Let W⊂GL(V) be a complex reflection group on the vector space V. Let φ be an element of GL(V) which normalizes W. Then the coset Wφ is called a reflection coset.
A reference for these cosets is BMM99; the main motivation is that in the case where W is a rational reflection group (a Weyl group) such cosets, that we will call Weyl cosets, model rational structures on finite reductive groups. Finally, when W is a so-called Spetsial group, they are the basic object for the construction of a Spetses, which is an object attached to a complex reflection group from which one can derive combinatorially some attributes shared with finite reductive groups, like unipotent degrees, etc....
We say that a reflection coset is irreducible if W is irreducible. A general coset is a direct product of descents of scalars, which is the case where φ is transitive on the irreducible components of W. The irreducible cosets have been classified in BMM99: up to multiplication of φ by a scalar, there is usually only one or two possible cosets for a given irreducible group.
In CHEVIE we deal only with finite order cosets, that is, we assume there is a (minimal) integer δ such that (Wφ)δ=Wφ. Then the group generated by W and φ is finite, of order δ|W|.
A subset C of a Wφ is called a conjugacy class if one of the following equivalent conditions is fulfilled:
• C is the orbit of an element in Wφ under the conjugation action of W.
• C is a conjugacy class of ⟨ W,φ ⟩ contained in Wφ.
• The set {w∈ W| wφ∈ C} is a φ-conjugacy class of W (two elements v,w∈ W are called φ-conjugate, if and only if there exists x∈ W with v = xwφ(x-1)).
An irreducible character of ⟨ W,φ ⟩ has some non-zero values on Wφ if and only if its restriction to W is irreducible. Further, two characters χ1 and χ2 which have same irreducible restriction to W differ by a character of the cyclic group ⟨φ⟩ (which identifies to the quotient ⟨ W,φ ⟩/W). A set containing one extension to ⟨ W,φ ⟩ of each φ-invariant character of W is called a set of irreducible characters of Wφ. Two such characters are orthogonal for the scalar product on the class functions on Wφ given by
| ⟨ χ,ψ ⟩ := 1/| W|∑w∈ W χ(wφ)ψ(wφ). |
The character table of Wφ is the table of values of a set of irreducible characters on the conjugacy classes.
A subcoset Lwφ of Wφ is given by a reflection subgroup L of W and an element w of W such that wφ normalizes L.
We then have a natural notion of restriction of class functions on Wφ to class functions on Lwφ as well as of induction in the other direction. These maps are adjoint with respect to the scalar product defined above (see BMM99).
In CHEVIE the most general construction of a reflection coset is by
starting from a reflection datum, and giving in addition the matrix
phiMat of the map φ:V→ V (see the command
ReflectionCoset). However, at present, general cosets are only
implemented for groups represented as permutation groups on a set of roots,
and it is required that the automorphism given preserves this set up to a
scalar (it is allowed that these scalars depend on the pair of an
irreducible component and its image). If it also allowed to specify φ
by the permutation it induces on the roots; in this case it is assumed that
φ acts trivially on the orthogonal of the roots, but the roots could
be those of a parent group, generating a larger space. Thus in any case we
have a permutation representation of ⟨ W,φ ⟩ and we
consider the coset to be a set of permutations.
Reflection cosets are implemented in CHEVIE by a record which points to a
reflection group record and has additional fields holding phiMat and the
corresponding permutation phi. In the general case, on each component of
W which is a descent of scalars, phiMat will permute the components and
differ by a scalar on each component from an automorphism which preserves
the roots. In this case, we have a permutation phi and a scalar which
is stored for that component.
The most common situation where cosets with non-trivial phi arise is as
sub-cosets of reflection groups. Here is an ``exotic'' example, see the
next chapter for more classical examples involving Coxeter groups.
gap> W:=ComplexReflectionGroup(14);
ComplexReflectionGroup(14)
gap> PrintDiagram(W);
G14 1--8--2(3)
gap> R:=ReflectionSubgroup(W,[2,4]);
ReflectionSubgroup(ComplexReflectionGroup(14), [ 2, 4 ])
gap> PrintDiagram(R);
G5(ER(6)) 2(3)==4(3)
gap> Rphi:=ReflectionCoset(R,W.1);
2G5(ER(6))<2,4>
gap> PrintDiagram(Rphi);
phi acts as (2,4) on the component below
G5(ER(6)) 2(3)==4(3)
gap> ReflectionDegrees(Rphi);
[ [ 6, 1 ], [ 12, -1 ] ]
The last line shows for each reflection degree the corresponding factor of the coset, which is the scalar by which φ acts on the corresponding fundamental reflection invariant. The factors characterize the coset.
The variable CHEVIE.PrintSpets determines if a coset is printed in an
abbreviated form which describes its type, as above (G5 twisted by 2,
with a Cartan matrix which differs from the standard one by a factor of
√ 6), or in a form which could be input back in GAP3. The above
example was for the default value CHEVIE.PrintSpets=rec(). With the same
data we have:
gap> CHEVIE.PrintSpets:=rec(GAP:=true);;
gap> Rphi;
Spets(ReflectionSubgroup(ComplexReflectionGroup(14), [ 2, 4 ]), (1,3)(\
2,4)(5,9)(6,10)(7,11)(8,12)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(\
19,27)(20,28)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(\
37,49)(38,50)(39,51)(40,52)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(\
59,77)(60,78)(62,79)(64,80)(65,81)(66,82)(67,69)(68,70)(83,100)(84,101\
)(85,102)(87,103)(89,99)(90,97)(91,98)(92,96)(93,104)(94,95)(105,113)(\
106,114)(109,111)(110,112)(115,118)(116,117)(119,120))
gap> CHEVIE.PrintSpets:=rec();;
ReflectionCoset( W[, phiMat ] )
ReflectionCoset( W[, phiPerm] )
This function returns a reflection coset as a GAP3 object. The argument
W must be a reflection group (created by ComplexReflectionGroup,
CoxeterGroup, PermRootGroup or ReflectionSubgroup). In the first form
the argument phiMat must be an invertible matrix with Rank(W) rows,
which normalizes the parent of W (if any) as well as W. In the second
form phiPerm is a permutation which describes the images of the roots
under phi (only the image of the roots corresponding to the generating
reflections need be given, since they already determine a unique phiMat).
This second form is only allowed if the semisimple rank of W equals the
rank (i.e., the roots are a basis of V). If there is no second argument
the default for phiMat is the identity matrix, so the result is the
trivial coset equal to W itself.
ReflectionCoset returns a record from which we document the following
components:
isDomain, isFinite:
group:
phiMat:
phi:phiMat.
gap> W := CoxeterGroup("A",3);;
gap> Wphi := ReflectionCoset( W, (1,3));
2A3
gap> m:=MatXPerm(W,(1,3));
[ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ]
gap> ReflectionCoset( W,m);
2A3
Spets is a synonym for ReflectionCoset. See ReflectionCoset.
ReflectionSubCoset( WF, r, [w] )
Returns the reflection subcoset of the reflection coset WF generated by
the reflections specified by r. r is a list of indices specifying a
subset of the roots of W where W is the reflection group Group(WF).
If specified, w must be an element of W such that w*WF.phi
normalizes up to scalars the subroot system generated by r. If absent,
the default value for w is (). If the subroot system is not normalized
then false is returned, with a warning message if InfoChevie=Print.
gap> W:=ComplexReflectionGroup(14);
ComplexReflectionGroup(14)
gap> Wphi:=ReflectionCoset(W);
G14
gap> ReflectionSubCoset(Wphi,[2,4],W.1);
2G5(ER(6))<2,4>
gap> WF:=ReflectionCoset(CoxeterGroup("A",4),(1,4)(2,3));
2A4
gap> ReflectionSubCoset(WF,[2,3]);
2A2<2,3>.(q-1)(q+1)
gap> ReflectionSubCoset(WF,[1,2]);
#I permutation for F0 must normalize set of roots.
false
SubSpets is a synonym for ReflectionSubCoset. See ReflectionSubCoset.
95.5 Functions for Reflection cosets
Group( WF ):Quite a few functions defined for domains, permutation groups or reflection groups have been implemented to work with reflection Cosets.
Size, Rank, SemisimpleRank:Group( WF ).
Elements, Random, Representative, in:Group( WF ) and multiply the
result by WF.phi.
ConjugacyClasses( WF ):Group(WF). Then the classes are defined to be the W-orbits on W
φ, where W acts by conjugation (they coincide with the W
φ-orbits, W φ acting by the conjugation); by the translation w
→ wφ-1 they are sent to the φ-conjugacy classes of W.
PositionClass( WF , x ):i such that x is an element of
ConjugacyClasses(WF)[i] (to work fast, the classification of
reflection groups is used).
FusionConjugacyClasses( WF1, WF ):ReflectionSubCoset.
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 GAP3.
InductionTable( HF, WF ):
gap> W := CoxeterGroup( "A", 4 );;
gap> Wphi := ReflectionCoset( W, (1,4)(2,3) );
2A4
gap> Display(InductionTable(ReflectionSubCoset(Wphi,[2,3 ]),Wphi));
Induction from 2A2<2,3>.(q-1)(q+1) to 2A4
|111 21 3
________________
11111 | 1 . .
2111 | . 1 .
221 | 1 . .
311 | 1 . 1
32 | . . 1
41 | . 1 .
5 | . . 1
InductionTable and FusionConjugacyClasses work only between cosets.
If the parent coset is the trivial coset it should still be given as
a coset and not as a group:
gap> Wphi:=ReflectionCoset(W);
A4
gap> L:=ReflectionSubCoset(Wphi,[2,3],LongestCoxeterElement(W));
A2<2,3>.(q-1)(q+1)
gap> InductionTable(L,W);
Error, A2<2,3>.(q-1)(q+1) is a coset but CoxeterGroup("A",4) is not in
S.operations.FusionConjugacyClasses( S, R ) called from
FusionConjugacyClasses( u, g ) called from
InductionTable( L, W ) called from
main loop
brk>
gap> InductionTable(L,Wphi);
InductionTable(A2<2,3>.(q-1)(q+1), A4)
ReflectionName( WF ):phi on the components is put in brackets if of length k
greater than 1, and is preceded by the order of phik on it, if this
is not 1. For example "2(A2xA2)" denotes 2 components of type
A2 permuted by φ, and such that phi2 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:=ReflectionCoset(CoxeterGroup("A",2,"G",2,"A",2),(1,5,2,6));
2(A2xA2)<1,2,5,6>xG2<3,4>
gap> ReflectionName( W );
"2(A2xA2)<1,2,5,6>xG2<3,4>"
PrintDiagram( WF ):ReflectionGroup(WF) together with the
information how WF.phi acts on it. Going from the above example:
gap> PrintDiagram( W );
phi permutes the next 2 components
phi^2 acts as (1,2) on the component below
A2 1 - 2
A2 5 - 6
G2 3 >>> 4
ChevieClassInfo( WF ), see the explicit description in
ChevieClassInfo for Reflection cosets.
CharParams( WF ):CharNames has also a special version for cosets.
GenericOrder( WF, q ):TorusOrder(WF,i,q) which is the
same as
GenericOrder(SubSpets(WF,[],Representative(ConjugacyClasses(WF)[i]))).
Note that some functions for elements of a Reflection group work
naturally for elements of a Reflection coset:
EltWord, ReflectionLength, ReducedInRightCoset, etc...
95.6 ChevieCharInfo for reflection cosets
ChevieCharInfo( WF )
ChevieCharInfo gives for a reflection coset WF a record similar to what
it gives for the corresponding group W, excepted that some fields which
do not make sense are omitted, and that two fields record information
allowing to relate characters of the coset to that of the group:
charRestriction:
nrGroupClasses:NrConjugacyClasses(Group(WF)).
gap> ChevieCharInfo(RootDatum("3D4"));
rec(
extRefl := [ 1, 5, 4, 6, 2 ],
charparams := [ [ [ [ ], [ 4 ] ] ], [ [ [ ], [ 1, 1, 1, 1 ] ] ],
[ [ [ ], [ 2, 2 ] ] ], [ [ [ 1, 1 ], [ 2 ] ] ],
[ [ [ 1 ], [ 3 ] ] ], [ [ [ 1 ], [ 1, 1, 1 ] ] ],
[ [ [ 1 ], [ 2, 1 ] ] ] ],
charRestrictions := [ 13, 4, 10, 5, 11, 3, 6 ],
nrGroupClasses := 13,
b := [ 0, 12, 4, 4, 1, 7, 3 ],
B := [ 0, 12, 8, 8, 5, 11, 9 ],
charnames := [ ".4", ".1111", ".22", "11.2", "1.3", "1.111", "1.21"
],
positionId := 1,
positionDet := 2,
a := [ 0, 12, 3, 3, 1, 7, 3 ],
A := [ 0, 12, 9, 9, 5, 11, 9 ] )
95.7 ReflectionType for reflection cosets
ReflectionType( WF )
returns the type of the Reflection coset WF. This consists of a list of
records, one for each orbit of WF.phi on the irreducible components of
the Dynkin diagram of Group(WF), which have two fields:
orbit:ReflectionType
for an irreducible untwisted reflection group. The components are
ordered according to the action of WF.phi, so WF.phi maps the
generating permutations with indices in the first type to indices in the
second type in the same order as stored in the type, etc ...
phi:WF.phik on
the simple roots of the first irreducible component in the orbit.
gap> W:=ReflectionCoset(CoxeterGroup("A",2,"A",2), (1,3,2,4));
2(A2xA2)
gap> ReflectionType( W );
[ rec(orbit := [ rec(rank := 2,
series := "A",
indices := [ 1, 2 ]), rec(rank := 2,
series := "A",
indices := [ 3, 4 ]) ],
twist := (1,2)) ]
95.8 ReflectionDegrees for reflection cosets
ReflectionDegrees( WF )
Let W be the Reflection group corresponding to the Reflection coset WF,
and let V be the vector space of dimension W.rank on which W acts as
a reflection group. Let f1,...,fn be the basic invariants of W on
the symmetric algebra SV of V; they can be chosen so they are
eigenvectors of the matrix WF.phiMat. The corresponding eigenvalues are
called the factors of φ acting on V; they characterize the coset
--- they are equal to 1 for the trivial coset. The generalized degrees of
WF are the pairs formed of the reflection degrees and the corresponding
factor.
gap> W := CoxeterGroup( "E", 6 );; WF := ReflectionCoset( W );
E6
gap> phi := EltWord( W,[6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1]);;
gap> HF := ReflectionSubCoset( WF, [ 2..5 ], phi );;
gap> PrintDiagram( HF );
phi acts as (2,3,5) on the component below
D4 2
\
4 - 5
/
3
gap> ReflectionDegrees( HF );
[ [ 1, E(3) ], [ 1, E(3)^2 ], [ 2, 1 ], [ 4, E(3) ], [ 6, 1 ],
[ 4, E(3)^2 ] ]
Twistings( W, L )
W should be a Reflection group record or a Reflection coset record, and
L should be a reflection subgroup of W (or of Group(W) for a coset),
or a sublist of the generating reflections of W (resp. Group(W)), in
which case the call is the same as
Twistings(W,ReflectionSubgroup(W,L)) (resp.
Twistings(W,ReflectionSubgroup(Group(W),L))).
The function returns a list of representatives, up to W-conjugacy, of reflection sub-cosets of W whose reflection group is L.
gap> W:=ComplexReflectionGroup(3,3,4);
ComplexReflectionGroup(3,3,4)
gap> Twistings(W,[1..3]);
[ G333.(q-1), 3'G333<1,2,3,76>.(q-E3^2), 3G333<1,2,3,76>.(q-E3) ]
95.10 ChevieClassInfo for Reflection cosets
ChevieClassInfo( WF )
returns information about the conjugacy classes of the Reflection coset
WF. The result is a record with three components: classtext contains
a list of reduced words for the representatives in
ConjugacyClasses(WF), classnames contains corresponding names for the
classes, and classparams gives corresponding parameters for the classes.
gap> W:=ReflectionCoset(ComplexReflectionGroup(14));
G14
gap> Rphi:=ReflectionSubCoset(W,[2,4],Group(W).1);
2G5(ER(6))<2,4>
gap> ChevieClassInfo(Rphi);
rec(
classtext :=
[ [ ], [ 2, 4, 4, 2, 4, 4, 2 ], [ 2, 4, 4, 2, 4, 4, 2, 2 ],
[ 2 ], [ 2, 2, 4, 2, 2 ], [ 2, 2, 4, 4, 2, 2 ], [ 2, 4 ],
[ 2, 4, 2 ], [ 4, 2, 4, 2 ] ],
classes := [ 12, 6, 6, 6, 12, 6, 6, 6, 12 ],
orders := [ 2, 24, 24, 24, 6, 8, 24, 8, 6 ],
classnames := [ "", "1221221", "12212211", "1", "11211", "112211",
"12", "121", "2121" ] )
95.11 CharTable for Reflection cosets
CharTable( WF )
This function returns the character table of the Reflection coset WF (see also the introduction of this Chapter). We call ``characters'' of the Reflection coset WF with corresponding Reflection group W the restriction to W φ of a set containing one extension of each φ-invariant character of W to the semidirect product of W with the cyclic group generated by φ. The choice of extension is always the same for a given coset, but rather arbitrary in general; for Weyl cosets it is the "preferred extension" of Lusztig.
The returned record contains almost all components present in the character
table of a Reflection group. But if φ is not trivial then there are no
components powermap (since powers of elements in the coset need not be in
the coset) and orders (if you really need them, use MatXPerm to
determine the order of elements in the coset).
gap> W := ReflectionCoset( CoxeterGroup( "D", 4 ), (1,2,4) );
3D4
gap> Display( CharTable( W ) );
3D4
2 2 2 2 2 2 3 3
3 1 1 1 . . 1 1
C3 ~A2 C3+A1 ~A2+A1 F4 ~A2+A2 F4(a1)
.4 1 1 1 1 1 1 1
.1111 -1 1 1 -1 1 1 1
.22 . 2 2 . -1 -1 -1
11.2 . . . . -1 3 3
1.3 1 1 -1 -1 . -2 2
1.111 -1 1 -1 1 . -2 2
1.21 . 2 -2 . . 2 -2
gap3-jm