# 77 Character tables for Coxeter groups

The ordinary complex character table of any finite Coxeter group is computed in CHEVIE using the function `CharTable`. This command first checks whether or not W is irreducible. For each irreducible Coxeter group a character table record is computed either using recursive formulas (for the classical types: these functions are part of GAP and described in detail in~Pfe94a) or read into the system from a library file (for the exceptional types: the tables can be found in the Cambridge ATLAS CCN85, for example). Thus, character tables can be obtained quickly even for very large groups (e.g., E_8) since they are obtained via the classification and special algorithms or stored tables for irreducible groups. The above record is a usual character table record as defined in GAP, but with some additional components.

It is important to note that the conjugacy classes and the irreducible characters of a finite Coxeter group each have a canonical labeling by certain combinatorial objects, and that such labelings are contained in a consistent way in the tables of CHEVIE. For the classes, these are partitions, pairs of partitions, or Carter's admissible diagrams Car72. For the characters, these are again partitions, pairs of partitions, or 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`. It is printed automatically when you display the character table in GAP.

The prototype for this is the symmetric group {mathfrak S}_{n+1} (type A_n) where the classes and characters are parametrized by partitions of n+1.

```    gap> W := CoxeterGroup( "A", 3 );;
gap> Display( CharTable( W ));
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 ```

Recall that our Coxeter groups acts a reflection group on the real vector space V (we do not assume that V is spanned by the roots, i.e., that `W.rank = W.semisimpleRank`). This reflection representation and its character contain some further useful information about~W.

Let SV be the symmetric algebra of V. The invariants of W in SV are called the polynomial invariants of W. They are generated as a polynomial ring by dim V homogeneous algebraically independent polynomials f_1,ldots,f_{dim V}. The polynomials f_i are not uniquely determined but their degrees are. The f_i are called the basic invariants of W, and their degrees the reflection degrees of W. Let I be the ideal generated by the homogeneous invariants of positive degree in SV. Then SV/I is isomorphic to the regular representation of W as a W-module. It is thus a graded (by the degree of elements of SV) version of the regular representation of W. The polynomial which gives the graded multiplicity of a character chi of W in the graded module SV/I is called the fake degree of chi. (See ReflectionCharValue, ReflectionDegrees and FakeDegree.)

Using these constructions and the generic degrees of the corresponding one-parameter generic Iwahori--Hecke algebra, one can associate four integers a,A,b,B with each irreducible character of W (see the functions LowestPowerGenericDegrees, HighestPowerGenericDegrees, LowestPowerFakeDegrees, HighestPowerFakeDegrees, and cite[Ch.11]Car85 for more details). These will also be used in the operations of truncated inductions in chapter Reflection subgroups.

Iwahori-Hecke algebras associated with finite Coxeter groups also have character tables, see Chapter Iwahori-Hecke algebras.

We now describe, for each type of irreducible finite Coxeter groups, our conventions about labeling the classes and characters. Assume that the Root systems and finite Coxeter groups.

smallskip noindent em Type A_n (n geq 0). In this case we have W cong {mathfrak S}_{n+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 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 every other character occuring in this induced character has a higher a-value. Thus, the sign character corresponds to the partition (1^{n+1}) and the trivial character to the partition (n+1). The character of the reflection representation of W is labeled by (n,1).

medskip noindent em Type B_n (n geq 2). In this case W=W(B_n) is isomorphic to the wreath product of the cyclic group of order~2 with the symmetric group {mathfrak S}_n. 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 (lambda,mu) is such a pair then lambda corresponds to the positive and mu to the negative cycles. Thus, (1^n,-) and (-,1^n) 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(B_n)=N.{mathfrak S}_n where N is the standard n-dimensional {FF}_2^n-vector space. For a,b geq 0 such that n=a+b let eta_{a,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 eta_{a,b} has the form T_{a,b}:=N.({mathfrak S}_a times {mathfrak S}_b) and we can extend eta_{a,b} trivially to an irreducible character tilde{eta}_{a,b} of T_{a,b}. Let alpha and beta be partitions of a and b, respectively. We take the tensor product of the corresponding irreducible characters of {mathfrak S}_a and {mathfrak S}_b and regard this as an irreducible character of T_{a,b}. Multiplying this character with tilde{eta}_{a,b} and inducing to W(B_n) yields an irreducible character chi= chi_{(alpha,beta)} of W(B_n). This defines the correspondence between irreducible characters and pairs of partitions as above.

For example, the pair ((n),-) labels the trivial character and (-,(1^n)) labels the sign character. The character of the natural reflection representation is labeled by ((n-1),(1)).

smallskip noindent em Type D_n (n geq 4). In this case W=W(D_n) can be embedded as a subgroup of index~2 into the Coxeter group W(B_n). The intersection of a class of W(B_n) with W(D_n) is either empty or a single class in W(D_n) or splits up into two classes in W(D_n). This also leads to a parametrization of the classes of W(D_n) by pairs of partitions (lambda,mu) as before but where the number of parts of mu is even and where there are two classes of this type if mu is empty and all parts of lambda are even. In the latter case we denote the two classes in W(D_n) by (lambda,+) and (lambda,-), where we use the convention that the class labeled by (lambda,+) contains a representative which can be written as a word in {s_1,s_3,ldots,s_n} and (lambda,-) contains a representative which can be written as a word in {s_2,s_3, ldots,s_n}.

By Clifford theory the restriction of an irreducible character of W(B_n) to W(D_n) is either irreducible or splits up into two irreducible components. Let (alpha,beta) be a pair of partitions with total sum of parts equal to n. If alpha neq beta then the restrictions of the irreducible characters of W(B_n) labeled by (alpha,beta) and (beta, alpha) are irreducible and equal. If alpha=beta then the restriction of the character labeled by (alpha,alpha) splits into two irreducible components which we denote by (alpha,+) and (alpha,-). 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 (lambda,+) in terms of the character values of the symmetric group {mathfrak S}_{n/2}. Recall that it is implicit in the notation (lambda,+) that all parts of lambda are even. Let lambda^prime be the partition of n/2 obtained by dividing each part by~2. Then the value of [chi_(alpha,-)-chi_(alpha,+)] on an element in the class (lambda,+) is given by 2^{k(lambda)} times the value of the irreducible character of {mathfrak S}_{n/2} labeled by alpha on the class of cycle type lambda^prime. (Here, k(lambda) denotes the number of non-zero parts of lambda.)

The labels for the trivial, the sign and the natural reflection character are the same as for W(B_n), since these characters are restrictions of the corresponding characters of W(B_n).

smallskip em Types G_2 and F_4. The matrices of character values and the orderings and labelings of the irreducible characters are exactly the same as in cite[p.412/413]Car85. Note, however, that in CHEVIE we have reversed the labeling of the Dynkin diagrams to be in accordance with the conventions in cite[(4.8) and (4.10)]Lus85.

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 F_4 the result of `ChevieCharInfo` contains an additional component `kondo` which contains the labels originally given by Kondo (and which are also used in cite[(4.10)]Lus85). The reflection character is labeled by (4,1) or 4_2 (Kondo).

smallskip em Types E_6,E_7,E_8. The character tables are re-constructed from the compound tables in the Cambridge ATLAS CCN85, p.26, p.46 and p.85, respectively (or, they can be recomputed with GAP by the general Dixon-Schneider algorithm). 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 E_6, E_7, E_8, respectively, the reflection character is the one with label (6,1), (7,1), (8,1) or 6_p, 7_a^prime, 8_z (Frame).

smallskip em Non-crystallographic types I_2(m), H_3, H_4. In these cases we do not have canonical labelings for the classes.

Each character for type H_3 is uniquely determined by the pair (n,b) where n is the degree and b the corresponding b-invariant. For type H_4 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 q^{10}+q^{12}+ higher terms, while the second has fake degree q^{12}+q^{14}+ higher terms. The characters in the CHEVIE-table for type H_4 are ordered in the same way as in AL82.

Finally, the characters of degree~2 for type I_2(m) are ordered as follows. Let varepsilon be a primitive m-th root of unity. Then matrix representations affording the characters of degree~2 are given by: [ varphi_j colon s_1s_2 mapsto left( beginarraycc varepsilon^j & 0
0 & varepsilon^-j endarray right), quad s_1 mapsto left( beginarraycc 0 & 1
1 & 0 endarray right),] where 1 leq j leq (m-1)/2 for m odd, and 1 leq j leq (m-2)/2 for m even. The natural reflection representation determined by the root system is rho=varphi_1.

In GAP we take varepsilon as `E( m )`. Then the characters in the CHEVIE-table are ordered as varphi_1,varphi_2,ldots.

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GAP 3.4.4
April 1997