extended_misc.at

x_rep type

set_type x_rep = (ratvec v, mat theta)

Fields: v, theta

ExtParam type

set_type ExtParam =
  ( InnerClass ic, mat delta
  , ratvec gamma, vec lambda_rho, mat theta
  , ratvec g, vec l, mat omega
  , vec tau, vec t
  )

Fields: ic, delta, gamma, lambda_rho, theta, g, l, omega, tau, t

canonical_g 4 overloads

canonical_g(x) returns a dominant regular element g such that e(g)=x^2
This only depends on the real_form(x), in fact only on its square class
the implementation depends on base_grading_vector(G) being dominant.
An alternative that does not depend on that (recently ensured) circumstance:
canonical_g(RealForm G)= ratvec: shift_g_regular(G,square(G))

cross

also defined in basic.at, coherent_irreducible.at, synthetic.at, coherent_characters.at

Cayley

also defined in basic.at

Cayley_set 3 overloads

Cayley transform as a set (actually an array, but order doesn't matter)

doubly iterated Cayley

uniform syntax for single or iterated Cayley

twist 2 overloads

action of delta on KGB for general distinguished delta
if delta=distinguished_involution(real_form(x)) then twist(delta,x)=twist(x)

also defined in basic.at, nilpotent_orbits.at

is_fixed 4 overloads

whether KGBElt (pair) is fixed by twist by matrix delta

also defined in synthetic.at

in_A

in_A@x_rep: test if e(v) is in A:=(H^{-theta})^0 (validity test)
concretely: whether rational vector (1+theta)v lies in lattice (1+theta)X_*

is_equivalent

$H$-conjugation of elements determined by |x_rep| values
implicit: $\exp_1(v_i)\theta_i=\exp_1(v_i)\sigma_i*\xi_0$ (for $i=1,2$)
in same extended group for $G$
test: $\theta_1=\theta_2$, and $\exp_1(v_1-v_2)\in A=(H^{\theta_1})^0$

square_is_central

use the function defined in parameters.at

also defined in synthetic.at, parameters.at

root_datum

also defined in basic.at, W_reps.at, K_Nilpotent.at, sommers.at, parabolics.at, K.at, nilpotent_orbits.at, W_classes.at, character_tables.at, springer_tables.at, disconnected.at, cyclotomic_Lie_algebra.at, bracket.at, exceptional.at

rho

also defined in basic.at, nilpotent_orbit_partitions.at, nilpotent_orbits.at

rho_check

also defined in basic.at

dual_parameter

log_z

in PFTR: z=i^<tau,(delta_0-1)l>(-1)^<lambda,t>
PFTR:delta_0 is ^delta here

modulo 1

also in this file at line 121

log_zeta

modulo 1

also in this file at line 123

log_z

also in this file at line 116

log_zeta

also in this file at line 118

x

also defined in basic.at

y

assumes integral |E.gamma|

parameter

recover ordinary Param value from ExtParam; could be defined as
set parameter (ExtParam E) = Param: parameter(x(E),y(E),E.gamma)

actually this is better: use |(E.x,E.lambda,E.gamma)|

also defined in basic.at, L_packet.at, K.at, parameters.at

integrality_datum

also defined in basic.at, nilpotent_orbits.at

translate_to_dominant

assume gamma(E) is integrally dominant
translate by sum of roots to make it dominant
return (new extended parameter, sum of roots)
see nonintegral.at
*** should not need this any more, now that gamma is allowed to be
    integrally dominant
However, since not all functions allow that, it is still needed

also defined in Weylgroup.at

valid

for debugging:
test that an extended parameter satisfies the defining conditions

fixed

select delta-fixed elements from B

filter delta-fixed elements

fixed_block_of

select delta-fixed elements from block_of(p)

fixed_block

format

also defined in cyclotomic.at, polynomial.at, cyclotomicMat.at, bracket.at

short_format

also defined in G2_unitary_dual.at

long_format

display

also defined in extParamPol.at

display_one_line