In this chapter we describe functions for dealing with general Coxeter groups.
A suitable reference for the general theory is, for example, the volume of Bourbaki Bou68.
A Coxeter group is a group which has the presentation W= 〈
s1, ..., sn | (si sj)m(i,j)=1 〉 for some
symmetric integer matrix m(i,j) called the Coxeter matrix, where
m(i,j)>1 for i ≠ j and m(i,i)=1. It is true (but a non-trivial
theorem) that in a Coxeter group the order of si sj is exactly
m(i,j), thus a Coxeter group is the same as a Coxeter system, that is a
pair (W,S) of a group W and a set S={s1,...,sn} of
involutions, such that the group is presented by relations describing the
order of the product of two generating elements. A Coxeter group has a
natural representation on a real vector space V of dimension the number
of generators, where each generator acts as a reflection, its reflection
representation (see CoxeterGroupByCoxeterMatrix
); the faithfulness of
this representation in the main argument to prove that the order of si
sj is exactly m(i,j). Thus Coxeter groups are real reflection groups.
The converse need not be true if the set of reflecting hyperplanes has bad
topological properties, but it turns out that finite Coxeter groups are the
same as finite real reflection groups. The possible Coxeter matrices for
finite Coxeter groups have been completely classified; the corresponding
finite groups play a deep role in several areas of mathematics.
Coxeter groups have a nice solution to the word problem. The length
l(w) of an element w of W is the minimum number of elements of S of
which it is a product (since the elements of S are involutions, we do not
need inverses). An expression of w of minimal length is called a reduced
word for w. The main property of reduced words is the exchange lemma
which states that if s1... sk is a reduced word for w where k=l(w)
and s∈ S is such that l(sw) ≤ l(w) then one of the si in the word
for w can be deleted to obtain a reduced word for sw. Thus given s∈
S and w∈ W, either l(sw)=l(w)+1 or l(sw)=l(w)-1 and we say in this
last case that s belongs to the left descent set of w. The
computation of a reduced word for an element, and other word problems, are
easily done if we know the left descent sets. For most Coxeter groups that
we will be able to build in CHEVIE, this left descent set can be easily
determined (see e.g. CoxeterGroupSymmetricGroup
below), so this suggests
how to deal with Coxeter groups in CHEVIE. They are reflection groups,
so the following fields are defined in the group record:
.nbGeneratingReflections
:
.reflections
:W.reflections{[1..W.nbGeneratingReflections]}
is the set S.
the above names are used instead of names like CoxeterGenerators
and
CoxeterRank
since the Coxeter groups are reflection groups and we
want the functions for reflection groups applicable to them (similarly,
if you have read the chapter on reflections and reflection groups, you
will realize that there is also a field .OrdersGeneratingReflections
which contains only 2's). The main additional function which allows to
compute within Coxeter groups is:
.operations.IsLeftDescending(W,w,i)
:
For Coxeter groups constructed in CHEVIE an IsLeftDescending
operation
is provided, but you can construct your own Coxeter groups just by filling
the above fields (see the function CoxeterGroupSymmetricGroup
below for
an example). It should be noted than you can make into a Coxeter group
any kind of group: finitely presented groups, permutation groups or
matrix groups, if you fill appropriately the above fields; and the given
generating reflection do not have to be W.generators
--- all functions
for Coxeter group and Hecke algebras will then work for your Coxeter groups
(using your function IsLeftDescending
).
A common occurrence in CHEVIE code for Coxeter groups is a loop like:
First([1..W.semisimpleRank],x->IsLeftDescending(W,w,x))
which for a reflection subgroup becomes
First(W.rootRestriction{[1..W.semisimpleRank]},x->IsLeftDescending(W,w,x))
where the overhead is quite large, since dispatching on the group type is
done in IsLeftDescending
. To improve this code, if you provide a function
FirstLeftDescending(W,w)
it will be called instead of the above loop (if
you do not provide one the above loop will be used). Such a function
provided by CHEVIE for finite Coxeter groups represented as permutation
groups of the roots is 3 times more efficient than the above loop.
Because of the easy solution of the word problem in Coxeter groups,
a convenient way to represent their elements is as words in the
Coxeter generators. They are represented in CHEVIE as lists of
labels for the generators. By default these labels are given as
the index of a generator in S, so a Coxeter word is just a
list of integers which run from 1 to the length of S. This
can be changed to reflect a more conventional notation for some
groups, by changing the field .reflectionsLabels
of the Coxeter group
which contains the labels used for the Coxeter words (by default it
contains [1..W.nbGeneratingReflections]
). For a Coxeter group with 2
generators, you could for instance set this field to "st"
to use
words such as "sts"
instead of [1,2,1]
. For reflection subgroups,
this is used in CHEVIE by setting the reflection labels to the
indices of the generators in the set S of the parent group (which is
given by .rootInclusion
).
The functions CoxeterWord
and EltWord
will do the conversion between
Coxeter words and elements of the group.
gap> W := CoxeterGroup( "D", 4 );; gap> p := EltWord( W, [ 1, 3, 2, 1, 3 ] ); ( 1,14,13, 2)( 3,17, 8,18)( 4,12)( 5,20, 6,15)( 7,10,11, 9)(16,24) (19,22,23,21) gap> CoxeterWord( W, p ); [ 1, 3, 1, 2, 3 ] gap> W.reflectionsLabels:="stuv"; "stuv" gap> CoxeterWord(W,p); "sustu"
We notice that the word we started with and the one that we
ended up with, are not the same. But of course, they represent
the same element of W. The reason for this difference is that
the function CoxeterWord
always computes a reduced word which is
the lexicographically smallest among all possible expressions of an
element of W as a word in the fundamental reflections. The function
ReducedCoxeterWord
does the same but with a word as input (rather than
an element of the group). Below are some other possible computations
with the same Coxeter group as above:
gap> LongestCoxeterWord( W ); # the (unique) longest element in W [ 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4 ] gap> w0 := LongestCoxeterElement( W ); # = EltWord( W, last ) ( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21) (10,22)(11,23)(12,24) gap> CoxeterLength( W, w0 ); 12 gap> List( Reflections( W ), i -> CoxeterWord( W, i ) ); [ "s", "t", "u", "v", "sus", "tut", "uvu", "stust", "suvus", "tuvut", "stuvust", "ustuvustu" ] gap> l := List( [0 .. W.N], x -> CoxeterElements( W, x ) );; gap> List( l, Length ); [ 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1 ]
The above line tells us that there is 1 element of length 0, there are 4 elements of length 4, etc.
For many basic functions (like Bruhat
, CoxeterLength
, etc.) we have
chosen the convention that the input is an element of a Coxeter group
(rather than a Coxeter word). The reason is that for a Coxeter group which
is a permutation group, if in some application one has to do a lot of
computations with Coxeter group elements then using the low level GAP
functions for permutations is usually much faster than manipulating lists
of reduced expressions.
Before describing functions applicable to Coxeter groups and Coxeter words we describe functions which build two familiar examples.
CoxeterGroupSymmetricGroup( n )
returns the symmetric group on n letters as a Coxeter group. We give the code of this function as it is a good example on how to make a Coxeter group:
CoxeterGroupSymmetricGroup := function ( n ) local W; W := SymmetricGroup( n ); W.reflections := List( [ 1 .. n - 1 ], i->(i,i + 1) ); W.operations.IsLeftDescending := function ( W, w, i ) return i ^ w > (i + 1) ^ w; end; AbsCoxOps.CompleteCoxeterGroupRecord( W ); return W; end;
In the above, we first set the generating reflections of
W to be the elementary transpositions (i,i+1)
(which are
reflections in the natural representation of the symmetric group
permuting the standard basis of an n-dimensional vector space),
then give the IsLeftDescending
function (which just checks
if (i,i+1)
is an inversion of the permutation). Finally,
AbsCoxOps.CompleteCoxeterGroupRecord
is a service routine which fills
other fields from the ones we gave. We can see what it did by doing:
gap> PrintRec(CoxeterGroupSymmetricGroup(3));Print("\n"); rec( isDomain := true, isGroup := true, identity := (), generators := [ (1,3), (2,3) ], operations := ..., isPermGroup := true, isFinite := true, 1 := (1,3), 2 := (2,3), degree := 3, reflections := [ (1,2), (2,3) ], nbGeneratingReflections := 2, generatingReflections := [ 1 .. 2 ], EigenvaluesGeneratingReflections := [ 1/2, 1/2 ], isCoxeterGroup := true, reflectionsLabels := [ 1 .. 2 ], coxeterMat := [ [ 1, 3 ], [ 3, 1 ] ], orbitRepresentative := [ 1, 1 ], longestElm := (1,3), longestCoxeterWord := [ 1, 2, 1 ], N := 3 )
We do not indicate all the fields here. Some are there for
technical reasons and may change from version to version of
CHEVIE. Among the added fields, we see nbGeneratingReflections
(taken to be Length(W.reflections)
if we do not give it),
.OrdersGeneratingReflections
, the Coxeter matrix .coxeterMat
, a
description of conjugacy classes of the generating reflections given in
.orbitRepresentative
(whose i-th entry is the smallest index of a
reflection conjugate to .reflections[i]
), .reflectionsLabels
(the
default labels used for Coxeter word). At the end are 3 fields which are
computed only for finite Coxeter groups: the longest element, as an
element and as a Coxeter word, and in W.N
the number of reflections in
W (which is also the length of the longest Coxeter word).
75.2 CoxeterGroupHyperoctaedralGroup
CoxeterGroupHyperoctaedralGroup( n )
returns the hyperoctaedral group of rank n as a Coxeter group. It is
given as a permutation group on 2n letters, with Coxeter generators
the permutations (i,i+1)(2n+1-i,2n-i)
and (n,n+1)
.
gap> CoxeterGroupHyperoctaedralGroup(2); Group( (2,3), (1,2)(3,4) )
This function requires the package "chevie" (see RequirePackage).
CoxeterMatrix( W )
return the Coxeter matrix of the Coxeter group W, that is the matrix
whose entry m[i][j]
contains the order of gi*gj where gi is
the i-th Coxeter generator of W. An infinite order is represented by
the entry 0.
gap> W:=CoxeterGroupSymmetricGroup(4); Group( (1,4), (2,4), (3,4) ) gap> CoxeterMatrix(W); [ [ 1, 3, 2 ], [ 3, 1, 3 ], [ 2, 3, 1 ] ]
This function requires the package "chevie" (see RequirePackage).
75.4 CoxeterGroupByCoxeterMatrix
CoxeterGroupByCoxeterMatrix( m )
returns the Coxeter group whose Coxeter matrix is m.
The matrix m should be a symmetric integer matrix such that m[i,i]=1
and m[i,j]>=2
(or m[i,j]=0
to represent an infinite entry).
The group is constructed as a matrix group, using the standard reflection
representation for Coxeter groups. This is the representation on a real
vector space V of dimension Length(m)
defined as follows : if es
is a basis of V indexed by the lines of m, we make the s-th
reflection act by s(x)=x-2〈 x, es〉 es where
〈,〉 is the bilinear form on V defined by 〈
es,et〉=-cos(π/m[s,t]) (where by convention π/m[s,t]=0 if
m[s,t]=∞, which is represented in CHEVIE by setting
m[s,t]:=0
). In the example below the affine Weyl group ~ A2 is
constructed, and then ~ A1.
gap> m:=[[1,3,3],[3,1,3],[3,3,1]];; gap> W:=CoxeterGroupByCoxeterMatrix(m); CoxeterGroupByCoxeterMatrix([ [ 1, 3, 3 ], [ 3, 1, 3 ], [ 3, 3, 1] ]) gap> CoxeterWords(W,3); [ [ 1, 3, 2 ], [ 1, 2, 3 ], [ 1, 2, 1 ], [ 1, 3, 1 ], [ 2, 1, 3 ], [ 3, 1, 2 ], [ 2, 3, 2 ], [ 2, 3, 1 ], [ 3, 2, 1 ] ] gap> CoxeterGroupByCoxeterMatrix([[1,0],[0,1]]); CoxeterGroupByCoxeterMatrix([ [ 1, 0 ], [ 0, 1 ] ])
This function requires the package "chevie" (see RequirePackage).
75.5 CoxeterGroupByCartanMatrix
CoxeterGroupByCartanMatrix( m )
m should be a square matrix of real cyclotomic numbers. It returns the
reflection group whose Cartan matrix is m. This is a matrix group
constructed as follows. Let V be a real vector space of dimension
Length(m)
, and let 〈,〉 be the bilinear form defined by
〈 ei, ej〉=m[i,j] where ei is the canonical basis of V.
Then the result is the matrix group generated by the reflections
si(x)=x-2〈 x, ei〉 ei.
This function is used in CoxeterGroupByCoxeterMatrix
, using also the
function CartanMatFromCoxeterMatrix
.
gap> CartanMatFromCoxeterMatrix([[1,0],[0,1]]); [ [ 2, -2 ], [ -2, 2 ] ] gap> CoxeterGroupByCartanMatrix(last); CoxeterGroupByCoxeterMatrix([ [ 1, 0 ], [ 0, 1 ] ])
Above is another way to construct ~ A1.
This function requires the package "chevie" (see RequirePackage).
75.6 CartanMatFromCoxeterMatrix
CartanMatFromCoxeterMatrix( m )
The argument is a CoxeterMatrix for a finite Coxeter group W and the
result is a Cartan Matrix for the standard reflection representation of W
(see CartanMat). Its diagonal terms are 2 and the coefficient between
two generating reflections s and t is -2cos(π/m[s,t]) (where by
convention π/m[s,t]=0 if m[s,t]=∞, which is represented in
CHEVIE by setting m[s,t]:=0
).
gap> m:=[[1,3],[3,1]]; [ [ 1, 3 ], [ 3, 1 ] ] gap> CartanMatFromCoxeterMatrix(m); [ [ 2, -1 ], [ -1, 2 ] ]
This function requires the package "chevie" (see RequirePackage).
75.7 Functions for general Coxeter groups
Some functions take advantage of the fact a group is a Coxeter group to use a better algorithm. A typical example is:
Elements(W)
For finite Coxeter groups, uses a recursive algorithm based on the construction of elements of a chain of parabolic subgroups
ReflectionSubgroup(W, J)
When I is a subset of [1..W.nbGeneratingReflections]
then the
reflection subgroup of W generated by W.reflections{I}
can
be generated abstractly (without any specific knowledge about the
representation of W) as a Coxeter group. This is what this routine
does: implement a special case of ReflectionSubgroup
which works
for arbitrary Coxeter groups (see ReflectionSubgroup). The actual
argument J should be reflection labels for W, i.e. be a subset of
W.reflectionsLabels
.
Similarly, the functions ReducedRightCosetRepresentatives
,
PermCosetsSubgroup
, work for reflection subgroups of the above form.
See the chapter on reflection subgroups for a description of these
functions.
CartanMat(W)
Returns CartanMatFromCoxeterMatrix(CoxeterMatrix(W))
(see
CartanMatFromCoxeterMatrix).
The functions ReflectionType
, ReflectionName
and all functions
depending on the classification of finite Coxeter groups work for finite
Coxeter groups. See the chapter on reflection groups for a description
of these functions.
BraidRelations(W)
returns the braid relations implied by the Coxeter matrix of W.
These functions require the package "chevie" (see RequirePackage).
IsLeftDescending( W , w, i )
returns true
if and only if the i-th generating reflection
W.reflections[i]
is in the left descent set of the element w of W.
gap> W:=CoxeterGroupSymmetricGroup(3); Group( (1,3), (2,3) ) gap> IsLeftDescending(W,(1,2),1); true
This function requires the package "chevie" (see RequirePackage).
FirstLeftDescending( W , w )
returns the index in the list of generating reflections of W of the
first element of the left descent set of the element w of W (i.e.,
the first i such that if s=W.reflections[i]
then l(sw)<l(w)). It
is quite important to think of using this function rather than write
a loop like First([1..W.nbGeneratingReflections],IsLeftDescending)
,
since for particular classes of groups (e.g. finite Coxeter groups) the
function is much optimized compared to such a loop.
gap> W:=CoxeterGroupSymmetricGroup(3); Group( (1,3), (2,3) ) gap> FirstLeftDescending(W,(2,3)); 2
This function requires the package "chevie" (see RequirePackage).
LeftDescentSet( W, w )
The set of generators s such that l(sw)<l(w), given as a list of labels for the corresponding generating reflections.
gap> W:=CoxeterGroupSymmetricGroup(3); Group( (1,3), (2,3) ) gap> LeftDescentSet( W, (1,3)); [ 1, 2 ]
See also RightDescentSet.
This function requires the package "chevie" (see RequirePackage).
RightDescentSet( W, w )
The set of generators s such that l(ws)< l(w), given as a list of labels for the corresponding generating reflections.
gap> W := CoxeterGroup( "A", 2 );; gap> w := EltWord( W, [ 1, 2 ] );; gap> RightDescentSet( W, w ); [ 2 ]
See also LeftDescentSet.
This function requires the package "chevie" (see RequirePackage).
EltWord( W , w )
returns the element of W which corresponds to the Coxeter word w. Thus it returns a permutation if W is a permutation group (the usual case for finite Coxeter groups) and a matrix for matrix groups (such as affine Coxeter groups).
gap> W:=CoxeterGroupSymmetricGroup(4); Group( (1,4), (2,4), (3,4) ) gap> EltWord(W,[1,2,3]); (1,4,3,2)
See also CoxeterWord.
This function requires the package "chevie" (see RequirePackage).
CoxeterWord( W , w )
returns a reduced word in the standard generators of the Coxeter group W for the element w (represented as the GAP list of the corresponding reflection labels).
gap> W := CoxeterGroup( "A", 3 );; gap> w := ( 1,11)( 3,10)( 4, 9)( 5, 7)( 6,12);; gap> w in W; true gap> CoxeterWord( W, w ); [ 1, 2, 3, 2, 1 ]
The result of CoxeterWord
is the lexicographically smallest reduced
word for w (for the ordering of the Coxeter generators given by
W.reflections
).
See also EltWord and ReducedCoxeterWord.
This function requires the package "chevie" (see RequirePackage).
CoxeterLength( W , w )
returns the length of the element w of W as a reduced expression in the standard generators.
gap> W := CoxeterGroup( "F", 4 );; gap> p := EltWord( W, [ 1, 2, 3, 4, 2 ] ); ( 1,44,38,25,20,14)( 2, 5,40,47,48,35)( 3, 7,13,21,19,15) ( 4, 6,12,28,30,36)( 8,34,41,32,10,17)( 9,18)(11,26,29,16,23,24) (27,31,37,45,43,39)(33,42) gap> CoxeterLength( W, p ); 5 gap> CoxeterWord( W, p ); [ 1, 2, 3, 2, 4 ]
This function requires the package "chevie" (see RequirePackage).
ReducedCoxeterWord( W , w )
returns a reduced expression for an element of the Coxeter group W, which is given as a GAP list of reflection labels for the standard generators of W.
gap> W := CoxeterGroup( "E", 6 );; gap> ReducedCoxeterWord( W, [ 1, 1, 1, 1, 1, 2, 2, 2, 3 ] ); [ 1, 2, 3 ]
This function requires the package "chevie" (see RequirePackage).
BrieskornNormalForm( W , w )
Brieskorn Bri71 has noticed that if L(w) is the left descent
set of w (see LeftDescentSet), and if wL(w) is the longest
Coxeter element (see LongestCoxeterElement) of the reflection subgroup
WL(w) (not that this element is an involution), then wL(w)
divides w, in the sense that l(wL(w))+l(wL(w)-1w)=l(w).
We can now divide w by wL(w) and continue this process
with the quotient. In this way, we obtain a reduced expression
w=wL1 ... wLr where Li=L(wLi ... wLr) for
all i, which we call the Brieskorn normal form of w. The
function BrieskornNormalForm
will return a description of this form,
by returning the list of sets L(w) which describe the above
decomposition.
gap> W:=CoxeterGroup("E",8); CoxeterGroup("E",8) gap> w:=[ 2, 3, 4, 2, 3, 4, 5, 4, 2, 3, 4, 5, 6, 5, 4, 2, 3, 4, > 5, 6, 7, 6, 5, 4, 2, 3, 4, 5, 6, 7, 8 ];; gap> BrieskornNormalForm(W,EltWord(W,w)); [ [ 2, 3, 4, 5, 6, 7 ], [ 8 ] ] gap> EltWord(W,w)=Product(last,x->LongestCoxeterElement(W,x)); true
This function requires the package "chevie" (see RequirePackage).
LongestCoxeterElement( W [,I])
If W is finite, returns the unique element of maximal length of the Coxeter group W. May loop infinitely otherwise.
gap> LongestCoxeterElement( CoxeterGroupSymmetricGroup( 4 ) ); (1,4)(2,3)
If a second argument I is given, returns the longest element of the
parabolic subgroup generated by the reflections in I (where I is given
as .reflectionsLabels
).
gap> LongestCoxeterElement(CoxeterGroupSymmetricGroup(4),[2,3]); (2,4)
This function requires the package "chevie" (see RequirePackage).
LongestCoxeterWord( W )
If W is finite, returns a reduced expression in the standard generators for the unique element of maximal length of the Coxeter group W. May loop infinitely otherwise.
gap> LongestCoxeterWord( CoxeterGroupSymmetricGroup( 5 ) ); [ 1, 2, 1, 3, 2, 1, 4, 3, 2, 1 ]
This function requires the package "chevie" (see RequirePackage).
CoxeterElements( W[, l] )
Given with a second argument l returns the list of all elements of W of Coxeter length l.
gap> W := CoxeterGroup( "G", 2 );; gap> e := CoxeterElements( W, 6 ); [ ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] gap> e[1] = LongestCoxeterElement( W ); true
After the call to CoxeterElements(W,l)
, the list of elements of W
of
length l
is stored in the component elts[l+1]
of the record of W.
Thus W.elts[lw+1]
contains the list of all elements of length lw in
W. There are a number of programs (like criticalPairs
, see Chapter
Kazhdan-Lusztig polynomials and bases) which refer to the ordering of
the elements in these lists in W.elts
.
If only one argument is given, returns all elements of W, in the same order as
Concatenation(List([0..W.N],i->CoxeterElements(W,i)))
this only makes sense for finite Coxeter groups.
This function requires the package "chevie" (see RequirePackage).
CoxeterWords( W[, l] )
With second argument the integer l returns the list of CoxeterWord
s for
all elements of CoxeterLength
l in the Coxeter group W.
If only one argument is given, returns all elements of W as Coxeter words, in the same order as
Concatenation(List([0..W.N],i->CoxeterWords(W,i)))
this only makes sense for finite Coxeter groups.
gap> CoxeterWords( CoxeterGroup( "G", 2 ) ); [ [ ], [ 2 ], [ 1 ], [ 2, 1 ], [ 1, 2 ], [ 2, 1, 2 ], [ 1, 2, 1 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2 ], [ 2, 1, 2, 1, 2 ], [ 1, 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1, 2 ] ]
This function requires the package "chevie" (see RequirePackage).
Bruhat( W, y, w )
returns true
, if the element y is less than or equal to the element w
of the Coxeter group W for the Bruhat order, and false
otherwise (y
is less than w if a reduced expression for y can be extracted from one
for w). See Hum90, (5.9) and (5.10) for properties of the Bruhat
order.
gap> W := CoxeterGroup( "H", 3 );; gap> w := EltWord( W, [ 1, 2, 1, 3 ] );; gap> b := Filtered( Elements( W ), x -> Bruhat( W, x, w) );; gap> List( b, x -> CoxeterWord( W, x ) ); [ [ ], [ 3 ], [ 2 ], [ 1 ], [ 2, 1 ], [ 2, 3 ], [ 1, 3 ], [ 1, 2 ], [ 2, 1, 3 ], [ 1, 2, 1 ], [ 1, 2, 3 ], [ 1, 2, 1, 3 ] ]
This function requires the package "chevie" (see RequirePackage).
BruhatSmaller( W, w)
Returns a list whose i-th element is the list of elements of W
smaller for the Bruhat order that w and of Length i-1. Thus the
first element of the returned list contains only W.identity
and the
CoxeterLength(W,w)
-th element contains only w.
gap> W:=CoxeterGroupSymmetricGroup(3); Group( (1,3), (2,3) ) gap> BruhatSmaller(W,(1,3)); [ [ () ], [ (2,3), (1,2) ], [ (1,2,3), (1,3,2) ], [ (1,3) ] ]
This function requires the package "chevie" (see RequirePackage).
ReducedInRightCoset( W, w)
Let w be an element of a parent group of W whose action by
conjugation induces an automorphism of Coxeter groups on W, that is
sends the Coxeter generators of W to a conjugate set (but may not send
the tuple of generators to a conjugate tuple). ReducedInRightCoset
returns the unique element in the right coset W.w which is
W-reduced, that is which preserves the set of Coxeter generators of
W.
gap> W:=CoxeterGroupSymmetricGroup(6); Group( (1,6), (2,6), (3,6), (4,6), (5,6) ) gap> H:=ReflectionSubgroup(W,[2..4]); Subgroup( Group( (1,6), (2,6), (3,6), (4,6), (5,6) ), [ (2,3), (3,4), (4,5) ] ) gap> ReducedInRightCoset(H,(1,6)(2,4)(3,5)); (1,6)
This function requires the package "chevie" (see RequirePackage).
ForEachElement( W, f)
This functions calls f(x)
for each element x of the finite Coxeter
group W. It is quite useful when the Size
of W would make impossible
to call Elements(W)
. For example,
gap> i:=0;; gap> W:=CoxeterGroup("E",7);; gap> ForEachElement(W,function(x)i:=i+1; > if i mod 1000000=0 then Print("*\c");fi; > end);Print("\n"); **
prints a *
about every 10 seconds on a 1Ghz computer, so takes about
30 seconds to enumerate the whole group E7.
This function requires the package "chevie" (see RequirePackage).
StandardParabolicClass( W, I)
I should be a subset of W.reflectionsLabels
describing a subset of the
generating reflections for W. The function returns the list of subsets of
W.reflectionsLabels
corresponding to sets of reflections conjugate to the
given subset.
gap> StandardParabolicClass(CoxeterGroup("E",8),[7,8]); [ [ 1, 3 ], [ 2, 4 ], [ 3, 4 ], [ 4, 5 ], [ 5, 6 ], [ 6, 7 ], [ 7, 8 ] ]
This function requires the package "chevie" (see RequirePackage).
75.26 ParabolicRepresentatives
ParabolicRepresentatives(W)
Returns a list of subsets of W.reflectionsLabels
describing representatives
of orbits of parabolic subgroups under conjugation by W.
gap> ParabolicRepresentatives(Affine(CoxeterGroup("A",3))); [ [ ], [ 1 ], [ 1, 2 ], [ 1, 2, 3 ], [ 1, 2, 3, 4 ], [ 1, 2, 4 ], [ 1, 3 ], [ 1, 3, 4 ], [ 2, 3, 4 ], [ 2, 4 ] ]
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4