The CharTable
of a finite complex reflection group W is computed in
CHEVIE using the decomposition of W in irreducible groups (see
ReflectionType). For each irreducible group the character table is
either computed using recursive formulas for the infinite series, or
read into the system from a library file for the exceptional types.
Thus, character tables can be obtained quickly even for very large
groups (e.g., E8). Similar remarks apply for conjugacy classes.
The conjugacy classes and irreducible characters of irreducible
finite complex reflection groups have canonical labelings by certain
combinatorial objects; such labelings are contained in a consistent way
in the tables of CHEVIE. For the classes, these are partitions or
partition tuples for the infinite series, or, for exceptional Coxeter
groups, Carter's admissible diagrams Car72 (for other primitive
complex reflection groups we just use words in the generators to specify
the classes). For the characters, these are again partitions or
partition tuples for the infinite series, and for the others they are
pairs of two integers (n,e) where n is the degree of the character
and e is the smallest symmetric power of the natural reflection
representation containing the given character as a constituent. This
information is obtained by using the functions ChevieClassInfo
and
ChevieCharInfo
(and some of it is also available more directly via the
functions CharParams
, CharName
, HighestPowerFakeDegrees
). When you
display the character table in GAP, the canonical names for classes and
characters are those displayed.
A typical example is CoxeterGroup("A",n)
, the symmetric group
Sn+1 where classes and characters are parametrized by partitions
of n+1.
gap> W := CoxeterGroup( "A", 3 );; gap> Display( CharTable( W )); A3 2 3 2 3 . 2 3 1 . . 1 . 1111 211 22 31 4 2P 1111 1111 1111 31 22 3P 1111 211 22 1111 4 1111 1 -1 1 1 -1 211 3 -1 -1 . 1 22 2 . 2 -1 . 31 3 1 -1 . -1 4 1 1 1 1 1
The charTable
record (computed the first time the function CharTable
is called) is a usual character table record as defined in GAP,
but with some additional components. The components classtext
,
classnames
contain information as described for ChevieClassInfo
(see
ChevieClassInfo). There is also a field irredinfo
, which is a
list of records for each irreducible character which have components
charname
and charparam
as described for ChevieCharInfo
(see
ChevieCharInfo).
gap> W := CoxeterGroup( "G", 2);; gap> ct := CharTable( W ); CharTable( "G2" ) gap> ct.classtext; [ [ ], [ 2 ], [ 1 ], [ 1, 2 ], [ 1, 2, 1, 2 ], [ 1, 2, 1, 2, 1, 2 ] ] gap> ct.classnames; [ "A_0", "~A_1", "A_1", "G_2", "A_2", "A_1+~A_1" ] gap> ct.irredinfo; [ rec( charparam := [ [ 1, 0 ] ], charname := "phi{1,0}" ), rec( charparam := [ [ 1, 6 ] ], charname := "phi{1,6}" ), rec( charparam := [ [ 1, 3, "'" ] ], charname := "phi{1,3}'" ), rec( charparam := [ [ 1, 3, "''" ] ], charname := "phi{1,3}''" ), rec( charparam := [ [ 2, 1 ] ], charname := "phi{2,1}" ), rec( charparam := [ [ 2, 2 ] ], charname := "phi{2,2}" ) ]
Recall that our groups acts a reflection group on the vector space V, so have fake degrees (see FakeDegree). Using these one can associate two integers b,B with each irreducible character of W (see LowestPowerFakeDegrees and HighestPowerFakeDegrees). For finite Coxeter groups, using the generic degrees of the corresponding one-parameter generic Hecke algebra, one can associate two more integers a,A (see the functions LowestPowerGenericDegrees, HighestPowerGenericDegrees, and Car85, Ch.11 for more details). These will also be used in the operations of truncated inductions explained in the chapter Reflection subgroups.
Iwahori-Hecke algebras and cyclotomic Hecke algebras also have character tables, see the corresponding chapters.
We now describe, for each type our conventions about labeling the classes and characters.
Type An (n ≥ 0). In this case we have W ≅
Sn+1. The classes and characters are labeled by partitions of
n+1. The partition corresponding to a class describes the cycle type
for the elements in that class; the representative in .classtext
is
the concatenation of the words corresponding to each part, and to a part
i is associated the product of i-1 consecutive generators (starting
one higher that the last generator used for the previous parts). The
partition corresponding to a character describes the type of the Young
subgroup such that the trivial character induced from this subgroup
contains that character with multiplicity 1 and such that every other
character occurring in this induced character has a higher a-value.
Thus, the sign character corresponds to the partition (1n+1) and
the trivial character to the partition (n+1). The character of the
reflection representation of W is labeled by (n,1).
Type Bn (n ≥ 2). In this case W=W(Bn) is isomorphic to the wreath product of the cyclic group of order 2 with the symmetric group Sn. Hence the classes and characters are parametrized by pairs of partitions such that the total sum of their parts equals n. The pair corresponding to a class describes the signed cycle type for the elements in that class, as in Car72. We use the convention that if (λ,μ) is such a pair then λ corresponds to the positive and μ to the negative cycles. Thus, (1n,-) and (-,1n) label the trivial class and the class containing the longest element, respectively. The pair corresponding to an irreducible character is determined via Clifford theory, as follows.
We have a semidirect product decomposition W(Bn)=N.Sn where N is the standard n-dimensional F2n-vector space. For a,b ≥ 0 such that n=a+b let ηa,b be the irreducible character of N which takes value 1 on the first a standard basis vectors and value -1 on the next b standard basis vectors of N. Then the inertia subgroup of ηa,b has the form Ta,b:=N.(Sa x Sb) and we can extend ηa,b trivially to an irreducible character ~ηa,b of Ta,b. Let α and β be partitions of a and b, respectively. We take the tensor product of the corresponding irreducible characters of Sa and Sb and regard this as an irreducible character of Ta,b. Multiplying this character with ~ηa,b and inducing to W(Bn) yields an irreducible character χ= χ(α,β) of W(Bn). This defines the correspondence between irreducible characters and pairs of partitions as above.
For example, the pair ((n),-) labels the trivial character and (-,(1n)) labels the sign character. The character of the natural reflection representation is labeled by ((n-1),(1)).
Type Dn (n ≥ 4). In this case W=W(Dn) can be embedded as a subgroup of index 2 into the Coxeter group W(Bn). The intersection of a class of W(Bn) with W(Dn) is either empty or a single class in W(Dn) or splits up into two classes in W(Dn). This also leads to a parameterization of the classes of W(Dn) by pairs of partitions (λ,μ) as before but where the number of parts of μ is even and where there are two classes of this type if μ is empty and all parts of λ are even. In the latter case we denote the two classes in W(Dn) by (λ,+) and (λ,-), where we use the convention that the class labeled by (λ,+) contains a representative which can be written as a word in {s1,s3,...,sn} and (λ,-) contains a representative which can be written as a word in {s2,s3, ...,sn}.
By Clifford theory the restriction of an irreducible character of W(Bn) to W(Dn) is either irreducible or splits up into two irreducible components. Let (α,β) be a pair of partitions with total sum of parts equal to n. If α ≠ β then the restrictions of the irreducible characters of W(Bn) labeled by (α,β) and (β, α) are irreducible and equal. If α=β then the restriction of the character labeled by (α,α) splits into two irreducible components which we denote by (α,+) and (α,-). Note that this can only happen if n is even. In order to fix the notation we use a result of Ste89 which describes the value of the difference of these two characters on a class of the form (λ,+) in terms of the character values of the symmetric group Sn/2. Recall that it is implicit in the notation (λ,+) that all parts of λ are even. Let λ′ be the partition of n/2 obtained by dividing each part by 2. Then the value of
χ(α,-)-χ(α,+) |
The labels for the trivial, the sign and the natural reflection character are the same as for W(Bn), since these characters are restrictions of the corresponding characters of W(Bn).
The groups G(d,1,n). They are isomorphic to the wreath product of the cyclic group of order d with the symmetric group Sn. Hence the classes and characters are parametrized by d-tuples of partitions such that the total sum of their parts equals n. The words chosen as representatives of the classes are, when d>2, computed in a slightly different way than for Bn, in order to agree with the words on which Ram and Halverson compute the characters of the Hecke algebra. First the parts of the d partitions merged in one big partition and sorted in increasing order. Then, to a part i coming from the j-th partition is associated the word (l+1...1... l+1)j-1l=2... l+i where l is the highest generator already used.
The d-tuple corresponding to an irreducible character is determined via Clifford theory in a similar way than for the Bn case. The identity character has the first partition with one part equal n and the other ones empty. The character of the reflection representations has the first two partitions with one part equal respectively to n-1 and to 1, and the other partitions empty.
The groups G(de,e,n). They are normal subgroups of index e in G(de,1,n). The quotient is cyclic, generated by the image g of the first generator of G(de,1,n). The classes are parametrized as the classes of G(de,e,n) with an extra information for a component of a class which splits.
According to Hugues On decompositions in complex imprimitive
reflection groups, Indagationes 88 (1985) 207--219, a class C of
G(de,1,n) parametrized by a de-partition (S0,...,Sde-1) is
in G(de,e,n) if e divides ∑i i ∑p∈ Sip. It splits in
d classes for the largest d dividing e and all parts of all Si
and such that Si is empty if d does not divide i. If w is in
C then g^i w g^-i
for i in [0..d-1]
are representatives of the
classes of G(de,e,n) which meet C. They are described by appending
the integer i to the label for C.
The characters are described by Clifford theory. We make g act on
labels for characters of G(de,1,n) . The action of g permutes
circularly by d the partitions in the de-tuple. A character has same
restriction to G(de,e,n) as its transform by g. The number of
irreducible components of its restriction is equal to the order k of
its stabilizer under powers of g. We encode a character of G(de,e,n)
by first, choosing the smallest for lexicographical order label of a
character whose restriction contains it; then this label is periodic
with a motive repeated k times; we represent the character by one of
these motives, to which we append E(k)i for i in [0..k-1]
to
describe which component of the restriction we choose.
Types G2 and F4. The matrices of character values and the orderings and labelings of the irreducible characters are exactly the same as in Car85, p.412/413. Note, however, that in CHEVIE we have reversed the labeling of the Dynkin diagrams to be in accordance with the conventions in Lus85, (4.8) and (4.10).
The classes are labeled by Carter's admissible diagrams Car72. A character is labeled by a pair (n,b) where n denotes the degree and b the corresponding b-invariant. If there are several characters with the same pair (n,b) we attach a prime to them, as in Car85.
For type F4 the result of ChevieCharInfo
contains an additional
component kondo
which contains the labels originally given by Kondo
(and which are also used in Lus85, (4.10)). The reflection
character is labeled by (4,1) or 42 (Kondo).
Types E6,E7,E8. The character tables are obtained
by specialization of those of the Hecke algebra. The classes are labeled
by Carter's admissible diagrams Car72. A character is labeled
by the pair (n,b) where n denotes the degree and b is the
corresponding b-invariant. For these types, this gives a unique
labeling of the characters. The result of ChevieCharInfo
contains an
additional component frame
which contains the labels originally given
by Frame (and which are used in Lus85, (4.11), (4.12), and
(4.13)). For type E6, E7, E8, respectively, the reflection
character is the one with label (6,1), (7,1), (8,1) or 6p,
7a′, 8z (Frame).
Non-crystallographic types I2(m), H3, H4. In these cases we do not have canonical labelings for the classes.
Each character for type H3 is uniquely determined by the pair (n,b) where n is the degree and b the corresponding b-invariant. For type H4 there are just two characters (those of degree 30) for which the corresponding pairs are the same. These two characters are nevertheless distinguished by their fake degrees: the first of these (in the CHEVIE-table) has fake degree q10+q12+ higher terms, while the second has fake degree q12+q14+ higher terms. The characters in the CHEVIE-table for type H4 are ordered in the same way as in AL82.
Finally, the characters of degree 2 for type I2(m) are ordered as follows. Let ε be a primitive m-th root of unity. Then matrix representations affording the characters of degree 2 are given by:
φj : s1s2 → ( |
| ), s1→( |
| ), |
In GAP we take ε as E( m )
. Then the characters in the
CHEVIE-table are ordered as φ1,φ2,....
CharParams
:A
, double partitions for type B
,
etc... CharName
also has a special version which knows how to
display nicely such labels.
returns information about the conjugacy classes of the finite reflection group W. The result is a record with three components:
classtext
:WordsClassRepresentatives(W)
and the
representatives given are of minimal length (the representatives
taken are explained in GM97).
classparams
:
classnames
:classparams
.
gap> ChevieClassInfo(CoxeterGroup( "D", 4 )); rec( classtext := [ [ ], [ 1, 2 ], [ 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4 ], [ 1 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 4 ], [ 1, 3, 1, 2, 3, 4 ], [ 1, 3 ], [ 1, 2, 3, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ] ], classnames := [ "1111.", "11.11", ".1111", "211.", "1.21", "2.11", "22.+", "22.-", ".22", "31.", ".31", "4.+", "4.-" ], classparams := [ [ [ [ 1, 1, 1, 1 ], [ ] ] ], [ [ [ 1, 1 ], [ 1, 1 ] ] ], [ [ [ ], [ 1, 1, 1, 1 ] ] ], [ [ [ 2, 1, 1 ], [ ] ] ], [ [ [ 1 ], [ 2, 1 ] ] ], [ [ [ 2 ], [ 1, 1 ] ] ], [ [ [ 2, 2 ], '+' ] ], [ [ [ 2, 2 ], '-' ] ], [ [ [ ], [ 2, 2 ] ] ], [ [ [ 3, 1 ], [ ] ] ], [ [ [ ], [ 3, 1 ] ] ], [ [ [ 4 ], '+' ] ], [ [ [ 4 ], '-' ] ] ]) gap> ChevieClassInfo(ComplexReflectionGroup(3,1,2)); rec( classtext := [ [ ], [ 1 ], [ 1, 1 ], [ 1, 2, 1, 2 ], [ 1, 1, 2, 1, 2 ], [ 1, 1, 2, 1, 2, 2, 1, 2 ], [ 2 ], [ 1, 2 ], [ 1, 1, 2 ] ], classnames := [ "11..", "1.1.", "1..1", ".11.", ".1.1", "..11", "2..", ".2.", "..2" ], classparams := [ [ [ [ 1, 1 ], [ ], [ ] ] ], [ [ [ 1 ], [ 1 ], [ ] ] ], [ [ [ 1 ], [ ], [ 1 ] ] ], [ [ [ ], [ 1, 1 ], [ ] ] ], [ [ [ ], [ 1 ], [ 1 ] ] ], [ [ [ ], [ ], [ 1, 1 ] ] ], [ [ [ 2 ], [ ], [ ] ] ], [ [ [ ], [ 2 ], [ ] ] ], [ [ [ ], [ ], [ 2 ] ] ] ] )
See also the introduction of this section.
This function requires the package "chevie" (see RequirePackage).
79.2 WordsClassRepresentatives
WordsClassRepresentatives( W )
returns a list of representatives of the conjugacy classes of the
group W. Each element in this list is given as a word in the
standard generators, where the generator si is represented by the
number i in a list. For finite Coxeter groups, it is the same
as List(ConjugacyClasses(W),x->CoxeterWord(W,Representative(x)))
, and
each representative given by CHEVIE has the property that it is of
minimal length in its conjugacy class and is a "good" element in the
sense of GM97.
gap> WordsClassRepresentatives( CoxeterGroup( "F", 4 ) ); [ [ ], [ 1, 2, 1, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4 ], [ 2, 3, 2, 3 ], [ 2, 1 ], [ 2, 3, 2, 3, 4, 3, 2, 1, 3, 4 ], [ 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1 ], [ 4, 3 ], [ 1, 2, 1, 4, 3, 2, 1, 3, 2, 3 ], [ 3, 2, 1, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2 ], [ 3, 2, 4, 3, 2, 1, 3, 2 ], [ 4, 3, 2, 1 ], [ 1 ], [ 2, 3, 2, 3, 4, 3, 2, 3, 4 ], [ 1, 4, 3 ], [ 4, 3, 2 ], [ 2, 3, 2, 1, 3 ], [ 3 ], [ 1, 2, 1, 3, 2, 1, 3, 2, 3 ], [ 2, 1, 4 ], [ 3, 2, 1 ], [ 2, 4, 3, 2, 3 ], [ 1, 3 ], [ 3, 2 ], [ 2, 3, 2, 3, 4, 3, 2, 1, 3, 2, 4, 3, 2, 1 ], [ 2, 4, 3, 2, 1, 3 ] ]
See also ChevieClassInfo.
This function requires the package "chevie" (see RequirePackage).
ChevieCharInfo( W )
returns information about the irreducible characters of the finite reflection group W. The result is a record with the following components:
charparams
:CharParams(W)
. The
parameters are tuples with one component for each irreducible
irreducible component of W (as given by ReflectionType
). For an
irreducible component which is an imprimitive reflection group the
component of the charparam
is a tuple of partitions, and for a
primitive irreducible group it is a pair (n,e) where n is the
degree of the character and e is the smallest symmetric power of
the character of the reflection representation which contains the
given character as a component.
charnames
:charparams
.
positionId
:PositionId
).
positionRefl
:
a
:LowestPowerFakeDegrees(W)
.
A
:HighestPowerFakeDegrees(W)
.
b
:LowestPowerGenericDegrees(W)
.
B
:HighestPowerGenericDegrees(W)
.
positionSgn
:PositionSgn
).
gap> ChevieCharInfo(ComplexReflectionGroup(4)); rec( charparams := [ [ [ 1, 0 ] ], [ [ 1, 4 ] ], [ [ 1, 8 ] ], [ [ 2, 5 ] ], [ [ 2, 3 ] ], [ [ 2, 1 ] ], [ [ 3, 2 ] ] ], charnames := [ "phi{1,0}", "phi{1,4}", "phi{1,8}", "phi{2,5}", "phi{2,3}", "phi{2,1}", "phi{3,2}" ], positionId := 1, positionRefl := 6 )
gap> ChevieCharInfo( CoxeterGroup( "G", 2 ) ); rec( charparams := [ [ [ 1, 0 ] ], [ [ 1, 6 ] ], [ [ 1, 3, "'" ] ], [ [ 1, 3, "''" ] ], [ [ 2, 1 ] ], [ [ 2, 2 ] ] ], charnames := [ "phi{1,0}", "phi{1,6}", "phi{1,3}'", "phi{1,3}''", "phi{2,1}", "phi{2,2}" ], a := [ 0, 6, 1, 1, 1, 1 ], A := [ 0, 6, 5, 5, 5, 5 ], b := [ 0, 6, 3, 3, 1, 2 ], B := [ 0, 6, 3, 3, 5, 4 ], positionId := 1, positionSgn := 2, positionRefl := 5 )
If W is irreducible of type F4 or of type En (n=6,7,8) then
there is an additional component kondo
or frame
, respectively, which
gives the labeling of the characters as determined by Kondo and Frame.
gap> W := CoxeterGroup( "E", 6 );; gap> ChevieCharInfo( W ).frame; [ "1_p", "1_p'", "10_s", "6_p", "6_p'", "20_s", "15_p", "15_p'", "15_q", "15_q'", "20_p", "20_p'", "24_p", "24_p'", "30_p", "30_p'", "60_s", "80_s", "90_s", "60_p", "60_p'", "64_p", "64_p'", "81_p", "81_p'" ]
This function requires the package "chevie" (see RequirePackage).
FakeDegrees( W, q )
returns a list holding the fake degrees of the reflection group W
on the vector space V, evaluated at q. These are the graded
multiplicities of the irreducible characters of W in the quotient
SV/I where SV is the symmetric algebra of V and I is the ideal
generated by the homogeneous invariants of positive degree in SV. The
ordering of the result corresponds to the ordering of the characters in
CharTable(W)
.
gap> q := X( Rationals );; q.name := "q";; gap> FakeDegrees( CoxeterGroup( "A", 2 ), q ); [ q^3, q^2 + q, q^0 ]
This function requires the package "chevie" (see RequirePackage).
FakeDegree( W, phi, q )
returns the fake degree of the character of parameter phi (see CharParams) of the reflection group W, evaluated at q (see FakeDegrees for a definition of the fake degrees).
gap> q := X( Rationals );; q.name := "q";; gap> FakeDegree( CoxeterGroup( "A", 2 ), [ [ 2, 1 ] ], q ); q^2 + q
This function requires the package "chevie" (see RequirePackage).
LowestPowerFakeDegrees( W )
return a list holding the b-function for all irreducible characters of
W, that is, for each character χ, the valuation of the fake
degree of χ. The ordering of the result corresponds to the ordering
of the characters in CharTable(W)
. The advantage of this function
compared to calling FakeDegrees
is that one does not have to provide
an indeterminate, and that it may be much faster to compute than the
fake degrees.
gap> LowestPowerFakeDegrees( CoxeterGroup( "D", 4 ) ); [ 6, 6, 7, 12, 4, 3, 6, 2, 2, 4, 1, 2, 0 ]
This function requires the package "chevie" (see RequirePackage).
HighestPowerFakeDegrees( W )
returns a list holding the B-function for all irreducible characters
of W, that is, for each character χ, the degree of the fake
degree of χ. The ordering of the result corresponds to the ordering
of the characters in CharTable(W)
. The advantage of this function
compared to calling FakeDegrees
is that one does not have to provide
an indeterminate, and that it may be much faster to compute than the
fake degrees.
gap> HighestPowerFakeDegrees( CoxeterGroup( "D", 4 ) ); [ 10, 10, 11, 12, 8, 9, 10, 6, 6, 8, 5, 6, 0 ]
This function requires the package "chevie" (see RequirePackage).
Representations( W, l )
returns a list holding, for each irreducible character of W, a list of matrices images of the generating reflections of W in a model of the corresponding representation. This function is based on the classification, and is not implemented for all irreducible types (most primitive non-real complex reflection groups of Shepard-Todd index greater that 15 are missing).
If there is a second argument, it must be a list of indices, and only the representations with these indices in the list of all representations are returned.
gap> Representations(CoxeterGroup("B",2)); [ [ [ [ 1 ] ], [ [ -1 ] ] ], [ [ [ 1, 0 ], [ -1, -1 ] ], [ [ 1, 2 ], [ 0, -1 ] ] ], [ [ [ -1 ] ], [ [ -1 ] ] ], [ [ [ 1 ] ], [ [ 1 ] ] ], [ [ [ -1 ] ], [ [ 1 ] ] ] ]
This function requires the package "chevie" (see RequirePackage).
79.9 LowestPowerGenericDegrees
LowestPowerGenericDegrees( W )
returns a list holding the a-function for all irreducible characters of
the Coxeter group W, that is, for each character χ, the valuation
of the generic degree of χ (in the one-parameter Iwahori-Hecke
algebra Hecke(W,X(Rationals))
corresponding to W). The ordering of
the result corresponds to the ordering of the characters in
CharTable(W)
.
gap> LowestPowerGenericDegrees( CoxeterGroup( "D", 4 ) ); [ 6, 6, 7, 12, 3, 3, 6, 2, 2, 3, 1, 2, 0 ]
This function requires the package "chevie" (see RequirePackage).
79.10 HighestPowerGenericDegrees
HighestPowerGenericDegrees( W )
returns a list holding the A-function for all irreducible characters of
the Coxeter group W, that is, for each character χ, the degree of
the generic degree of χ (in the one-parameter Iwahori-Hecke algebra
Hecke(W,X(Rationals))
corresponding to W). The ordering of the
result corresponds to the ordering of the characters in CharTable(W)
.
gap> HighestPowerGenericDegrees( CoxeterGroup( "D", 4 ) ); [ 10, 10, 11, 12, 9, 9, 10, 6, 6, 9, 5, 6, 0 ]
This function requires the package "chevie" (see RequirePackage).
PositionSgn( W )
return the position of the sign character in the character table of the group W.
gap> W := CoxeterGroup( "D", 4 );; gap> PositionSgn( W ); 4
See also ChevieCharInfo
(ChevieCharInfo).
This function requires the package "chevie" (see RequirePackage).
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GAP 3.4.4