Classes/characters of reflection groups

Chevie.Chars — Module

Characters and conjugacy classes of complex reflection groups.

The CharTable of a finite complex reflection group W is computed using the decomposition of W in irreducible groups (see refltype). 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 we give. For the classes, these are partitions or partition tuples for the infinite series, or, for exceptional Coxeter groups, Carter's admissible diagrams Carter1972; 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 (d,e) where d is the degree of the character and e is the smallest symmetric power of the reflection representation containing the given character as a constituent (the b-invariant of the character). This information is given by the functions classinfo and charinfo. When you display the character table, the canonical labelings for classes and characters are displayed.

A typical example is coxgroup(:A,n), the symmetric group 𝔖ₙ₊₁ where classes and characters are parameterized by partitions of n+1 (this is also the case for coxsym(n+1)).

julia> W=coxgroup(:A,3)
A₃

julia> CharTable(W)
CharTable(A₃)
┌────┬─────────────────┐
│    │1111 211 22 31  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│
└────┴─────────────────┘

julia> W=coxgroup(:G,2)
G₂

julia> ct=CharTable(W)
CharTable(G₂)
┌─────┬────────────────────┐
│     │A₀ Ã₁ A₁ G₂ A₂ A₁+Ã₁│
├─────┼────────────────────┤
│φ₁‚₀ │ 1  1  1  1  1     1│
│φ₁‚₆ │ 1 -1 -1  1  1     1│
│φ′₁‚₃│ 1  1 -1 -1  1    -1│
│φ″₁‚₃│ 1 -1  1 -1  1    -1│
│φ₂‚₁ │ 2  .  .  1 -1    -2│
│φ₂‚₂ │ 2  .  . -1 -1     2│
└─────┴────────────────────┘

julia> ct.charnames
6-element Vector{String}:
 "\phi_{1,0}"
 "\phi_{1,6}"
 "\phi_{1,3}'"
 "\phi_{1,3}''"
 "\phi_{2,1}"
 "\phi_{2,2}"

julia> ct.classnames
6-element Vector{String}:
 "A_0"
 "\tilde A_1"
 "A_1"
 "G_2"
 "A_2"
 "A_1+\tilde A_1"

Reflection groups have fake degrees (see fakedegrees), whose valuation and degree give two integers b,B for each irreducible character of W. For spetsial groups (which include finite Coxeter groups), the valuation and degree of the generic degrees of the Hecke algebra give two more integers a,A (for Coxeter groups see Carter1985, Ch.11 for more details). These integers are also used in the operations of truncated induction, see j_induction_table and J_induction_table.

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 labelled by partitions of n+1. The partition labelling a class is the cycle type of the elements in that class; the representative in '.classtext' is the concatenation of the words corresponding to each part, where the word for a part i is the product of i-1 consecutive generators (starting one higher than the last generator used for the previous parts). The partition labelling 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 is labelled by the partition (1ⁿ⁺¹) and the trivial character by the partition (n+1). The character of the reflection representation of W is labelled 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 Carter1972. 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 respectively the trivial class and the class of the longest element.

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 $η_{a,b}$ be the irreducible character of N which takes value 1 on the first a standard basis vectors and value -1 on the last b standard basis vectors of N. Then the inertia subgroup of $η_{a,b}$ has the form $T_{a,b}=N.(𝔖_a × 𝔖_b)$ and we can extend $η_{a,b}$ trivially to an irreducible character $η̃_{a,b}$ of $T_{a,b}$. Let α and β be partitions of a and b, respectively. We take the tensor product of the corresponding irreducible characters of 𝔖_a and 𝔖_b and regard this as an irreducible character of $T_{a,b}$. Multiplying this character with $η̃_{a,b}$ 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 Stembridge1989 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 𝔖_{n/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,+)} on an element in the class (λ,+) is given by 2^{k(λ)} times the value of the irreducible character of 𝔖_{n/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 are 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 used to express the previous part.

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 Hugues1985, a class C of G(de,1,n) parameterized by a de-partition $(S₀,…,S_{de-1})$ is in G(de,e,n) if e divides $∑ᵢ i ∑_{p∈ Sᵢ}p$. 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 'gⁱ 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 Carter1985, 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 we put the long root at the left of the Dynkin diagrams to be in accordance with the conventions in Lusztig1985, (4.8) and (4.10).

The classes are labeled by Carter's admissible diagrams Carter1972. A character is labeled by a pair (d,b) where d denotes the degree and b the corresponding b-invariant. If there are several characters with the same pair (d,b) we attach a prime to them, as in Carter1985.

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 Carter1972. A character is labeled by the pair (d,b) where d denotes the degree and b is the corresponding b-invariant. For these types, this gives a unique labeling of the characters.

Non-crystallographic types I₂(m), H₃, H₄. In these cases we do not have canonical labelings for the classes. We label them by reduced expressions.

Each character for type H₃ is uniquely determined by the pair (d,b) where d 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 table for type H₄ are ordered in the same way as in Alvis and Lusztig1982.

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: $ρ_j : s₁s₂ ↦ \begin{pmatrix}\zeta_m^j&0\\0&\zeta_m^{-j}\end{pmatrix}, s₁↦\begin{pmatrix}0&1\\1&0\end{pmatrix},$ where 1 ≤ j ≤ ⌊(m-1)/2⌋. The reflection representation is ρ₁. The characters in the table are ordered by listing first the characters of degree 1 and then ρ₁,ρ₂,….

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 (d,b) where d 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 (d,b) value. We disambiguate these according to the conventions of Malle2000 for G₂₇, G₂₉, G₃₁, G₃₃ 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 generated by the first and third generator (it is asserted mistakenly in Malle2000 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 generated by the first, second and fourth generators.

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 two 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 generated by the generators of G₃₄ with the third one omitted.

For the groups G₅ and G₇ we adopt the following conventions. For G₅ they are compatible with those of MalleRouquier2003 and BroueMalleMichel2014.

For G₅:

We let W=complex_reflection_group(5), so the generators are W(1) and W(2).

The character φ₁,₄' (resp. φ₁,₁₂', φ₂,₃') takes the value 1 (resp. ζ₃, -ζ₃) 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' takes the value -1 on W(1). The character φ₂,₇' is complex conjugate to φ₂,₅'.

For G₇:

We let W=complex_reflection_group(7), so the generators 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 ζ₃ 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 -ζ₃ 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 (d,b). We give labels uniformly to these characters by applying in order the following rules :

If the two characters have different fake degrees, label φ_{d,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 (d,b)-invariant, label it φ_{d,b}'.
For the not yet labeled pairs, if the minimum of the (d,b)-value (for the lexicographic order (d,b)) in the Galois orbits of the two character is different, label φ_{d,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 (d,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 (d(χ),b(χ),d(ψ),b(ψ),m).

For the not yet labeled pairs, if the t-invariants are different, label φ_{d,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 Malle2000 and for G₈ with that described in MalleRouquier2003.

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 outer automorphisms preserving the set of reflections, 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).