Reflection cosets and Spets

Let R be a root system in the real vector space V. We say that F₀∈ GL(V) is an automorphism of R if it permutes R and is of finite order (finite order is automatic if R generates V). It follows by chap. VI, §1.1, lemme 1 Bourbaki1968 that the dual F₀*∈ GL(V*) permutes the coroots R*⊂ V*; thus F₀ normalizes the reflection group W associated to R, that is w↦ F₀wF₀⁻¹ is an automorphism of W. Thus we get a reflection coset WF₀, which we call a Coxeter coset.

The motivation for introducing Coxeter cosets comes from automorphisms of algebraic reductive groups, giving rise to non-split reductive groups over finite fields. Let 𝐆 be a connected reductive algebraic group 𝐆 over an algebraic closure 𝔽̄_q of a finite field 𝔽_q, defined over 𝔽_q; this corresponds to a Frobenius endomorphism F so that the finite group of rational points 𝐆(𝔽_q) identifies to the subgroup 𝐆^F of fixed points under F.

Let 𝐓 be a maximal torus of 𝐆, and Φ (resp. Φ*) be the roots (resp. coroots) of 𝐆 with respect to 𝐓 in the character group X(𝐓) (resp. the group of one-parameter subgroups Y(𝐓)). Then 𝐆 is determined up to isomorphism by (X(𝐓),Φ,Y(𝐓),Φ*); this corresponds to give a root system in the vector space V=ℚ ⊗ X(𝐓) and a rational reflection group W=N_𝐆(𝐓)/𝐓 acting on it.

If 𝐓 is F-stable the Frobenius endomorphism F acts also naturally on X(T) and defines thus an endomorphism of V, which is of the form q F₀, where F₀∈ GL(V) is of finite order and normalizes W. We get thus a Coxeter coset WF₀⊂GL(V). The data (X(𝐓), Φ, Y(𝐓), Φ*, F₀), and the integer q completely determine up to isomorphism the associated reductive finite group 𝐆^F. Thus these data is a way of representing in the essential information which determines a finite reductive group. Indeed, all properties of Chevalley groups can be computed from that datum: symbols representing characters, conjugacy classes, and finally the whole character table of 𝐆^F.

It turns out that many interesting objects attached to this datum depend only on (V,W, F₀): the order of the maximal tori, the fake degrees, the order of 𝐆^F, symbols representing unipotent characters, Deligne-Lusztig induction in terms of almost characters, the Fourier matrix relating characters and almost characters, etc… (see, e.g., Broue-Malle-Michel1993). It is thus possible to extend their construction to non-crystallographic groups (or even to more general complex reflection groups, see spets); this is why we did not include a root system in the definition of a reflection coset. However, unipotent conjugacy classes for instance depend on the root system.

We assume now that 𝐓 is contained in an F-stable Borel subgroup of 𝐆. This defines an order on the roots, and there is a unique element ϕ∈ W F₀, the reduced element of the coset, which preserves the set of positive roots. It thus defines a diagram automorphism, that is an automorphism of the Coxeter system (W,S). This element is stored in the component .phi of the coset record. It may be defined without mentioning the roots, as follows: (W,F₀(S)) is another Coxeter system, thus conjugate to S by a unique element of W, thus there is a unique element ϕ∈ WF₀ which stabilizes S (a proof follows from Theoreme 1, chap. V, §3 Bourbaki1968). We consider thus cosets of the form Wϕ where ϕ stabilizes S. The coset W ϕ is completely defined by the permutation .phi when 𝐆 is semi-simple –- equivalently when Φ generates V; in this case we just need to specify phi to define the coset.

There is a slight generalisation of the above setup, covering in particular the case of the Ree and Suzuki groups. We consider 𝐆^F where F not a Frobenius endomorphism, but an isogeny such that some power F^n is a Frobenius endomorphism. Then F still defines an endomorphism of V which normalizes W; we define a real number q such that F^n is attached to an 𝔽_{qⁿ}-structure. Then we still have F=q F₀ where F₀ is of finite order but q is no more an integer. Thus F₀∈ GL(V⊗ ℝ) but F₀∉ GL(V). For instance, for the Ree and Suzuki groups, F₀ is an automorphism of order 2 of W, which is of type G₂, B₂ or F₄, and q=√2 for B₂ and F₄ and q=√3 for G₂ This can be constructed starting from root systems for G₂, B₂ or F₄ where all the roots have the same length. This kind of root system is not crystallographic. Such non-crystallographic root systems exist for all finite Coxeter groups such as the dihedral groups, H₃ and H₄. We will call here Weyl cosets the cosets corresponding to rational forms of algebraic groups, which include thus some non-rational roots systems for B₂, G₂ and F₄.

Spets

We now extend the above notions to general complex reflection groups. Let W⊂ GL(V) be a complex reflection group on the vector space V. Let ϕ be an element of GL(V) which normalizes W. Then the coset Wϕ is called a reflection coset.

A reference for these cosets is Broue-Malle-Michel 1999. When W is a so-called Spetsial group, they are the basic object for the construction of a Spetses, which is an object attached to a complex reflection group from which one can derive combinatorially some attributes shared with finite reductive groups, like unipotent degrees, etc….

We say that a reflection coset is irreducible if W is irreducible. A general coset is a direct product of descents of scalars, which is the case where ϕ is transitive on the irreducible components of W. The irreducible cosets have been classified in Broue-Malle-Michel 1999: up to multiplication of ϕ by a scalar, there is usually only one or two possible cosets for a given irreducible group.

We deal only with finite order cosets, that is, we assume there is a (minimal) integer δ such that (Wϕ)^δ=W. Then the group generated by W and ϕ is finite, of order δ|W|.

A subset C of a Wϕ is called a conjugacy class if one of the following equivalent conditions is fulfilled:

• C is the orbit of an element in Wϕ under the conjugation action of W.

• C is a conjugacy class of ⟨W,ϕ⟩ contained in Wϕ.

• The set {w∈ W|wϕ∈ C} is a ϕ-conjugacy class of W (two elements

v,w∈ W are called ϕ-conjugate, if and only if there exists x∈ W with v=xwϕ(x⁻¹)).

An irreducible character of ⟨W,ϕ⟩ has some non-zero values on Wϕ if and only if its restriction to W is irreducible. Further, two characters χ₁ and χ₂ which have same irreducible restriction to W differ by a character of the cyclic group ⟨ϕ⟩ (which identifies to the quotient ⟨W,ϕ⟩/W). A set containing one extension to ⟨W,ϕ⟩ of each ϕ-invariant character of W is called a set of irreducible characters of Wϕ. Two such characters are orthogonal for the scalar product on the class functions on Wϕ given by $⟨χ,ψ⟩:=|W|¹∑_{w∈ W}χ(wϕ)\overline{ψ(wϕ)}.$ For rational groups (Weyl groups), Lusztig has defined a choice of a set of irreducible characters for Wϕ (called the preferred extensions), but for more general reflection cosets we have made some rather arbitrary choices, which however have the property that their values lie in the smallest possible field.

The character table of Wϕ is the table of values of a set of irreducible characters on the conjugacy classes.

A subcoset Lwϕ of Wϕ is given by a reflection subgroup L of W and an element w of W such that wϕ normalizes L.

We then have a natural notion of restriction of class functions on Wϕ to class functions on Lwϕ as well as of induction in the other direction. These maps are adjoint with respect to the scalar product defined above (see Broue-Malle-Michel 1999).

In this package the most general construction of a reflection coset is by starting from a reflection datum, and giving in addition the matrix F of the map ϕ:V→ V (see the command spets). However, at present, general cosets are only implemented for groups represented as permutation groups on a set of roots, and it is required that the automorphism given preserves this set up to a scalar (it is allowed that these scalars depend on the pair of an irreducible component and its image). It is also allowed to specify ϕ by the permutation it induces on the roots; in this case it is assumed that ϕ acts trivially on the orthogonal of the roots, but the roots could be those of a parent group, generating a larger space. Thus in any case we have a permutation representation of ⟨W,ϕ⟩ and we consider the coset to be a set of permutations.

Reflection cosets are implemented in by a struct which points to a reflection group record and has additional fields holding F and the corresponding permutation phi. In the general case, on each component of W which is a descent of scalars, F will permute the components and differ by a scalar on each component from an automorphism which preserves the roots. In this case, we have a permutation phi and a scalar which is stored for that component.

The most common situation where cosets with non-trivial phi arise is as sub-cosets of reflection groups. Here is an "exotic" example, see the next chapter for more classical examples involving Coxeter groups.

julia> W=complex_reflection_group(14)
G₁₄

julia> R=reflection_subgroup(W,[2,4])
G₁₄₍₂₄₎=G₅

julia> RF=spets(R,W(1)) # should be ²G₅(√6)
G₁₄₍₂₄₎=²G₅

julia> diagram(RF)
ϕ acts as (1,2) on the component below
③ ══③ G₅
1   2

julia> degrees(RF)
2-element Vector{Tuple{Int64, Cyc{Int64}}}:
 (6, 1)
 (12, -1)

The last line shows for each reflection degree the corresponding factor of the coset, which is the scalar by which ϕ acts on the corresponding fundamental reflection invariant. The factors characterize the coset.

A spets by default is printed in an abbreviated form which describes its type, as above (G₅ twisted by 2, with a Cartan matrix which differs from the standard one by a factor of √6). The function repr gives a form which could be input back in Julia. With the same data as above we have:

julia> print(RF)
spets(reflection_subgroup(complex_reflection_group(14),[2, 4]),perm"(1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(62,79)(64,80)(65,81)(66,82)(67,69)(68,70)(83,100)(84,101)(85,102)(87,103)(89,99)(90,97)(91,98)(92,96)(93,104)(94,95)(105,113)(106,114)(109,111)(110,112)(115,118)(116,117)(119,120)")

Conjugacy classes and irreducible characters of Coxeter cosets are defined as for general reflection cosets. For irreducible characters of Weyl cosets, we choose (following Lusztig) for each ϕ-stable character of W a particular extension to a character of W⋊ ⟨ϕ⟩, which we will call the preferred extension. The character table of the coset Wϕ is the table of the restrictions to Wϕ of the preferred extensions. The question of finding the conjugacy classes and character table of a Coxeter coset can be reduced to the case of irreducible root systems R.

• The automorphism ϕ permutes the irreducible components of W, and Wϕ is a direct product of cosets where ϕ permutes cyclically the irreducible components of W. The preferred extension is defined to be the direct product of the preferred extension in each of these situations.

• Assume now that Wϕ is a descent of scalars, that is the decomposition in irreducible components W=W₁× ⋯ × Wₖ is cyclically permuted by ϕ. Then there are natural bijections from the ϕ-conjugacy classes of W to the ϕᵏ-conjugacy classes of W₁ as well as from the ϕ-stable characters of W to the ϕᵏ-stable characters of W₁, which reduce the definition of preferred extensions on Wϕ to the definition for W₁ϕᵏ.

• Assume now that W is the Coxeter group of an irreducible root system. ϕ permutes the simple roots, hence induces a graph automorphism on the corresponding Dynkin diagram. If ϕ=1 then conjugacy classes and characters coincide with those of the Coxeter group W.

The nontrivial cases for crystallographic roots systems are (the order of ϕ is written as left exponent to the type): ²Aₙ, ²Dₙ, ³D₄, ²E₆. For non-crystallographic root systems where all the roots have the same length the additional cases ²B₂, ²G₂, ²F₄ and ²I₂(k) arise.

• In case ³D₄ the group W⋊ ⟨ϕ⟩ can be embedded into the Coxeter group of type F₄, which induces a labeling for the conjugacy classes of the coset. The preferred extension is chosen as the (single) extension with rational values.

• In case ²Dₙ the group W⋊ ⟨ϕ⟩ is isomorphic to a Coxeter group of type Bₙ. This induces a canonical labeling for the conjugacy classes of the coset and allows to define the preferred extension in a combinatorial way using the labels (pairs of partitions) for the characters of the Coxeter group of type Bₙ.

• In the remaining crystallographic cases ϕ identifies to -w₀ where w₀ is the longest element of W. So, there is a canonical labeling of the conjugacy classes and characters of the coset by those of W. The preferred extensions are defined by describing the signs of the character values on -w₀.

The most general construction of a Coxeter coset is by starting from a Coxeter datum specified by the matrices of simpleRoots and simplecoroots, and giving in addition the matrix F0Mat of the map F₀:V→ V (see the commands CoxeterCoset and CoxeterSubCoset). As for Coxeter groups, the elements of Wϕ are uniquely determined by the permutation they induce on the set of roots R. We consider these permutations as elements of the Coxeter coset.

Coxeter cosets are implemented by a struct which points to a Coxeter datum record and has additional fields holding F0Mat and the corresponding element phi. Functions on the coset (for example, classinfo) are about properties of the group coset W ϕ ; however, most definitions for elements of untwisted Coxeter groups apply without change to elements in W ϕ: e.g., if we define the length of an element wϕ∈ Wϕ as the number of positive roots it sends to negative ones, it is the same as the length of w, i.e., ϕ is of length 0, since ϕ has been chosen to preserve the set of positive roots. Similarly, the Coxeter word describing wϕ is the same as the one for w, etc…

We associate to a Coxeter coset Wϕ a twisted Dynkin diagram, consisting of the Dynkin diagram of W and the graph automorphism induced by ϕ on this diagram (this specifies the group W⋊ ⟨F⟩, mentioned above, up to isomorphism). See the functions ReflectionType, ReflectionName and diagram for Coxeter cosets.

Below is an example showing first how to not define, then how to define, the Weyl coset for a Suzuki group:

julia> W=coxgroup(:B,2)
B₂

julia> spets(W,Perm(1,2))
ERROR: matrix F must preserve the roots
Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] spets(::Chevie.Weyl.FCG{Int16,Int64,PRG{Int64,Int16}}, ::Matrix{Int64}) at /home/jmichel/julia/Chevie/src/Cosets.jl:241 (repeats 2 times)
 [3] top-level scope at REPL[19]:1

julia> W=coxgroup(:Bsym,2)
Bsym₂

julia> WF=spets(W,Perm(1,2))
²Bsym₂

julia> CharTable(WF)
CharTable(²Bsym₂)
┌───┬─────────┐
│   │    1 121│
├───┼─────────┤
│2. │1   1   1│
│.11│1  -1  -1│
│1.1│. -√2  √2│
└───┴─────────┘

A subcoset Hwϕ of Wϕ is given by a reflection subgroup H of W and an element w of W such that wϕ induces an automorphism of the root system of H. For algebraic groups, this corresponds to a rational form of a reductive subgroup of maximal rank. For example, if Wϕ corresponds to the algebraic group 𝐆 and H is the trivial subgroup, the coset Hwϕ corresponds to a maximal torus 𝐓_w of type w.

julia> W=coxgroup(:Bsym,2)
Bsym₂

julia> WF=spets(W,Perm(1,2))
²Bsym₂

julia> subspets(WF,Int[],W(1))
²Bsym₂₍₎=Φ‴₈

A subgroup H which is a parabolic subgroup corresponds to a rational form of a Levi subgroup of 𝐆. The command twistings gives all rational forms of such a Levi.

julia> W=coxgroup(:B,2)
B₂

julia> twistings(W,[1])
2-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 B₂₍₁₎=Ã₁Φ₁
 B₂₍₁₎=Ã₁Φ₂

julia> twistings(W,[2])
2-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 B₂₍₂₎=A₁Φ₁
 B₂₍₂₎=A₁Φ₂

Notice how we distinguish between subgroups generated by short roots and by long roots. A general H corresponds to a reductive subgroup of maximal rank. Here we consider the subgroup generated by the long roots in B₂, which corresponds to a subgroup of type SL₂× SL₂ in SP₄, and show its possible rational forms.

julia> W=coxgroup(:B,2)
B₂

julia> twistings(W,[2,4])
2-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 B₂₍₂₄₎=A₁×A₁
 B₂₍₂₄₎=(A₁A₁)

rootdatum(type::String or Symbol[,dimension or bond::Integer]) root datum from type. The known types are

2B2, 2E6, 2E6sc, 2F4, 2G2, 2I2, 3D4, 3D4sc, 3gpin8, CE6, CE7, E6, E6sc, E7, E7sc, E8, F4, G2, cso, csp, gl, gpin, gpin-, halfspin, pgl, pso, pso-, psp, psu, ree, sl, slmod, so, so-, sp, spin, spin-, su, suzuki, tgl, triality, u

degrees(WF::Spets)

Let W be the group of the reflection coset WF, and let V be the vector space of dimension rank(W) on which W acts as a reflection group. Let f₁,…,fₙ be the basic invariants of W on the symmetric algebra SV of V; they can be chosen so they are eigenvectors of the matrix WF.F. The corresponding eigenvalues are called the factors of F acting on V; they characterize the coset –- they are equal to 1 only for the trivial coset. The generalized degrees of WF are the pairs formed of the reflection degrees and the corresponding factor.

julia> W=coxgroup(:E,6)
E₆

julia> WF=spets(W)
E₆

julia> phi=W(6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1);

julia> HF=subspets(WF,2:5,phi)
E₆₍₂₃₄₅₎=³D₄Φ₃

julia> diagram(HF)
ϕ acts as (1,2,4) on the component below
  O 2
  ￨
O—O—O D₄
1 3 4

julia> degrees(HF)
6-element Vector{Tuple{Int64, Cyc{Int64}}}:
 (1, ζ₃)
 (1, ζ₃²)
 (2, 1)
 (4, ζ₃)
 (6, 1)
 (4, ζ₃²)

spets(W::ComplexReflectionGroup, F::Matrix=I(rank(W)))

This function returns a or a CoxeterCoset or a Spets. F must be an invertible matrix, representing an automorphism of the vector space V of dimension of dimension rank(W) which for a finite Coxeter group induces an automorphism of the root system of parent(W), or for a more general complex reflection group just stabilizes W.

The returned struct has in particular the following fields:

.W: the group W

.F: the matrix acting on V which represents the unique element phi in WF which preserves the positive roots (for finite Coxeter groups) or some "canonical" representative of the coset for more general complex reflection groups.

.phi: a Perm, the permutation of the roots of W induced by .F (for general complex reflection groups this may be a permutation up to scalars) (also for Coxeter groups the element of smallest length in the NormalCoset W .phi).

In the first example we create a Coxeter coset corresponding to the general unitary group GU_3(q) over the finite field FF(q).

julia> W=rootdatum(:gl,3)
gl₃

julia> gu3=spets(W,-reflrep(W,W()))
²A₂Φ₂

julia> F4=coxgroup(:F,4);D4=reflection_subgroup(F4,[1,2,16,48])
F₄₍₉‚₂‚₁‚₁₆₎=D₄₍₃₂₁₄₎

julia> spets(D4,[1 0 0 0;0 1 2 0;0 0 0 1;0 0 -1 -1])
F₄₍₉‚₁₆‚₁‚₂₎=³D₄₍₃₄₁₂₎

spets(W::ComplexReflectionGroup,p::Perm)

In this version F is defined by the permutation of the simple roots it does.

julia> W=coxgroup(:A,3)
A₃

julia> spets(W,Perm(1,3))
²A₃

spets(s::String) builds a few of the exceptional spets

julia> spets("3G422")
³G₄‚₂‚₂

julia> spets("2G5")
²G₅

julia> spets("3G333")
G₃‚₃‚₃₍₁‚₂‚₃‚₄₄₎=³G₃‚₃‚₃₍₁‚₂‚₃‚₄₄₎

julia> spets("3pG333")
G₃‚₃‚₃₍₁‚₂‚₃‚₄₄₎=³G₃‚₃‚₃₍₁‚₂‚₃‚₄₄₎

julia> spets("4G333")
G₃‚₃‚₃₍₂‚₁₂‚₁₁‚₁₆‚₅₃‚₁₀‚₄₃‚₃₆₎=⁴G₃‚₃‚₃₍₁‚₂‚₃‚₃₂‚₁₆‚₃₆‚₃₀‚₁₀₎

twistings(W,I)

W should be a complex reflection group.

The function returns the list, up to W-conjugacy, of subspets of W whose group is reflection_subgroup(W,I) –- In the case of Weyl groups, it corresponds to representatives of the possible twisted forms of the reductive subgroup of maximal rank L defined by reflection_subgroup(W,I).

W could also be a coset Wϕ; then the subgroup L must be conjugate to ϕ(L) for a rational form to exist. If ϕ normalizes L, then the rational forms are classified by the the ϕ-classes of N_W(L)/L.

julia> W=coxgroup(:E,6)
E₆

julia> WF=spets(W,Perm(1,6)*Perm(3,5))
²E₆

julia> twistings(W,2:5)
3-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 E₆₍₂₃₄₅₎=D₄Φ₁²
 E₆₍₂₃₄₅₎=³D₄Φ₃
 E₆₍₂₃₄₅₎=²D₄Φ₁Φ₂


julia> twistings(WF,2:5)
3-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 ²E₆₍₂₅₄₃₎=²D₄₍₁₄₃₂₎Φ₁Φ₂
 ²E₆₍₂₅₄₃₎=³D₄₍₁₄₃₂₎Φ₆
 ²E₆₍₂₃₄₅₎=D₄Φ₂²

twistings(W)

W should be a Coxeter group which is not a proper reflection subgroup of another reflection group (so that inclusion(W)==eachindex(roots(W))). The function returns all spets representing twisted forms of algebraic groups of type W.

julia> twistings(coxgroup(:A,3)*coxgroup(:A,3))
8-element Vector{Spets{FiniteCoxeterGroup{Perm{Int16},Int64}}}:
 A₃×A₃
 A₃ײA₃
 ²A₃×A₃
 ²A₃ײA₃
 (A₃A₃)
 ²(A₃A₃)
 ²(A₃A₃)₍₁₂₃₆₅₄₎
 (A₃A₃)₍₁₂₃₆₅₄₎

julia> twistings(coxgroup(:D,4))
6-element Vector{Spets{FiniteCoxeterGroup{Perm{Int16},Int64}}}:
 D₄
 ²D₄₍₂₄₃₁₎
 ²D₄
 ³D₄
 ²D₄₍₁₄₃₂₎
 ³D₄₍₁₄₃₂₎

julia> W=rootdatum(:so,8)
so₈

julia> twistings(W)
2-element Vector{Spets{FiniteCoxeterGroup{Perm{Int16},Int64}}}:
 D₄
 ²D₄

torus(m::AbstractMatrix)

m should be a matrix of finite order. The function returns the coset T of the trivial group such that T.F==m. When m is integral his corresponds to an algebraic torus 𝐓 of rank size(m,1), with an isogeny which acts by m on X(𝐓).

julia> torus([0 -1;1 -1])
Φ₃

torus(W,i)

where W is a Spets or a ComplexReflectionGroup. This returns the torus twisted by a representative of the i-th conjugacy class of W. This is the same as twistings(W,Int[])[i].

julia> W=coxgroup(:A,3)
A₃

julia> twistings(W,Int[])
5-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 A₃₍₎=Φ₁³
 A₃₍₎=Φ₁²Φ₂
 A₃₍₎=Φ₁Φ₂²
 A₃₍₎=Φ₁Φ₃
 A₃₍₎=Φ₂Φ₄

julia> torus(W,2)
A₃₍₎=Φ₁²Φ₂

julia> WF=spets(W,Perm(1,3))
²A₃

julia> twistings(WF,Int[])
5-element Vector{Spets{FiniteCoxeterSubGroup{Perm{Int16},Int64}}}:
 ²A₃₍₎=Φ₂³
 ²A₃₍₎=Φ₁Φ₂²
 ²A₃₍₎=Φ₁²Φ₂
 ²A₃₍₎=Φ₂Φ₆
 ²A₃₍₎=Φ₁Φ₄

julia> torus(WF,2)
²A₃₍₎=Φ₁Φ₂²

graph_automorphisms(t::Vector{TypeIrred})

Given the refltype of a finite Coxeter group, returns the group of all Graph automorphisms of t as a group of permutations of indices(t).

julia> W=coxgroup(:D,4)
D₄

julia> graph_automorphisms(refltype(W*W))
Group((1,5)(2,6)(3,7)(4,8),(1,2),(1,4))

subspets(WF,I,w=one(Group(WF)))

Returns the reflection subcoset of the coset WF with group reflection_subgroup(Group(WF),I) and torsion w*WF.phi. w must be an element of Group(WF) such that w*WF.phi normalizes the subroot system generated by I.

julia> WF=spets(coxgroup(:F,4))
F₄

julia> w=transporting_elt(Group(WF),[1,2,9,16],[1,9,16,2],ontuples);

julia> LF=subspets(WF,[1,2,9,16],w)
F₄₍₉‚₁₆‚₁‚₂₎=³D₄₍₃₄₁₂₎

julia> diagram(LF)
ϕ acts as (2,3,4) on the component below
  O 4
  ￨
O—O—O D₄
3 1 2

Frobenius(WF)(x,i=1)

If WF is a Coxeter coset associated to the Coxeter group W, Frobenius(WF) returns a function F such that x↦ F(x,i=1) does the automorphism induced by WF.phi^i on the object x.

julia> W=coxgroup(:D,4)
D₄

julia> WF=spets(W,Perm(1,2,4))
³D₄

julia> u=unichar(W,2)
[D₄]:<11->

julia> F=Frobenius(WF);F(u)
[D₄]:<.211>

julia> F(u,-1)
[D₄]:<11+>

julia> F(1)
4