KTP = (void timed_out | KTypePol done)

Fields: timed_out, done

the next type is the same as |Maybe<KTypePol>|, but injector names differ

map

also defined in basic.at

c_form_irreducible

also in this file at line 15

twisted_c_form_irreducible

also in this file at line 56

hermitian_form_irreducible

also in this file at line 74, line 253

is_unitary

-1: no, 0:timed out, 1: yes

also in this file at line 81

c_form_irreducible_long

same as |c_form_irreducible|, but also return a second component that
exposes the linear combination of contributions from standard representations
that produced the result (parameter, coefficient, c_form on this standard)

formulas: write
  J(y)  =sum_x               (-1)^{ell(x)-ell(y)}P(x,y)(q=1)I(x)
then
  J(y)_c=sum_x (-1)^{on(x,y)}(-1)^{ell(x)-ell(y)}P(x,y)(q=s)I(x)_c
where
  P(x,y) is a cumulated KL polynomial
  (-1)^{ell(x)-ell(y)}P(x,y) is given by signed_KL_col(y)[i] with
  indices[i]=x on(x,y)=orientation number given by orientation_nr_term()
  I(x)_c given as combination of standards x' with nu(x')=0 by c_form_std(x)
algorithm: compute the sum for J(y)_c, using signed_KL_col and c_form_std

c-form on an irreducible, with extra output.

twisted_c_form_irreducible_as_sum_of_standards

starting formula in the c-form calculation: J(p)_c=\sum w(q)I(q)_c

add twisted and untwisted

twisted_c_form_irreducible_long

c_form_irrecible with extra information.

twist_orbits

print_twisted_c_form_irreducible_long

mixed

print only terms with "mixed" coefficient (a+bs), i.e., both a,b\ne 0

mixed_terms

analyse

hermitian_dual

Hermitian dual.

hermitian_form_std

Hermitian form on a standard module, canonical normalization.

hermitian_form_irreducible

Hermitian form on a irreducible module, normalization from p0.

also in this file at line 74, line 118

hermitian_form_irreducible_long 2 overloads

Hermitian form on an irreducible, with extra information.

print_hermitian_form_irreducible 4 overloads

nice output of hermitian_form_irreducible.

nice output of Hermitian forms on list of parameters.

print_hermitian_form_irreducible_long 2 overloads

print Hermitian form on irreducible, plus extra information.

analyse_hermitian_form_irreducible

cut the red tape and tell about the hermitian form analysis directly

hermitian_form_is_pure

compute |p.hermitian_form_irreducible.is_pure|; try to get |false| rapidly

print_is_unitary

is_weakly_unitary 2 overloads

no mixed terms

compute Hermitian form on p, and report if it is weakly unitary

test_line

weak_test

strong_test

branch_c_form_irreducible