Weylgroup.at

sgn

also defined in misc.at

length

also defined in K.at, extended.at

lengthens 2 overloads

whether multiplication by simple reflection lengthens or shortens M

for right multiplication

for left multiplication

also defined in bruhat.at

from_simple

write given root (assumed positive) as |w*beta| with |beta| simple

root_reflection

make_dominant_using_roots 2 overloads

|gamma| is integrally dominant, need to add a sum of roots to it to make it
dominant; this does not change the integral Weyl chamber of |gamma|,
so when acting on parameters it is another parameter with the same types of
roots, generating an isomorphic block
this algorithm is crude: it adds a big enough multiple of |2*rho(rd)|

if vector was in $X^*$ it remains so, so return it as |vec|

dom

shift

translate_to_dominant

assume infinitesimal character |gamma| of |p| is integrally dominant
translate by sum of roots to make it dominant
return (new parameter, sum of roots)

dom

shift

also defined in extended_misc.at

integrally_from_dominant

integrally_dominant

invariant_form_matrix

Each coroot alpha^v defines defines a rank-1 bilinear form on X^* by
multiplying its evaluations at the pair of argument weights. Since alpha^v
and -\alpha^v in this way give the same form, one can obtain a W-invariant
biliear form by summing these rank-1 forms over all positive coroots; the
radical of this form is the coradical of the root datum. In formula
$$
  (\mu,\nu)=\sum_{\alpha^v\in\Delta^{\vee+}} <\alpha^v,\mu><\alpha^v\nu>
$$
The summation is conveniently done by multiplying a wide matrix M by ^M

invariant_form

nondegenerate_invariant_form

co_invariant_form_matrix

a W-invariant form on X_*\otimes Q; not directly related to invariant_form

co_invariant_form

is_elliptic

|eigen_lattice| empty means no eigenvalue 1

is_relatively_elliptic

a nonzero central torus makes it impossible to be elliptic; therefore a more
practical condition, the weaker |is_relatively_ellptic|, ignores that torus

from_no_Cminus

use complex cross actions to find a KGBElt x0 in a fiber with no C- roots
by descending through complex descents until none are left
return (w,x0) where x=cross(w,x0)

also in this file at line 119; also defined in Levi_subgroups.at

from_no_Cplus

the same, but ascending through complex descents until no C+ roots are left

also in this file at line 123; also defined in Levi_subgroups.at

from_no_Cminus

the same for parameters

also in this file at line 107; also defined in Levi_subgroups.at

from_no_Cplus

also in this file at line 113; also defined in Levi_subgroups.at

from_dlambda_dom

to_no_Cminus

also in this file at line 134; also defined in K.at

to_no_Cplus

also in this file at line 135

to_no_Cminus

also in this file at line 132; also defined in K.at

to_no_Cplus

also in this file at line 133

* 2 overloads

action of invertible matrix on RootDatum

also defined in basic.at, extParamPol.at, complex.at, modules.at, hodge_tensor.at, tits_centralizer.at, stable.at, synthetic.at, group_operations.at, ratmat.at, tits.at, twisted_root_datum.at, cyclotomic.at, polynomial.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, hodge_functions.at, sparse.at, lazy_lists.at

order_W_simple 2 overloads

order_W 2 overloads

also defined in character_tables.at

factors

factor w into elements for subgroups of W(rd) for simple factors of rd

also defined in basic.at

factor_strong

w in rd.W, factor_strong(w)=[w_1,...,w_n]:
w_i is in very_simple_factor(rd)[i]
these are distinct root data and not directly related

char_poly

also defined in basic.at

coxeter_element

Coxeter element of W

coxeter_number_simple

Coxeter number

coxeter_number

coxeter_torus_element

torus element representing the Coxeter Weyl group element

lift_sc

lift Weyl group element to simply connected group

w 2 overloads

also defined in complex.at

is_long

is_short

root_lengths