CoxeterGroup( WF )
:Quite a few functions defined for domains, permutation groups or Coxeter groups have been implemented to work with Coxeter cosets.
Elements
, Random
, Representative
, Size
, in
:CoxeterGroup( WF )
.
ConjugacyClasses( WF )
:CoxeterGroup(WF)
. Then the classes are defined to be the
W-orbits on W F_0, where W acts by conjugation (they coincide
with the W F_0-orbits, W F_0 acting by the conjugation); by the
translation wmapsto wphi^{-1} they are sent to the
phi-conjugacy classes of W.
PositionClass( WF , x )
:i
such that x is an element of
ConjugacyClasses(WF)[i]
(to work fast, the classification of
Coxeter groups is used).
FusionConjugacyClasses( WF1, WF )
:CoxeterSubCoset
.
Print( WF )
:WF.name
is bound then this is printed, else
this function prints the coset in a form which can be input back
into GAP.
InductionTable( HF, WF )
:Harish-Chandra induction in the basis of almost characters:
gap> WF := CoxeterCoset( CoxeterGroup( "A", 4 ), (1,4)(2,3) ); CoxeterCoset(CoxeterGroup("A", 4), (1,4)(2,3)) gap> Display( InductionTable( CoxeterSubCoset( WF, [ 2, 3 ] ), WF ) );tt |
111 21 3 ________________ 11111tt |
1 . . 2111tt |
. 1 . 221tt |
1 . . 311tt |
1 . 1 32tt |
. . 1 41tt |
. 1 . 5tt |
. . 1
Lusztig induction from a diagonal Levi:
gap> HF := CoxeterSubCoset( WF, [1, 2], > LongestCoxeterElement( CoxeterGroup( WF ) ) );; gap> Display( InductionTable( HF, WF ) );tt |
111 21 3 _________________ 11111tt |
-1 . . 2111tt |
-2 -1 . 221tt |
-1 -2 . 311tt |
1 2 -1 32tt |
. -2 1 41tt |
. 1 -2 5tt |
. . 1
A descent of scalars:
gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "A", 2 ), (1,3)(2,4) ); CoxeterCoset(CoxeterGroup("A", 2, "A", 2), (1,3)(2,4)) gap> Display( InductionTable( CoxeterSubCoset( W, [ 1, 3 ] ), W ) );tt |
11 2 __________ 111tt |
1 . 21tt |
1 1 3tt |
. 1
CartanName( WF )
:phi
on the components is put in brackets if of length
k greater than 1, and is preceded by the order of phi^k on it,
if this is not 1. For example "2(A2xA2)"
denotes 2 components
of type A_2 permuted by F_0, and such that phi^2 induces the
non-trivial diagram automorphism on any of them, while 3D4
denotes
an orbit of length 1 on which phi is of order 3.
gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "G", 2, "A", 2 ), > (1,5,2,6) ); CoxeterCoset(CoxeterGroup("A", 2, "G", 2, "A", 2), (1,5,2,6)) gap> CartanName( W ); "2(A2xA2)xG2"
PrintDynkinDiagram( WF )
:CoxeterGroup(WF)
together with the
information how WF.phi
acts on it.
gap> W := CoxeterCoset( CoxeterGroup( "A", 2, "A", 2 ), (1,3,2,4) ); CoxeterCoset(CoxeterGroup("A", 2, "A", 2), (1,3,2,4)) gap> PrintDynkinDiagram( W ); phi permutes the next 2 components phi^2 acts as (1,2) on the component below A2 1 - 2 A2 3 - 4
ChevieClassInfo( WF )
, see the explicit description in
ChevieClassInfo for Coxeter cosets.
ChevieCharInfo
:
CharParams( WF )
CharName( WF )
Note that some functions for elements of a Coxeter group work naturally
for elements of a Coxeter coset: CoxeterWord
, PermCoxeterWord
,
CoxeterLength
, ReducedInCoxeterCoset
, LeftDescentSet
,
RightDescentSet
, etcldots
GAP 3.4.4