Finite reflection groups

Chevie.PermRoot — Module

Let V be a vector space over a subfield K of the complex numbers (in Julia the elements of K will be usually represented by Rationals or Cycs).

A complex reflection is an element s∈ GL(V) of finite order whose fixed point set is a hyperplane, the reflecting hyperplane Hₛ of s.

Most of the time we abbreviate "complex reflection" to reflection; in some literature "reflection" is reserved for s of order 2 and otherwise pseudo-reflection is used.

A reflection s has a unique eigenvalue ζₛ not equal to 1, which is a root of unity; if K is a subfield of the real numbers, ζₛ is necessarily equal to -1 and s is of order 2. An eigenvector r∈ V of s for the eigenvalue ζₛ is called a root of s and a linear form rᵛ on V of kernel Hₛ is called a coroot of s. If r and rᵛ are normalized so that rᵛ(r)=1-ζ, then s is the map on V defined by s(x)=x-rᵛ(x)r. Such a normalized root and coroot for s are unique up to scaling the root and inversely scaling the coroot.

A finite complex reflection group W is a finite group generated by complex reflections. Since whenever W contains a reflection s it contains its powers, W is always generated by reflections s such that ζₛ==E(d) where d is the order of s; we can also assume that s is not a power of another reflection with larger order d. Such a reflection is called a distinguished reflection of W; it is a canonical generator of the cyclic subgroup of W centralizing Hₛ. The generators of the complex reflection groups we construct are always distinguished reflections; the set of distinguished reflections are in bijection with the reflecting hyperplanes. In a real reflection group all reflections are distinguished.

It is much more efficient to compute with permutation groups than with matrix groups; hence we use for computing with a finite complex reflection group W a permutation representation on a W-invariant set of root and coroot vectors for reflections of W; that is, a set R of pairs (r,rᵛ)∈ V×Vᵛ invariant by W and such that every distinguished reflection in W is determined by some pair in R (see reflectionMatrix). Using a W-invariant set implies that there may be several pairs determining the same reflection, which differ by roots of unity. This generalizes the usual construction for Coxeter groups (the case K=ℝ) where to each reflection of W is associated two roots, one positive and one negative.

A complex reflection group is irreducible if the representation V is irreducible. For irreducible complex reflection groups, there are at least as many roots for a given reflection as the order of the center of W.

The following methods are always defined for finite complex reflection groups:

gens(W): the (distinguished) reflections which generate W, given as permutations of the roots. reflrep(W) gives them as matrices.
roots(W): the list of roots of W.
coroots(W): the list of coroots, in the same order as the roots.
refls(W): a list of distinguished reflections of W, as permutations. This list is of same length as the roots, and its i-th element is the distinguished reflection for the i-th root, so it will have repetitions. unique_refls(W) returns a sublist of 1:length(roots(W)) such that the distinguished reflections around the corresponding roots reach one time only each distinguished reflection of W.

The lists roots, coroots and refls starts with the "generating" ones (those corresponding to gens(W)).

simpleroots(W): the generating roots as a matrix.
simplecoroots(W): the generating coroots as a matrix.
reflections(W) gives a list without repetitions of all reflections of W (not only the distinguished ones), as objects of type Reflection. These objects contain more information on each reflection.

The finite irreducible complex reflection groups have been completely classified by Shephard and Todd, see ShephardTodd1954. They consist of one infinite family G(de,e,r) depending on 3 parameters (which includes the infinite families of Weyl groups), and 34 exceptional groups which have been given by Shephard and Todd names which range from G₄ to G₃₇. These exceptional groups include the exceptional Weyl groups, e.g., coxgroup(:E,8) is the same as G₃₇.

We provide functions to build any finite reflection group, either by giving a list of roots and coroots defining the generating reflections (see PRG), or in terms of the classification (see complex_reflection_group). In the context e.g. of Weyl groups, one wants to describe the particular root system chosen in term of the traditional classification of crystallographic root systems. This is done via calls to the function coxeter_group (see the module Weyl). There is not yet a general theory on how to construct a nice set of roots for a non-real reflection group; the roots chosen here where obtained case-by-case; however, they satisfy several important properties:

The generating reflections satisfy braid relations which present the braid group associated to W (see diagram).
The field of definition of W is the field K generated by the traces of the elements of W acting on V. It is a theorem that W may be realized as a reflection group over K.
The Cartan matrix rᵛ(r') where r and r' runs over the roots and coroots for the generating reflections has entries in the ring ℤₖ of integers of K, and the other roots (resp. coroots) are linear combinations with coefficients in ℤₖ of a linearly independent subset of them.

It turns out that all representations of W are defined over the field of definition K (see. Benard1976 and D.~Bessis' thesis). This has been known for a long time in the case K=ℚ, the case of Weyl groups, whose representations are all integral.

julia> W=complex_reflection_group(4)
G₄

julia> gens(W) # as permutations of the 24 roots
2-element Vector{Perm{Int16}}:
 (1,3,9)(2,4,7)(5,10,18)(6,11,16)(8,12,19)(13,15,20)(14,17,21)(22,23,24)
 (1,5,13)(2,6,10)(3,7,14)(4,8,15)(9,16,22)(11,12,17)(18,19,23)(20,21,24)

julia> length(unique(refls(W)))
4

julia> length(refls(W)) # 24=4*(number of roots of unity in ℚ (ζ₃))
24

julia> reflrep(W) # the generators as matrices
2-element Vector{Matrix{Cyc{Rational{Int64}}}}:
 [1 0; 0 ζ₃]
 [√-3/3 -ζ₃²√-3/3; (-3+√-3)/3 -ζ₃√-3/3]

For almost all irreducible complex reflection groups, the generating matrices for W we give have coefficients in the field of definition K. Further, the set of matrices for all elements of W is globally invariant under the Galois group of K/ℚ, thus the Galois action induces automorphisms of W. The exceptions are G₂₂, G₂₇ where the matrices are in a degree two extension of K (this is needed to have a globally invariant model, see MarinMichel2010) and some dihedral groups, as well as H₃ and H₄, where the matrices given (the usual Coxeter reflection representation over K) are not globally invariant (but one can get invariant ones by chevieget(:H4,:InvariantModel)).

julia> braid_relations(W) # as tuples of equal words in the generators
1-element Vector{Tuple{Vector{Int64}, Vector{Int64}}}:
 ([1, 2, 1], [2, 1, 2])

julia> diagram(W) # the same in pictures
③ ——③ G₄
1   2

julia> cartan(W)
2×2 Matrix{Cyc{Rational{Int64}}}:
 ζ₃²√-3     ζ₃²
   -ζ₃²  ζ₃²√-3

julia> simpleroots(W)
2×2 Matrix{Cyc{Rational{Int64}}}:
    0  ζ₃²√-3
 2ζ₃²     ζ₃²

julia> simplecoroots(W)
2×2 Matrix{Cyc{Rational{Int64}}}:
     0      1
 √-3/3  √-3/3

Let SV be the symmetric algebra of V. The invariants of W in SV are called the polynomial invariants of W. If n=dim V, they are generated as a polynomial ring by n homogeneous algebraically independent polynomials f₁,…,fₙ. The polynomials fᵢ are not uniquely determined but their degrees are. The fᵢ are called the basic invariants of W, and their degrees the reflection degrees of W; the function degrees(W) returns these invariants.

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 φ of W in the graded module SV/I is called the fake degree of φ. The function fakedegrees returns these polynomials.

julia> degrees(W)
2-element Vector{Int64}:
 4
 6

julia> fakedegrees(W,Pol(:x))
7-element Vector{Pol{Int64}}:
 1
 x⁴
 x⁸
 x⁷+x⁵
 x⁵+x³
 x³+x
 x⁶+x⁴+x²