79 Classes and representations for reflection groups

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; these labelings are used 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 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 labelings for classes and characters are those displayed.

A typical example is CoxeterGroup("A",n), the symmetric group Sn+1 where classes and characters are parameterized 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;
    [ "A0", "~A1", "A1", "G2", "A2", "A1+~A1" ]
    gap> ct.irredinfo;
    [ rec(
          charparam := [ [ 1, 0 ] ],
          charname := "phi{1,0}" ), rec(
          charparam := [ [ 1, 6 ] ],
          charname := "phi{1,6}" ), rec(
          charparam := [ [ 1, 3, 1 ] ],
          charname := "phi{1,3}'" ), rec(
          charparam := [ [ 1, 3, 2 ] ],
          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). The valuation and degree of these give two integers b,B for each irreducible character of W (see LowestPowerFakeDegrees and HighestPowerFakeDegrees). For finite Coxeter groups, the valuation and degree of the generic degrees of the one-parameter generic Hecke algebra give 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 for 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 parameterized 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 × 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,-)(α,+) on an element in the class (λ,+) is given by 2k(λ) times the value of the irreducible character of Sn/2 labeled by α on the class of cycle type λ'. (Here, k(λ) denotes the number of non-zero parts of λ.)

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 parameterized 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 parameterized as the classes of G(de,e,n) with an extra information for a component of a class which splits.

According to Hu85, a class C of G(de,1,n) parameterized 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: in type G2 the character φ1,3' takes the value -1 on the reflection associated to the long simple root; in type F4, the characters φ1,12', φ2,4', φ4,7', φ8,9' and φ9,6' occur in the induced of the identity from the A2 corresponding to the short simple roots; the pairs (φ2,16', φ2,4'') and (φ8,3', φ8,9'') are related by tensoring by sign; and finally φ6,6'' is the exterior square of the reflection representation. Note, however, that in CHEVIE we put the long root at the left 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)).

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)).

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 character φ30,10' has fake degree q10+q12+ higher terms, while φ30,10'' has fake degree q10+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. The matrix representations affording the characters of degree 2 are given by:

ρj : s1s2 → (
E(m)j0
0E(m)-j
),   s1→(
01
10
),
where 1 ≤ j ≤ ⌊ (m-1)/2⌋. The reflection representation is ρ1. The characters in the CHEVIE-table are ordered as ρ12,....

Primitive complex reflection groups G4 to G34. The groups G23=H3, G28=F4, G30=H4 are exceptional Coxeter groups and have been explained above. Similarly for the other groups labels for characters consist primarily of the pair (n,b) where n denotes the degree and b is the corresponding b-invariant. This is sufficient for G4, G12, G22 and G24. For other groups there are pairs or triples of characters which have the same (n,b) value. We disambiguate these according to the conventions of Mal00 for G27, G29, G31 and G34:

• For G27:\ The fake degree of φ3,5' (resp. φ3,20', φ8,9'') has smaller degree that of φ3,5'' (resp. φ3,20'', φ8,9'). The characters φ5,15' and φ5,6' occur with multiplicity 1 in the induced from the trivial character of the parabolic subgroup of type A2 (it is asserted mistakenly in Mal00 that φ5,6'' does not occur in this induced; it occurs with multiplicity 2).

• For G29:\ The character φ6,10′′′ is the exterior square of φ4,1; its complex conjugate is φ6,10′′′′. The character φ15,4'' occurs in φ4,1⊗φ4,3; the character φ15,12'' is tensored by the sign character from φ15,4''. Finally φ6,10' occurs in the induced from the trivial character of the standard parabolic subgroup of type A3.

• For G31:\ The characters φ15,8', φ15,20' and φ45,8'' occur in φ4,1⊗φ20,7; the character φ20,13' is complex conjugate of φ20,7; the character φ45,12' is tensored by sign of φ45,8'. The two terms of maximal degree of the fakedegree of φ30,10' are q50+q46 while for φ30,10'' they are q50+2q46.

• For G33:\ The terms of maximal degree of the fakedegree of φ10,8' are q28+q26 while for φ10,8' they are q28+q24. The terms of maximal degree of the fakedegree of φ40,5' are q31+q29 while for φ40,5'' they are q31+2q29. The character φ10,17' is tensored by sign of φ10,8' and φ40,14' is tensored by sign of φ40,5'.

• For G34:\ The character φ20,33' occurs in φ6,1⊗φ15,14. The character φ70,9' is rational. The character φ70,9'' occurs in φ6,1⊗φ15,14. The character φ70,45' is rational.The character φ70,45'' is tensored by the determinant character of φ70,9''. The character φ560,18' is rational. The character φ560,18′′′ occurs in φ6,1⊗φ336,17. The character φ280,12' occurs in φ6,1⊗φ336,17. The character φ280,30'' occurs in φ6,1⊗φ336,17. The character φ540,21' occurs in φ6,1⊗φ105,20. The character φ105,8' is complex conjugate of φ105,4, and φ840,13' is complex conjugate of φ840,11. The character φ840,23' is complex conjugate of φ840,19. Finally φ120,21' occurs in induced from the trivial character of the standard parabolic subgroup of type A5.

For the groups G5 and G7 we adopt the following conventions. For G5 they are compatible with those of MR03 and BMM14.

• For G5:\ We let W:=ComplexReflectionGroup(5), so the generators in CHEVIE are W.1 and W.2.

The character φ1,4' (resp. φ1,12', φ2,3') takes the value 1 (resp. E(3), -E(3)) on W.1. The character φ1,8'' is complex conjugate to φ1,16, and the character φ1,8' is complex conjugate to φ1,4' . The character φ2,5'' is complex conjugate to φ2,1; φ2,5' take the value -1 on W.1. The character φ2,7' is complex conjugate to φ2,5'.

• For G7:\ We let W:=ComplexReflectionGroup(7), so the generators in CHEVIE are W.1, W.2 and W.3.

The characters φ1,4' and φ1.10' take the value 1 on W.2. The character φ1,8'' is complex conjugate to φ1,16 and φ1,8' is complex conjugate to φ1,4'. The characters φ1,12' and φ1.18' take the value E(3) on W.2. The character φ1,14'' is complex conjugate to φ1,22 and φ1,14' is complex conjugate to φ1,10'. The character φ2,3' takes the value -E(3) on W.2 and φ2,13' takes the value -1 on W.2. The characters φ2,11'', φ2,5'', φ2,7′′′ and φ2,1 are Galois conjugate, as well as the characters φ2,7', φ2,13', φ2,11' and φ2,5'. The character φ2,9' is complex conjugate to φ2,15 and φ2,9′′′ is complex conjugate to φ2,3'.

Finally, for the remaining groups G6, G8 to G11, G13 to G21, G25, G26, G32 and G33 there are only pairs of characters with same value (n,b). We give labels uniformly to these characters by applying in order the following rules :\

• If the two characters have different fake degrees, label φn,b' the one whose fake degree is minimal for the lexicographic order of polynomials (starting with the highest term).

• For the not yet labeled pairs, if only one of the two characters has the property that in its Galois orbit at least one character is distinguished by its (n,b)-invariant, label it φn,b'.

• For the not yet labeled pairs, if the minimum of the (n,b)-value (for the lexicographic order (n,b)) in the Galois orbits of the two character is different, label φn,b' the character with the minimal minimum.

• We define now a new invariant for characters: consider all the pairs of irreducible characters χ and ψ uniquely determined by their (n,b)-invariant such that φ occurs with non-zero multiplicity m in χ⊗ψ. We define t(φ) to be the minimal (for lexicographic order) possible list (n(χ),b(χ),n(ψ),b(ψ),m).

For the not yet labeled pairs, if the t-invariants are different, label φn,b' the character with the minimal t-invariant.

After applying the last rule all the pairs will be labelled for the considered groups. The labelling obtained is compatible for G25, G26, G32 and G33 with that of Mal00 and for G8 with that described in MR03.

We should emphasize that for all groups with a few exceptions, the parameters for characters do not depend on any non-canonical choice. The exceptions are G(de,e,n) with e>1, and G5, G7, G27, G28, G29 and G34, groups which admit ``diagram'' automorphisms, so choices of a particular value on a particular generator must be made for characters which are not invariant by these automorphisms.

Labels for the classes. For the exceptional complex reflection groups, the labels for the classes represent the decomposition of a representative of the class as a product of generators, with the additional conventions that z represents the generator of the center and for well-generated groups c represents a Coxeter element (a product of the generators which is a regular element for the highest reflection degree).

CharParams:

Returns the list of parameters for irreducible characters of W as partitions for type A, double partitions for type B, etc... CharName also has a special version which knows how to display nicely such labels.

Subsections

  1. ChevieClassInfo
  2. WordsClassRepresentatives
  3. ChevieCharInfo
  4. FakeDegrees
  5. FakeDegree
  6. LowestPowerFakeDegrees
  7. HighestPowerFakeDegrees
  8. Representations
  9. LowestPowerGenericDegrees
  10. HighestPowerGenericDegrees
  11. PositionDet
  12. DetPerm

79.1 ChevieClassInfo

ChevieClassInfo( W )

returns information about the conjugacy classes of the finite reflection group W. The result is a record with three components:\

classtext:

contains words in the generators describing representatives of each conjugacy class; this is the same as WordsClassRepresentatives(W) and for finite Coxeter groups the representatives given are of minimal length (the representatives taken are explained in GM97).

classparams:

The elements of this list are tuples which have one component for each irreducible component of W. These components for the infinite series, contain partitions or partition tuples describing the class (see the introduction). For the exceptional Coxeter groups they contain Carter's admissible diagrams, see Car72. For exceptional complex reflection groups they contain in general the same information as in classtext.

classnames:

Contains strings describing the conjugacy classes, made out of the information in 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, 4, 3 ],
          [ 2, 4, 3 ] ],
      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 ], '-' ] ] ],
      classnames := [ "1111.", "11.11", ".1111", "211.", "1.21", "2.11",
          "22.+", "22.-", ".22", "31.", ".31", "4.+", "4.-" ])
    gap> ChevieClassInfo(ComplexReflectionGroup(3,1,2));
      rec(
        classparams :=
         [ [ [ [ 1, 1 ], [  ], [  ] ] ], [ [ [ 1 ], [ 1 ], [  ] ] ],
            [ [ [ 1 ], [  ], [ 1 ] ] ], [ [ [  ], [ 1, 1 ], [  ] ] ],
            [ [ [  ], [ 1 ], [ 1 ] ] ], [ [ [  ], [  ], [ 1, 1 ] ] ],
            [ [ [ 2 ], [  ], [  ] ] ], [ [ [  ], [ 2 ], [  ] ] ],
            [ [ [  ], [  ], [ 2 ] ] ] ],
        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" ])

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 complex reflection group W. Each element in this list is given as a positive word in the standard generators, which is represented as a list of integers where the generator si is represented by the integer i. 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 "very good" element in the sense of GM97.

    gap> WordsClassRepresentatives( CoxeterGroup( "F", 4 ) );
   [ [  ],
     [ 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1,
      2, 3, 4 ], [ 2, 3, 2, 3 ], [ 2, 1 ],                                    
     [ 1, 2, 3, 4, 2, 3, 2, 3, 4, 3 ],                                         
     [ 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4 ], [ 4, 3 ],                         
     [ 1, 2, 1, 3, 2, 3, 1, 2, 3, 4 ],                                          
     [ 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4 ],                       
     [ 1, 2, 3, 4, 1, 2, 3, 4 ], [ 1, 2, 3, 4 ], [ 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 ],           
     [ 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 2, 3 ], [ 1, 2, 3, 4, 2, 3 ] ] 

See also ChevieClassInfo.

This function requires the package "chevie" (see RequirePackage).

79.3 ChevieCharInfo

ChevieCharInfo( W )

returns information about the irreducible characters of the finite reflection group W. The result is a record with the following components:\

charparams:

contains parameters for the irreducible characters as described in the introduction or returned by 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:

strings describing the irreducible characters, computed from the charparams.

positionId:

the position of the trivial character in the character table of W (which is also returned by the function PositionId).

extRefl:

Only present if W is irreducible, in which case the reflection representation of W and all its exterior powers are irreducible. It then contains the position of the exterior powers of the reflection representation in the character table.

a:

contains the result of LowestPowerFakeDegrees(W).

A:

contains the result of HighestPowerFakeDegrees(W).

b:

Only filled for Spetsial groups. Contains the result of LowestPowerGenericDegrees(W).

B:

Only filled for Spetsial groups. Contains the result of HighestPowerGenericDegrees(W).

positionSgn:

Contains the position of the determinant character in the character table of W (which is also returned by the function PositionDet). For Coxeter groups this is the sign character.

    gap> ChevieCharInfo(ComplexReflectionGroup(4));
    rec(
      charparams := [ [ [ 1, 0 ] ], [ [ 1, 4 ] ], [ [ 1, 8 ] ],
          [ [ 2, 5 ] ], [ [ 2, 3 ] ], [ [ 2, 1 ] ], [ [ 3, 2 ] ] ],
      extRefl := [1, 6, 2],
      a := [ 0, 4, 4, 4, 1, 1, 2 ],
      A := [ 0, 8, 8, 8, 5, 5, 6 ],
      b := [ 0, 4, 8, 5, 3, 1, 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,
      positionDet := 2,
      B := [ 0, 4, 8, 7, 5, 3, 6 ] ) 
    gap>  ChevieCharInfo( CoxeterGroup( "G", 2 ) );
    rec(
      charparams := [ [ [ 1, 0 ] ], [ [ 1, 6 ] ], [ [ 1, 3, 1 ] ],
          [ [ 1, 3, 2 ] ], [ [ 2, 1 ] ], [ [ 2, 2 ] ] ],
      extRefl := [1, 5, 2],
      a := [ 0, 6, 1, 1, 1, 1 ],
      A := [ 0, 6, 5, 5, 5, 5 ],
      b := [ 0, 6, 3, 3, 1, 2 ],
      charnames := [ "phi{1,0}", "phi{1,6}", "phi{1,3}'", "phi{1,3}''",
          "phi{2,1}", "phi{2,2}" ],
      positionId := 1,
      positionDet := 2,
      B := [ 0, 6, 3, 3, 5, 4 ] ) 

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).

79.4 FakeDegrees

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).

79.5 FakeDegree

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).

79.6 LowestPowerFakeDegrees

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).

79.7 HighestPowerFakeDegrees

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).

79.8 Representations

Representations( W[, l])

returns a list holding, for each irreducible character of the complex reflection group 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 yet fully implemented for G34; still missing are 3 representations of dim. 70, 4 of dim. 105, 4 of dim. 315, 6 of dim. 420 and those of dim. 90, 120, 140, 189, 280, 384 or greater than 420.

If there is a second argument, it can be a list of indices (or a single integer) and only the representations with these indices (or that index) 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 ] ] ] ]
    gap> Representation(ComplexReflectionGroup(4),7);
    [ [ [ E(3)^2, 0, 0 ], [ 2*E(3)^2, E(3), 0 ], [ E(3), 1, 1 ] ], 
      [ [ 1, -1, E(3) ], [ 0, E(3), -2*E(3)^2 ], [ 0, 0, E(3)^2 ] ] ]

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 or Spetsial reflection group W, that is, for each character χ, the valuation of the generic degree of χ (in the one-parameter Hecke algebra Hecke(W,X(Cyclotomics)) 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 or Spetsial reflection group W, that is, for each character χ, the degree of the generic degree of χ (in the one-parameter Hecke algebra Hecke(W,X(Cyclotomics)) 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).

79.11 PositionDet

PositionDet( W )

return the position of the determinant character in the character table of the group W (for Coxeter groups this is the sign character).

    gap> W := CoxeterGroup( "D", 4 );;
    gap> PositionDet( W );
    4

See also ChevieCharInfo (ChevieCharInfo).

This function requires the package "chevie" (see RequirePackage).

79.12 DetPerm

DetPerm( W )

return the permutation of the characters of the reflection group W which is effected when tensoring by the determinant character (for Coxeter groups this is the sign character).

    gap> W := CoxeterGroup( "D", 4 );;
    gap> DetPerm( W );
    [ 8, 9, 11, 13, 5, 6, 12, 1, 2, 10, 3, 7, 4 ]

This function requires the package "chevie" (see RequirePackage). Previous Up Next
Index

GAP 3.4.4
April 1997