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

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 *χ _{1}* and

⟨ 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
- Spets
- ReflectionSubCoset
- SubSpets
- Functions for Reflection cosets
- ChevieCharInfo for reflection cosets
- ReflectionType for reflection cosets
- ReflectionDegrees for reflection cosets
- Twistings
- ChevieClassInfo for Reflection cosets
- CharTable for Reflection cosets

`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(`

rows,
which normalizes the parent of `W`)`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`

:

true

`group`

:

the group`W`

`phiMat`

:

the matrix acting on*V*which represents*φ*.

`phi`

:

the permutation on the roots of`W`induced by`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(`

.
If specified, `WF`)`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.

`Group(`

:`WF`)

returns the reflection group of which`WF`is a coset.

Quite a few functions defined for domains, permutation groups or reflection groups have been implemented to work with reflection Cosets.

`Size`

,`Rank`

,`SemisimpleRank`

:

these functions use the corresponding functions for`Group(`

.`WF`)`Elements`

,`Random`

,`Representative`

,`in`

:

these functions use the corresponding functions for`Group(`

and multiply the result by`WF`)`WF.phi`

.

`ConjugacyClasses(`

:`WF`)

returns the conjugacy classes of the coset`WF`(see the introduction of this Chapter). Let`W`be`Group(`

. Then the classes are defined to be the`WF`)*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*they are sent to the^{-1}*φ*-conjugacy classes of*W*.

`PositionClass(`

:`WF`,`x`)

for any element`x`in`WF`this returns the number`i`

such that`x`is an element of`ConjugacyClasses(`

(to work fast, the classification of reflection groups is used).`WF`)[i]

`FusionConjugacyClasses(`

:`WF1`,`WF`)

works in the same way as for groups. See the section`ReflectionSubCoset`

.

`Print(`

:`WF`)

if

is bound then this is printed, else this function prints the coset in a form which can be input back into`WF`.name**GAP3**.

`InductionTable(`

:`HF`,`WF`)

works in the same way as for groups. It gives the induction table from the Reflection subcoset`HF`to the Reflection coset`WF`. If*H wφ*is a Reflection subcoset of*W φ*, restriction of characters is defined as restriction of functions from*W φ*to*H wφ*, and induction as the adjoint map for the natural scalar product*⟨ f, g⟩ =1/| W|∑*._{v∈ W}f(v φ)g(v φ)

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`)

returns a string which describes the isomorphism type of the group*W⋊⟨ F⟩*, associated to`WF`, as described in the introduction of this Chapter. An orbit of*φ=*`WF`.`phi`

on the components is put in brackets if of length*k*greater than*1*, and is preceded by the order of*phi*on it, if this is not^{k}*1*. For example`"2(A2xA2)"`

denotes 2 components of type*A*permuted by_{2}*φ*, and such that*phi*induces the non-trivial diagram automorphism on any of them, while^{2}`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`)

this is a purely descriptive routine (as was already the case for finite Reflection groups themselves). It prints the Dynkin diagram of`ReflectionGroup(`

together with the information how`WF`)

acts on it. Going from the above example:`WF`.phi

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( `

, see the explicit description in
ChevieClassInfo for Reflection cosets.
`WF` )

`CharParams(`

:`WF`)

This returns appropriate labels for the characters of the ReflectionCoset.`CharName`

has also a special version for cosets.

`GenericOrder(`

:`WF`,`q`)

Returns the generic order of the associated algebraic group (for a Weyl coset) or Spetses, using the generalized reflection degrees. We also have`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*...*

`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`

:

records for each character of`WF`the index of the character of`W`of which it is an extension.

`nrGroupClasses`

:

records`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 ], positionId := 1, positionDet := 2, a := [ 0, 12, 7, 1, 3, 3 ], A := [ 0, 12, 11, 5, 9, 9 ], charnames := [ ".4", ".1111", ".22", "11.2", "1.3", "1.111", "1.21" ] )

`ReflectionType( `

`WF` )

returns the type of the Reflection coset `WF`. This consists of a list of
records, one for each orbit of

on the irreducible components of
the Dynkin diagram of `WF`.phi`Group(`

, which have two fields:`WF`)

`orbit`

:

is a list of types of the irreducible components in the orbit. These types are the same as returned by the function`ReflectionType`

for an irreducible untwisted reflection group. The components are ordered according to the action of

, 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`WF`.phi*...*

`phi`

:

if*k*is the number of irreducible components in the orbit, this is the permutation which describes the action of`WF`.phion the simple roots of the first irreducible component in the orbit.^{k}

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)) ]

`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 *f _{1},...,f_{n}* be the basic invariants of

`WF.phiMat`

. The corresponding eigenvalues are
called the

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(`

for a coset),
or a sublist of the generating reflections of `W`)`W` (resp. `Group(W)`

), in
which case the call is the same as
`Twistings(`

(resp.
`W`,ReflectionSubgroup(`W`,`L`))`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) ]

`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" ] )

`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

24 Apr 2021