Guillaume Collet

Rumen Andonov

Nikola Yanev

Jean-François Gibrat

Protein Threading

Keywords: Integer programming, combinatorial optimization, protein threading problem, protein structure alignment

Introduction

The most important in silico methods, to exploit the amount of new genomic data, are based on the concept of homology. The principle of homology-based analysis is to identify a homology relationship between a new protein and a protein whose function is known. For remote homologs, sequence alignment methods fail. In such a case one aligns the sequence of a new protein with the 3D structures of known proteins. Such methods are called fold recognition methods or threading methods.

Lathrop & Smith [START_REF] Lathrop | Global optimum protein threading with gapped alignment and empirical pair potentials[END_REF] were the first to propose an algorithm based on a branch & bound technique providing the global alignment with the optimal score and to prove the problem to be NP-Hard. Since then, other methods have been developed that improved the efficiency of the sequence -structure global alignment algorithm ( [START_REF] Xu | RAPTOR: optimal protein threading by linear programming[END_REF][START_REF] Andonov | Protein Threading Problem: From Mathematical Models to Parallel Implementations[END_REF][START_REF] Andonov | Recent Advances in Solving the Protein Threading Problem[END_REF]).

This paper describes a new algorithm that expands upon algorithms proposed in previous works ( [START_REF] Andonov | Protein Threading Problem: From Mathematical Models to Parallel Implementations[END_REF][START_REF] Andonov | Recent Advances in Solving the Protein Threading Problem[END_REF]) to allow implementation of local sequence -structure alignments. This allows threading methods to cover the whole spectrum of alignment types needed to analyze homologous proteins.

Our definition of alignments is based on the definition of the Protein Threading Problem (PTP) given in [START_REF] Lathrop | Global optimum protein threading with gapped alignment and empirical pair potentials[END_REF].

Preprint submitted to CTW 2009 7 May 2009

Query Sequence and Structure Template: A query sequence is a string of length N over the 20-letter amino acid alphabet. A structure template is an ordered set M of m blocks which correspond to the secondary structure elements (SSEs). Block k has a fixed length of L k amino acids. Let I ⊆ {(k, l) | 1 ≤ k < l ≤ m} be the set of blocks interactions.

Alignment: An alignment of a structure template with a query sequence corresponds to positioning blocks of the template along the sequence. A global alignment requires that all blocks are aligned, preserve their order, and do not overlap. This alignement has been modelized by mixed integer programming (MIP) approach in [START_REF] Xu | RAPTOR: optimal protein threading by linear programming[END_REF][START_REF] Andonov | Protein Threading Problem: From Mathematical Models to Parallel Implementations[END_REF]. In this paper, we extent the model presented in [START_REF] Andonov | Protein Threading Problem: From Mathematical Models to Parallel Implementations[END_REF].

Local alignments : towards better PTP models

Global alignment assumes that all blocks are aligned with the query sequence. However, it sometimes happens that some members of a protein family do not share exactly the same number of SSEs. An alignment which permits to omit blocks is called a local alignment. To solve this local alignment, we propose two models: (1) A compact model (CM) where we modify constraints to omit blocks.

(2) An extended model (EM) where we add dummy positions for each block. When a dummy position is chosen, the block is omitted. These models are described very briefly below. For more details, the interested reader can refer to our research report [START_REF] Collet | Flexible Alignments for Protein Threading[END_REF].

Compact model

We define a digraph G(V, A) with vertex set V and arc set A. Each vertex (i, k) ∈ V represents block k at position i along the sequence. A block k can take n k = N -L k + 1 positions along the query sequence. A cost C ik (resp. D ikjl ) is associated to each vertex (i, k) (resp each arc ((i, k), (j, l))). Let y ik (resp. z ikjl ) be binary variables associated with vertices (resp. arcs). Based on these notations, we obtain the following model:

max m k=1 n k i=1 C ik y ik + ((i,k),(j,l))∈A D ikjl z ikjl (1)
y ik ∈ {0, 1}, k ∈ M, i ∈ [1, n] (2) 0 ≤ z ikjl ≤ 1, ((i, k), (j, l)) ∈ A (3) n k i=1 y ik ≤ 1, k ∈ M (4) n l j=i+L k z ikjl -y ik ≤ 0, (k, l) ∈ I, i ∈ [1, n k ] (5) min(j-L k ,n k ) i=1 z ikjl -y jl ≤ 0, (k, l) ∈ I, j ∈ [1, n l ] ( 6
)
y ik + min(n k ,i+L k -1) j=1 y jl ≤ 1, 1 ≤ k ≤ l ≤ m, i ∈ [1, n k ] (7) n k i=1 y ik + n l i=1 y il - n l j=L k +1 j-L k i=1 z ikjl ≤ 1, (k, l) ∈ I (8)
Constraints ( 4) allow a block be aligned or not. Constraints ( 5) and (6) allow an arc, leaving (resp. entering) an activated vertex, be activated or not. Constraints (7) preserve the order of blocks. Finaly, constraints (8) coerce the activation of an arc if its vertices are activated.

Extended Model

Denote by

d ik , i ∈ [1, N], k ∈ [1, m]
a variable which we call dummy variables. The objective function is given by (1). This model uses constraints (2), (3), ( 5), ( 6) and (8). Additional constraints are the following:

d ik ∈ {0, 1} k ∈ M, i ∈ [1, n k ] (9) n k i=1 y ik + N i=1 d ik = 1 k ∈ M (10) j i=1 d ik + min(j,n k ) i=1 y ik - j i=1 d ik-1 - j-L k-1 i=1 y i(k-1) ≤ 0 k ∈ [2, m], j ∈ N (11)
Constraints (10) state that exactly one vertex (either real or dummy), must be activated in a column. Constraints (11) preserve the order of the blocks.

Two indicators have been used, computation time and the relative gap (RG) between the solution of the relaxed problem (LP ) and the optimal solution (OP T ): RG = LP -OP T OP T . RG is a good indicator of the efficiency of the model since the smaller RG, the easier for the branch & bound algorithm to find the solution. Figure 1 shows that EM is faster than CM for 99% of the instances. Moreover, Figure 2 shows that EM always gives a smaller RG than CM. It must be noted that LP relaxation directly gives the integer solution in 41% of the cases for the CM model and 52% of the cases for the EM model.

Fig. 1 .

 1 Fig. 1. Comparison of the computation times obtained by EM and CM. Each point represents an alignment. Times are expressed in seconds and are plotted using a base 10 logarithm scale.

Fig. 2 .

 2 Fig. 2. Comparisons of relative gaps ( LP -OP T OP T ) between models EM and CM. Each point is a sequence -structure alignment.


Frédéric Butin
email: butin@math.univ-lyon1.fr

Gadi S Perets

McKay correspondence and the branching law for finite subgroups of SL 3 C

Given Γ a finite subgroup of SL3C, we determine how an arbitrary finite dimensional irreducible representation of SL3C decomposes under the action of Γ. To the subgroup Γ we attach a generalized Cartan matrix CΓ. Then, inspired by B. Kostant, we decompose the Coxeter element of the Kac-Moody algebra attached to CΓ as a product of reflections of a special form, thereby suggesting an algebraic form for the McKay correspondence in dimension 3.

1 Introduction

Framework and results

Let Γ be a finite subgroup of SL 3 C. In this paper, we determine how the finite dimensional irreducible representations of SL 3 C decompose under the action of the subgroup Γ. These representations are indexed by N 2 . For (m, n) ∈ N 2 , let V (m, n) denote the corresponding simple finite dimensional module. Let {γ 0 , . . . , γ l } be the set of irreducible characters of Γ. We determine the numbers m i (m, n) -the multiplicity of the character γ i in the representation V (m, n). For that effect we introduce the formal power series:

P Γ (t, u) i = ∞ m=0 ∞ n=0 m i (m, n)t m u n .
We show that m i (t, u) is a rational function. We determine the rational functions which are obtained in that way for all the finite subgroups of SL 3 C.

The proof uses an inversion of the recursion formula for the numbers m i (m, n). The recursion formula is obtained through the decomposition of the tensor product of V (m, n) with the natural representation of SL 3 C. The key observation which leads to this inversion is that a certain pair matrices are simultaneously diagonalizable. The eigenvalues of the matrices are values from the character table of the group Γ. This leads to the proof that the power series

P Γ (t, u) i = ∞ m=0 ∞ n=0 m i (m, n)t m u n
is rational. The actual calculation of this rational function then reduces to matrix multiplication. This method applies indeed to the SL 2 C case. It gives a complete (very short) proof of the results obtained by B. Kostant in [START_REF] Kostant | The McKay Correspondence, the Coxeter Element and Representation Theory, SMF, Astérisque[END_REF], [START_REF] Kostant | The Coxeter element and the branching law for the finite subgroups of SU(2), The Coxeter legacy[END_REF], and by Gonzalez-Sprinberg and Verdier in [START_REF] Gonzalez-Sprinberg | Construction géométrique de la correspondance de McKay, Annales scientifiques de l[END_REF], and leads to an explicit determination of all the above multiplicities for the finite subgroups of SL 2 C. Although the results for SL 2 C are not new, the explicit relation of the rational functions with the eigenvalues of the Cartan matrix attached to the finite subgroup of SL 2 C doesn't seem to appear in the literature. In [START_REF] Kostant | The McKay Correspondence, the Coxeter Element and Representation Theory, SMF, Astérisque[END_REF] this is established through the analysis of the orbit structure of the Coxeter element.

The construction of a minimal resolution of singularities of the orbifold C 3 /Γ centralizes a lot of interest. It is related to the geometric McKay correspondence, cf. (for example) [START_REF] Bridgeland | The McKay correspondence as an equivalence of derived categories[END_REF], [START_REF] Gonzalez-Sprinberg | Construction géométrique de la correspondance de McKay, Annales scientifiques de l[END_REF]. In this framework Gonzalez-Spriberg-Verdier [START_REF] Gonzalez-Sprinberg | Construction géométrique de la correspondance de McKay, Annales scientifiques de l[END_REF] use the Poincaré series determined above in their explicit construction of minimal resolution for singularities for V = C 2 /Γ when Γ is a finite subgroup of SL 2 C. Following that approach the results of our calculation could be eventually used to construct explicit synthetic minimal resolution of singularities for orbifolds of the form SL 3 C/Γ where Γ is a finite subgroup of SL 3 C. This might clarify the description of the exceptional fiber of the minimal resolution of SL 3 C/Γ (see [START_REF] Gomi | Coinvariant Algebras of Finite Subgroups of SL3C[END_REF]).

An essential ingredient of the approach of B. Kostant in [START_REF] Kostant | The McKay Correspondence, the Coxeter Element and Representation Theory, SMF, Astérisque[END_REF] is the decomposition of a Coxeter element in the Weyl group attached to the Lie algebra corresponding to a subgroup Γ of SL 2 C through the McKay correspondence as a product of simple reflections belonging to mutually orthogonal sets of roots. Inspired by this approach, we attach to each finite subgroup Γ of SL 3 C a generalized Cartan matrix C Γ . We then factorize this matrix as a product of elements in the Weyl group of the Kac-Moody Lie algebra corresponding to C Γ . These elements are products of simple reflections corresponding to roots in mutually orthogonal sets.

Organization of the paper

In Section 2 we treat the SL 2 C case. We show that the formal power series of the multiplicities is a rational function by showing that it is an entry in a vector obtained as product of three matrices, two of which are scalar matrices the third one being a matrix with rational entries, by a scalar vector. We calculate the matrices for each finite subgroup of SL 3 C. We give then the rational functions obtained.

In Section 3 we apply the above method for the finite subgroups of SL 3 C. Here we use the notations of [START_REF] Yau | Gorenstein Quotient Singularities in Dimension Three[END_REF] in which a classification of the finite subgroups of SL 3 C is presented. Here again we prove the rationality of the formal power series of the multiplicities by showing that each such a series is an entry in the product of three matrices, two of them are scalar matrices and the third being a matrix with rational entries, with a scalar vector. For each finite subgroup of SL 3 C we give the the matrices involved in the product. To each subgroup Γ we attach a generalized Cartan matrix C Γ (McKay correspondence in dimension 3) we show its graph and its decomposition as a product of elements in the Weyl group of the Kac-Moody Lie algebra g(C Γ ).

Then, for the series A, B, C ([YY93] notation) we give all the rational functions explicitly, As for the series D we give the results for some specific examples because the description of the matrices engaged, in full generality doesn't have a simply presentable form.

For the exceptional finite subgroups of SL 3 C the numerators of the rational functions tend to be very long and we give them explicitly only for the cases where they are reasonably presentable. In all the cases we give the denominators explicitly. This is done in Section 4.

2 Branching law for the finite subgroups of SL 2 C 2.1 The formal power series of the multiplicities is a rational function • Let Γ be a finite subgroup of SL 2 C and {γ 0 , . . . , γ l } the set of equivalence classes of irreducible finite dimensional complex representations of Γ, where γ 0 is the trivial representation. We denote by χ j the character associated to γ i . Consider γ : Γ → SL 2 C, the natural 2-dimensional representation. Its character is denoted by χ. We have then the decomposition γ j ⊗ γ = l i=0 a ij γ i for every j ∈ [[0, l]]. This defines an (l + 1) × (l + 1) square matrix A := (a ij ) (i,j)∈[[0, l]] 2 .

• Let h be a Cartan subalgebra of sl 2 C and ̟ 1 be the corresponding fundamental weight, and V (n̟ 1 ) be the simple sl 2 -module of highest weight n̟ 1 . This give rise to an irreducible representation π n :

SL 2 C -→ V (n̟ 1 ).
The restriction of π n to the subgroup Γ, is a representation of Γ, and by complete reducibility, we have a decomposition π n | Γ = l i=0 m i (n)γ i , where the m i (n)'s are non negative integers. Let E := (e 0 , . . . , e l ) be the canonical basis of C l+1 , and

v n := l i=0 m i (n)e i ∈ C l+1 .
As γ 0 is the trivial representation, we have v 0 = e 0 . Let us consider the vector (with elements of C[[t]] as coefficients)

P Γ (t) := ∞ n=0 v n t n ∈ (C[[t]]) l+1 ,
and denote by P Γ (t) j its j-th coordinate in the basis E. The series P Γ (t) 0 is the Poincaré series of the invariant ring. Note also that P Γ (t) can also be seen as a formal power series with coefficients in C l+1 . We proceed to calculate P Γ (t) .

• We get by Clebsch-Gordan formula that :

π n ⊗ π 1 = π n+1 ⊕ π n-1 , so we have Av n = v n+1 + v n-1 .
From this we deduce the relation (1

-tA + t 2 )P Γ (t) = v 0 .
Let us denote by {C 0 , . . . , C l } the set of conjugacy classes of Γ, and for any j ∈ [[0, l]], let g j be an element of C j . So the character table of Γ is the matrix

T Γ ∈ M l+1 C defined by (T Γ ) i,j := χ i (g j ).
For all the finite subgroups of SL 2 C we have that, T Γ is invertible, and

Λ := T -1 Γ A T Γ is diagonal, with Λ jj = χ(g j ).
Set Θ := (Λ 00 , . . . , Λ ll ). We deduce from the preceding formula that

T Γ (1 -tΛ + t 2 )T -1 Γ P Γ (t) = v 0 . Let us define the rational function f : C 2 → C(t) d → 1 1 -td + t 2 . Then P Γ (t) = T Γ ∆(t) T -1 Γ v 0 = (T Γ ∆(t) T Γ ) (T -2 Γ v 0 ), where ∆(t) ∈ M l+1 C(t) is the diagonal matrix with coefficients in C(t), defined by ∆ jj (t) = f (Λ jj ).
Consequently, the coefficients of the vector P Γ (t) are rational fractions in t.

Hence we get:

Proposition 1 For each i ∈ [[0, l]], the formal power series P Γ (t) i is a rational function.

2.2

The results for the finite subgroups of SL 2 C

• The complete classification up to conjugation of all finite subgroups of SL 2 C is given in [START_REF] Springer | Invariant theory[END_REF]. It consists of two infinite series (types A, D) and three exceptional cases (types E 6 , E 7 , E 8 ). We set ζ j := e 2iπ j . For σ ∈ S [[0, j-1]] , we then define the matrix

Q σ := δ σ(l) k (k,l)∈[[0, j-1]]
.

Type A -Cyclic groups

• Here, we take Γ = Z/jZ. The natural representation and the natural character of Γ are

γ : Z/jZ → SL 2 C χ : Z/jZ → SL 2 C k → ζ k j 0 0 ζ -k j k → ζ k j + ζ -k j .
The character table is the Vandermonde matrix

T Γ = ζ kl j (k,l)∈[[0, j-1]] . Let σ be the permutation σ ∈ S [[0, j-1]] defined by σ(0) = 0 and ∀ i ∈ [[0, j -1]], σ(i) = j -i. Then T 2 Γ = j Q σ , i.e. T -1 Γ = 1 j T Γ Q σ . The eigenvalues of A are the numbers χ(k) = ζ k j + ζ -k j , for k ∈ [[0, j -1]]. Then P Γ (t) i = 1 j (T Γ ∆(t)T Γ Q σ ) i0 = 1 j j-1 p=0 ζ ip j (1 -tζ p j )(1 -tζ -p j )
.

Note that (1 -t j )(1 -t 2 ) is a common denominator of all the terms of the preceding sum.

Type D -Binary dihedral groups

The binary dihedral group is the subgroup a n , b of SL 2 C, with

a n := ζ 2n 0 0 ζ -1 2n , b := 0 i i 0 .
The order of Γ is 4n. The n + 3 conjugacy classes of Γ are

Class id a n b b a n n a n a 2 n a 3 n • • • a n-1 n Cardinality 1 n n 1 2 2 2 • • • 2
The character table of Γ is

TΓ := 0 B B B B B B B B B B B B @ 1 1 1 1 1 1 • • • 1 1 i n -i n (-1) n -1 1 • • • (-1) n-1 1 -i n i n (-1) n -1 1 • • • (-1) n-1 1 -1 -1 1 1 1 • • • 1 2 0 0 -2 ζ2n + ζ -1 2n ζ 2 2n + ζ -2 2n • • • ζ n-1 2n + ζ -(n-1) 2n 2 0 0 2 ζ 2 2n + ζ -2 2n ζ 4 2n + ζ -4 2n • • • ζ 2(n-1) 2n + ζ -2(n-1) 2n . . . . . . . . . . . . . . . . . . . . . 2 0 0 (-1) n-1 2 ζ n-1 2n + ζ -(n-1) 2n ζ 2(n-1) 2n + ζ -2(n-1) 2n • • • ζ (n-1)(n-1) 2n + ζ -(n-1)(n-1) 2n 1 C C C C C C C C C C C C A
The natural character χ of Γ is given by (χ(g 0 ), . . . , χ(g l )) = Θ, with

Θ = Θ = [tr(id), tr(a n b), tr(b), tr(a n n ), tr(a n ), tr(a 2 n ), tr(a 3 n ), . . . , tr(a n-1 n )] = [2, 0, 0, -2, ζ 2n + ζ -1 2n , ζ 2 2n + ζ -2 2n , ζ 3 2n + ζ -3 2n , . . . , ζ n-1 2n + ζ -(n-1) 2n
].

Set Diag (d 1 , d 2 , d 3 , d 4 , δ 1 , δ 2 , . . . , δ n-1 ) := ∆(t). We deduce the formula for the series P Γ (t) that :

P Γ (t) 0 = 3n -1 8n 2 d 1 + d 2 + d 3 + d 4 + 2 n-1 k=1 δ k + (-1) n n -1 8n 2 d 1 + (-1) n (d 2 + d 3 ) + d 4 + 2 n-1 k=1 (-1) k δ k + n-1 l=1 (-1) l n -1 4n 2 d 1 + (-1) l (d 2 + d 3 ) + d 4 + n-1 k=1 (ζ lk 2n + ζ -lk 2n )δ k , P Γ (t) 1 = 3n -1 8n 2 d 1 + i n d 2 -i n d 3 + (-1) n d 4 + 2 n-1 k=1 (-1) k δ k +(-1) n n -1 8n 2 d 1 + (-1) n (i n d 2 -i n d 3 + d 4 ) + 2 n-1 k=1 δ k + n-1 l=1 (-1) l n -1 4n 2 d 1 + (-1) l (i n d 2 -i n d 3 ) + (-1) n d 4 + n-1 k=1 (-1) k (ζ lk 2n + ζ -lk 2n )δ k ,
P Γ (t) i+3 = 3n -1 8n 2 2d 1 + 2(-1) i d 4 + 2 n-1 k=1 (ζ ki 2n + ζ -ki 2n )δ k +(-1) n n -1 8n 2 2d 1 + 2(-1) i d 4 + 2 n-1 k=1 (-1) k (ζ ik 2n + ζ -ik 2n )δ k + n-1 l=1 (-1) l n -1 4n 2 2d 1 + 2(-1) i d 4 + n-1 k=1 (ζ ki 2n + ζ -ki 2n )δ k (ζ kl 2n + ζ -kl 2n ) .

Exceptional cases

Type E 6 -Binary tetrahedral group

The binary tetrahedral group is the subgroup a 2 , b, c of SL 2 C, with

a := ζ 8 0 0 ζ 7 8 , b := 0 i i 0 , c := 1 √ 2 ζ 7 8 ζ 7 8 ζ 5 8 ζ 8 .
The order of Γ is 24. The 7 conjugacy classes of Γ are

Class id a 4 = -id b c c 2 -c -c 2 Cardinality 1 1 6 4 4 4 4
The character table T Γ of Γ and the matrix A are

TΓ = 0 B B B B B B B B B B B B B B @ 1 1 1 1 1 1 1 1 1 1 j j 2 j j 2 1 1 1 j 2 j j 2 j 2 -2 0 1 -1 -1 1 2 -2 0 j -j 2 -j j 2 2 -2 0 j 2 -j -j 2 j 3 3 -1 0 0 0 0 1 C C C C C C C C C C C C C C A , A = 0 B B B B B B B B B B B B B B @
0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1

0 0 0 1 1 0 1 C C C C C C C C C C C C C C A ,
and the eigenvalues are Θ = (2, -2, 0, 1, -1, -1, 1).

The series P Γ (t) i = NΓ(t)i DΓ(t) are given by D Γ (t) = (1 -t 6 )(1 -t 8 ), and

N Γ (t) 0 = t 12 + 1, N Γ (t) 4 = t 9 + t 7 + t 5 + t 3 , N Γ (t) 1 = t 8 + t 4 , N Γ (t) 5 = t 9 + t 7 + t 5 + t 3 , N Γ (t) 2 = t 8 + t 4 , N Γ (t) 6 = t 10 + t 8 + 2t 6 + t 4 + t 2 . N Γ (t) 3 = t 11 + t 7 + t 5 + t,

Type E 7 -Binary octahedral group

The binary octahedral group is the subgroup a, b, c of SL 2 C, with a, b, c defined as in the preceding section. The order of Γ is 48. The 8 conjugacy classes of Γ are Class id a 4 = -id ab b c 2 c a a 3 Cardinality 1 1 12 6 8 8 6 6

The character table T Γ of Γ and the matrix A are

TΓ = 0 B B B B B B B B B B B B B B B B B B @ 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 2 2 0 2 -1 -1 0 0 2 -2 0 0 -1 1 √ 2 - √ 2 2 -2 0 0 -1 1 - √ 2 √ 2 3 3 -1 -1 0 0 1 1 3 3 1 -1 0 0 -1 -1 4 -4 0 0 1 -1 0 0 1 C C C C C C C C C C C C C C C C C C A , A = 0 B B B B B B B B B B B B B B B B B B @
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1

0 0 1 0 0 1 1 0 1 C C C C C C C C C C C C C C C C C C A ,
and the eigenvalues are Θ = (2, -2, 0, 0, -1,

1, √ 2, - √ 2).
The series P Γ (t) i = NΓ(t)i DΓ(t) are given by D Γ (t) = (1 -t 8 )(1 -t 12 ), and

N Γ (t) 0 = t 18 + 1, N Γ (t

Type E 8 -Binary icosahedral group

The binary icosahedral group is the subgroup a, b, c of SL 2 C, with

a := -ζ 3 5 0 0 -ζ 2 5 , b := 0 1 -1 0 , c := 1 ζ 2 5 + ζ -2 5 ζ 5 + ζ -1 1 1 1 1 1 1 1 1 1 2 -2 1- √ 5 2 -1+ √ 5 2 1+ √ 5 2 -1+ √ 5 2 1 -1 0 2 -2 1+ √ 5 2 -1+ √ 5 2 1- √ 5 2 -1+ √ 5 2 1 -1 0 3 3 1+ √ 5 2 1- √ 5 2 1- √ 5 2 1+ √ 5 2 0 0 -1 3 3 1- √ 5 2 1+ √ 5 2 1+ √ 5 2 1- √ 5 2 0 0 -1 4 4 -1 -1 -1 -1 1 1 0 4 -4 1 -1 1 -1 -1 1 0 5 5 0 0 0 0 -1 -1 1 6 -6 -1 1 -1 1 0 0 0 1 C C C C C C C C C C C C C C C C C C C C C C A , A = 0 B B B B B B B B B B B B B B B B B B B B B @
0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0

1 C C C C C C C C C C C C C C C C C C C C C A
, and the eigenvalues are Θ = 2, -2,

1+ √ 5 2 , -1+ √ 5 2 , 1- √ 5 2 , -1- √ 5 2 , 1, -1, 0 .
The series P Γ (t) i = NΓ(t)i DΓ(t) are given by D Γ (t) = (1 -t 12 )(1 -t 20 ), and 3 Branching law for the finite subgroups of SL 3 C

N Γ (t) 0 = t 30 + 1, N Γ (t
• Let Γ be a finite subgroup of SL 3 C and {γ 0 , . . . , γ l } the set of equivalence classes of irreducible finite dimensional complex representations of Γ, where γ 0 is the trivial representation. The character associated to γ j is denoted by χ j . Consider γ : Γ → SL 3 C the natural 3-dimensional representation, and γ * its contragredient representation. The character of γ is denoted by χ. By complete reducibility we get the decompositions

∀ j ∈ [[0, l]], γ j ⊗ γ = l i=0 a (1) ij γ i and γ j ⊗ γ * = l i=0 a (2) ij γ i .
This defines two square matrices A (1) := a

(1) ij (i,j)∈[[0, l]] 2 and A (2) := a (2) ij (i,j)∈[[0, l]] 2 of M l+1 N.
• Let h be a Cartan subalgebra of sl 3 C and let ̟ 1 , ̟ 2 be the corresponding fundamental weights, and V (m̟ 1 + n̟ 2 ) the simple sl 3 C module of highest weight m̟ 1 + n̟ 2 with (m, n) ∈ N 2 . Then we get an irreducible representation π m,n : SL 3 C → GL(V (m̟ 1 + n̟ 2 )). The restriction of π m,n to the subgroup Γ is a representation of Γ, and by complete reducibility, we get the decomposition

π m,n | Γ = l i=0 m i (m, n)γ i ,
where the m i (m, n)'s are non negative integers. Let E := (e 0 , . . . , e l ) be the canonical basis of C l+1 , and

v m,n := l i=0 m i (m, n)e i ∈ C l+1 .
As γ 0 is the trivial representation, we have v 0,0 = e 0 . Let us consider the vector (with elements of C[[t, u]] as coefficients)

P Γ (t, u) := ∞ m=0 ∞ n=0 v m,n t m u n ∈ (C[[t, u]]) l+1 ,
and denote by P Γ (t, u) j its j-th coordinate in the basis E. Note that P Γ (t, u) can also be seen as a formal power series with coefficients in C l+1 . The aim of this article is to compute P Γ (t, u).

The formal power series of the multiplicies is a rational function

Here we establish some properties of the series P Γ (t, u), in order to give an explicit formula for it. The first proposition follows from the uniqueness of the decomposition of a representation as sum of irreducible representations.

Proposition 2 • A (2) = t A (1) . • A (1) and A (2) commute, i.e. A (1) is a normal matrix.

Since A (1) is normal, we know that it is diagonalizable with eigenvectors forming an orthogonal basis. Now we will diagonalize the matrix A (1) by using the character table of the group Γ. Let us denote by {C 0 , . . . , C l } the set of conjugacy classes of Γ, and for any j ∈ [[0, l]], let g j be an element of C j . So the character table of Γ is the matrix

T Γ ∈ M l+1 C defined by (T Γ ) i,j := χ i (g j ). Proposition 3 For k ∈ [[0, l]], set w k := (χ 0 (g k ), . . . , χ l (g k )) ∈ C l+1 . Then w k is an eigenvector of A (2) associated to the eigenvalue χ(g k ).
Similarly, w k is an eigenvector of A (1) associated to the eigenvalue χ(g k ).

We will see in the sequel that W := (w 0 , . . . , w l ) is always a basis of eigenvectors of A (1) and A (2) , so that T -1 Γ A (1) T Γ and T -1 Γ A (2) T Γ are diagonal matrices. Now, we make use of the Clebsch-Gordan formula

π 1,0 ⊗ π m,n = π m+1,n ⊕ π m,n-1 ⊕ π m-1,n+1 , π 0,1 ⊗ π m,n = π m,n+1 ⊕ π m-1,n ⊕ π m+1,n-1 .
(1)

Proposition 4

The vectors v m,n satisfy the following recurrence relations

A (1) v m,n = v m+1,n + v m,n-1 + v m-1,n+1 , A (2) v m,n = v m,n+1 + v m-1,n + v m+1,n-1 . Proof: The definition of v m,n reads v m,n = l i=0 m i (m, n)e i , thus A (1) v m,n = l i=0 l j=0 m j (m, n)a (1) ij e i . Now (π m,n ⊗ π 1,0 )| Γ = π m,n | Γ ⊗ γ = l j=0 m j (m, n)γ j ⊗ γ = l i=0 l j=0 m j (m, n)a (1) ij γ i , and π m+1,n | Γ + π m,n-1 | Γ + π m-1,n+1 | Γ = l i=0 (m i (m + 1, n) + m i (m, n -1) + m i (m -1, n + 1)) γ i . By uniqueness, l j=0 m j (m, n)a (1) ij = m i (m + 1, n) + m i (m, n -1) + m i (m -1, n + 1).

Proposition 5

The series P Γ (t, u) satisfies the following relation

1 -tA (1) + t 2 A (2) -t 3 1 -uA (2) + u 2 A (1) -u 3 P Γ (t, u) = (1 -tu)v 0,0 .

Proof:

• Set x := P Γ (t, u). Set also v m,-1 := 0 and v -1,n := 0 for (m, n) ∈ N, such that, according to the Clebsch-Gordan formula, the formulae of the preceding corollary are still true for (m, n) ∈ N. We have

tuA (1) x = tu ∞ m=0 ∞ n=0 A (1) v m,n t m u n = ∞ m=0 ∞ n=0 (v m+1,n + v m,n-1 + v m-1,n+1 )t m+1 u n+1 . Now ∞ m=0 ∞ n=0 v m+1,n t m+1 u n+1 = u ∞ m=1 ∞ n=0 v m,n t m u n = ux -u ∞ n=0 v 0,n u n , ∞ m=0 ∞ n=0 v m,n-1 t m+1 u n+1 = tu 2 ∞ m=0 ∞ n=1 v m,n-1 t m u n-1 = tu 2 x, and ∞ m=0 ∞ n=0 v m-1,n+1 t m+1 u n+1 = t 2 ∞ m=1 ∞ n=0 v m-1,n+1 t m-1 u n+1 = t 2 ∞ m=0 ∞ n=0 v m,n+1 t m u n+1 = t 2 ∞ m=0 ∞ n=0 v m,n t m u n -t 2 ∞ m=0 v m,0 t m = t 2 x -t 2 ∞ m=0 v m,0 t m . Therefore tuA (1) x = (u + tu 2 + t 2 )x -u ∞ n=0 v 0,n u n -t 2 ∞ m=0 v m,0 t m . (2)
We proceed likewise to obtain tuA

(2) x = (t + tu 2 + u 2 )x -t ∞ m=0 v m,0 t m -u 2 ∞ n=0 v 0,n u n . (3)
• By using Equations ( 2) and (3), we have tuA

(2) x -tu 2 A (1) x = t(1 -u 3 )x + (t 2 u -t) ∞ m=0 v m,0 t m , i.e. 1 -uA (2) + u 2 A (1) -u 3 x = (1 -tu) ∞ m=0 v m,0 t m . (4) Besides A (1) v m,0 = v m+1,0 + v m-1,1 , and A (2) v m-1,0 = v m-1,1 + v m-2,0 , hence A (1) v m,0 = v m+1,0 + A (2) v m-1,0 -v m-2,0 . Set y := ∞ m=0 v m,0 t m . Then tA (1) y = ∞ m=0 v m+1,0 t m+1 + A (2) ∞ m=1 v m-1,0 t m+1 - ∞ m=2 v m-2,0 t m+1 = ∞ m=1 v m,0 t m + t 2 A (2) ∞ m=0 v m,0 t m -t 3 ∞ m=0 v m,0 t m = y -v 0,0 + t 2 A (2) y -t 3 y. So 1 -tA (1) + t 2 A (2) -t 3 y = v 0,0 . (5)
Combining Eq. 4 and 5, we have 1 -

tA (1) + t 2 A (2) -t 3 1 -uA (2) + u 2 A (1) -u 3 x = (1 -tu)v 0,0 .
We may inverse the relation obtained in Proposition 5 and obtain an explicit expression 3 for P Γ (t, u) as well as an explicit formula for the vector v m,n . But, for the explicit calculations of P Γ (t, u), we will use an other fundamental formula (we inverse complex numbers instead of matrices). We need the rational function f defined by

We deduce that

P Γ (t, u) = T Γ ∆(t, u) T -1 Γ v 0,0 = (T Γ ∆(t, u) T Γ ) (T -2 Γ v 0,0 ), (6)
where

∆(t, u) ∈ M l+1 C(t, u) is the diagonal matrix defined by ∆(t, u) jj = f (Λ jj , Λ jj ) = f (χ(g j ), χ(g j )). Let Θ := (Λ (1) 00 , . . . , Λ (1)
ll ) be the list of eigenvalues of A (1) .

As a corollary of the preceding formula we get:

3 P Γ (t, u) = (1 -tu) " P ∞ p=0 `u3 + uA (2) -u 2 A (1) ´p" " P ∞ q=0 `t3 + tA (1) -t 2 A (2) ´q" v 0,0 .
For z ∈ R, let ⌈z⌉ be the smallest integer that is greater or equal to z, and set {r, s} := {1, 2}. For m ∈ N, set

α (r) m := m X q=⌈ m 3 ⌉ 0 B @ min(3q-m, q) X j=⌈ 3q-m 2 ⌉ C j q C 3q-m-j j (-1) 3q-m A (r) 3q-m-j A (s) 2j-3q+m 1 C A Then vm,n = v 0,0 if m = n = 0; α (2) n v 0,0 if m = 0, n = 0; α (1) m v 0,0 if n = 0, m = 0; (α (2)
n α

(1)

m -α (2) n-1 α (1)
m-1 )v 0,0 otherwise.

Proposition 6

The coefficients of the vector P Γ (t, u) are rational fractions in t and u, hence the formal power series of the multiplicities is a rational function.

We will denote them by

P Γ (t, u) i := N Γ (t, u) i D Γ (t, u) i , i ∈ [[0, l]]
where N Γ (t, u) i and D Γ (t, u) i are elements of C[t, u] that will be explicitly computed in the sequel. Finally, we introduce a generalized Cartan matrix that we will study for every finite subgroup of SL 3 C.

Definition 7

For every finite subgroup of SL 3 C, we define a generalized Cartan matrix by the following formula:

C Γ := 2 I -A (1) -t A (1) + 2 Diag(A (1) ).
For k ∈ [[0, l]], the matrix of the reflection s k associated to the k-th root of g(C Γ ) the Kac Moody algebra attached to C Γ is defined by

(s k ) ij = δ j i -(C Γ ) k,j δ k i .
For each finite subgroup, we will give a decomposition of the set of simple reflections S = {s 0 , . . . , s l } in p sets (with p minimal), i.e.

S = S 0 ⊔ • • • ⊔ S p-1 ,
such that roots corresponding to reflections in those sets form a partition of the set of simple roots to mutually orthogonal sets. We denote by τ l the (commutative) product of the elements of S l . Then we deduce the following decomposition of C Γ :

C Γ = p I - p-1 k=0 τ k .

Remark 8

Along this section we will present matrices that have only -2, -1, 0, 1, 2 as entries. For a clearer exposition, we represent the non-zero entries by colored points. The correspondence is the following: dark grey = -2, light grey = -1, white = 1, black = 2, empty = 0.

3.2

Explicit results for the infinite series -Types A, B, C, D

The A Series

In this section, we consider Γ a finite diagonal abelian subgroup of SL 3 C. Then Γ is isomorphic to a product of cyclic groups:

Γ ≃ Z/j 1 Z × • • • × Z/j k Z.
If Γ is a finite subgroup of SL r C, then Γ is a small subgroup of GL r C, i.e. no element of Γ has an eigenvalue 1 of multiplicity r -1. In fact, if g ∈ Γ has an eigenvalue 1 of multiplicity r -1, then the last eigenvalue of g is different from 1 and the determinant of g is also different from 1, which is impossible.

Then, according to a lemma of [START_REF] Dais | On the Existence of Crepant Resolutions of Gorenstein Abelian Quotient Singularities in Dimensions[END_REF] (p.13), Γ has at most r -1 generators. So, for a subgroup Γ of type A, we may assume that k ≤ 2, i.e. we have two cases:

(A 1 ) Γ ≃ Z/jZ, (A 2 ) Γ ≃ Z/j 1 Z × Z/j 2 Z, with j 1 ≥ j 2 ≥ 2.

Type A 1

• Here, we take Γ = Z/jZ. The natural representation and the natural character of Γ are

γ : Z/jZ → SL 3 C χ : Z/jZ → SL 3 C k →   ζ k j 0 0 0 1 0 0 0 ζ -k j   k → 1 + ζ k j + ζ -k j .
The character table of Γ is

T Γ = ζ kl j (k,l)∈[[0, j-1]] . Let σ ∈ S [[0, j-1]] be the permutation defined by σ(0) = 0 and ∀ i ∈ [[0, j -1]], σ(i) = j -i. Then T -1 Γ = 1 j T Γ Q σ . The eigenvalues of A (1) are the numbers χ(k) = 1 + ζ k j + ζ -k j , for k ∈ [[0, j -1]]. According to Formula 6, P Γ (t, u) i = 1 j (T Γ ∆(t, u)T Γ Q σ ) i0 = 1 j j-1 p=0 ζ ip j (1 -tu) (1 -t)(1 -tζ p j )(1 -tζ -p j )(1 -u)(1 -uζ p j )(1 -uζ -p j ) . Note that (1 -t j )(1 -t 2 )(1 -u j )(1 -u 2
) is a common denominator of all the terms of the preceding sum, which is independent of i.

• The matrix A (1) ∈ M j C is A (1) = 1 2 2 1 if j = 2, A (1) =       1 1 1 1 . . . . . . . . . . . . 1 1 1 1       if j ≥ 3.
Then the set of reflections S may be decomposed in two (resp. three) sets if j is even (resp. odd). ⊲ If j is even, we have τ 0 = s 0 s 4 . . . s j-2 , τ 1 = s 1 s 3 . . . s j-1 , and

C A1 (j) = 2I j-1 -(τ 0 + τ 1 ). ⊲ If j is odd, we have τ 0 = s 2 s 4 . . . s j-1 , τ 1 = s 1 s 3 . . . s j-2 , τ 2 = s 0 , and C A1 (j) = 3I j -(τ 0 + τ 1 + τ 2 ).
The graph associated to C A1 (j) is a cyclic graph with j vertices and j edges.

Type A 2

• We now consider the case Γ = Z/j 1 Z × Z/j 2 Z, with j 1 ≥ j 2 ≥ 2. The natural representation and the natural character of Γ are

γ : Z/j 1 Z × Z/j 2 Z → SL 3 C χ : Z/j 1 Z × Z/j 2 Z → SL 3 C (k 1 , k 1 ) →    ζ k1 j1 0 0 0 ζ k2 j2 0 0 0 ζ -k1 j1 ζ -k2 j2    (k 1 , k 2 ) → ζ k1 j1 + ζ k2 j2 + ζ -k1 j1 ζ -k2 j2 .
The irreducible characters of Γ are the elements of the form χ 1 ⊗ χ 2 , where χ 1 and χ 2 are irreducible characters of Z/j 1 Z and Z/j 2 Z, i.e. the irreducible characters of Γ are, for (l

1 , l 2 ) ∈ [[0, j 1 -1]]×[[0, j 2 -1]], χ l1,l2 : Z/j 1 Z × Z/j 2 Z → SL 3 C (k 1 , k 2 ) → ζ k1l1 j1 ζ k2l2 j2 .
For k ∈ {1, 2}, let us denote by T k the character table of the group Z/j k Z. Then the character table

of Γ = Z/j 1 Z × Z/j 2 Z is the Kronecker product 4 T Γ = T 1 ⊗ T 2 . Let σ k ∈ S [[0, j k -1]] be the permutation defined by σ k (0) = 0 and ∀ i ∈ [[0, j k -1]], σ k (i) = j k -i. We have T -1 Γ = (T 1 ⊗ T 2 ) -1 = 1 j 1 j 2 (T 1 ⊗ T 2 )(Q σ1 ⊗ Q σ2 ) = 1 j 1 j 2 (T 1 Q σ1 ) ⊗ (T 2 Q σ2 ).
The eigenvalues of A (1) are the numbers

χ(k 1 , k 2 ) = ζ -k1 j1 + ζ -k2 j2 + ζ k1 j1 ζ k2 j2 , for (k 1 , k 2 ) ∈ [[0, j 1 -1]] × [[0, j 2 -1]].
Let us denote by Λ (1) := Diag Λ

(1) 0 , . . . , Λ

(1) j1 the diagonal block-matrix defined by (Λ

(1) k1 ) k2k2 = χ(k 1 , k 2 ) = ζ -k1 j1 + ζ -k2 j2 + ζ k1 j1 ζ k2 j2 .
According to Formula 6, for (m,

n) ∈ [[0, j 1 -1]] × [[0, j 2 -1]],
we have

P Γ (t, u) mj2+n = 1 j 1 j 2 j1-1 k=0 j2-1 l=0 ζ mk j1 ζ nl j2 (1 -tu) 1 -t(ζ -k j1 + ζ -l j2 + ζ k j1 ζ l j2 ) + t 2 (ζ k j1 + ζ l j2 + ζ -k j1 ζ -l j2 ) -t 3 -1 1 -u(ζ k j1 + ζ l j2 + ζ -k j1 ζ -l j2 ) + u 2 (ζ -k j1 + ζ -l j2 + ζ k j1 ζ l j2 ) -u 3 -1 .
• The matrix A (1) is a block-matrix with j 2 1 blocs of size j 2 , and we have

A (1) = 14,4-I4, if j1 = j2 = 2, and A (1) = 0 B B B B @ Q ρ 2 t Q ρ 2 Ij 2 Ij 2 . . . . . . . . . . . . t Q ρ 2 t Q ρ 2 Ij 2 Q ρ 2 1 C C C C A if j1 ≥ 2, Q ρ k := 0 B B B B @ 0 1 1 . . . . . . . . . 1 0 1 C C C C A .
So, we may write

A (1) = I j1 ⊗ Q ρ2 + Q ρ1 ⊗ I j2 + t Q ρ1 ⊗ t Q ρ2 . Note that Diag(A (1) ) = 0. Then C A2 (j 1 , j 2 ) = I j1 ⊗ W + Q τ1 ⊗ S + t Q τ1 ⊗ t S.
• Now, let us decompose the matrix A (1) :

If

j 1 = j 2 = 2, then the decomposition of C A2 (2, 2) is C A2 (2, 2) = 4 I 4 -(s 0 + s 1 + s 2 + s 3 ). Now, we assume that j 1 ≥ 3. For (i 1 , i 2 ) ∈ [[0, j 1 -1]] × [[0, j 2 -1]]
, let s i1,i2 be the reflection associated to the (i 1 j 2 + i 2 )-th root. Then the set S may be decomposed into p sets where p ∈ {4, 6, 9}.

For

i 1 ∈ [[0, j 1 -1]], define S i1 := {s i1,0 , . . . , s i1,j2-1 } . ⊲ If j 1 is odd, we set I 0 := {0, 2, . . . , j 1 -3}, I 1 := {1, 3, . . . , j 1 -2}, I 0 := {j 1 -1}.
⊲ If j 1 is even, we set I 0 := {0, 2, . . . , j 1 -2}, I 1 := {1, 3, . . . , j 1 -1}.

Then, the roots associated to the reflections of distinct S i1 's for i 1 belonging to a same I k are orthogonal. Now, we decompose each S i1 , i.e.

S i1 = S i1,0 ⊔• • •⊔ S i1,q-1 , such that q ∈ {2, 3} and for every k ∈ [[0, q-1]],
the roots associated to the reflections belonging to S i1,k are orthogonal:

⊲ If j 2 is odd, then S i1 = S i1,0 ⊔ S i1,1 ⊔ S i1,2 , with S i1,0 = {s i1,0 , s i1,2 , . . . , s i1,j2-3 }, S i1,1 = {s i1,1 , s i1,3 , . . . , s i1,j2-2 }, S i1,2 = {s i1,j2-1 }. ⊲ If j 2 is even, then S i1 = S i1,0 ⊔ S i1,1 , with S i1,0 = {s i1,0 , s i1,2 , . . . , s i1,j2-2 }, S i1,1 = {s i1,1 , s i1,3 , . . . , s i1,j2-1 }.
Finally, for (k, l) ∈ {0, 1, 2} 2 , we set S k,l :=

i1∈ b I k S i1,l ,
and we denote by p ∈ {4, 6, 9} the number of non-empty sets S k,l , and by τ k,l the commutative product of the reflections of S k,l . Then, we have

C A2 (j 1 , j 2 ) := 2 I j1j2 -A (1) -t A (1) + 2 Diag(A (1) ) = p I j1j2 - (k,l)∈{0, 1, 2} 2 τ k,l .
• If j 1 ≥ 3, the graph associated to Γ is a j 1 -gone, such that every vertex of this j 1 -gone is a j 2 -gone, and every vertex of each j 2 -gone is connected with exactly 2 vertices of both adjacent j 2 -gones (for j 1 = j 2 = 2, see Remark 9).

Remark 9

In the cases j 1 = 2, j 2 = 2 and j 1 = 3, j 2 = 2, we obtain full matrices and complete graphs. Moreover the complete graphs with 4 and 6 vertices are the unique complete graphs that we can obtain for the type A 2 (the complete graphs with 2 and 3 vertices are the unique complete graphs that we can obtain for the type A 1 ).

Example 10

We consider the case where j 1 = 6 and j 2 = 5. Then the decomposition of the Cartan matrix is

C A2 (6, 5) = 6 I 30 -(τ 0,0 + τ 0,1 + τ 0,2 + τ 1,0 + τ 1,1 + τ 1,2 ),
where the matrix C A2 (6, 5), the τ i,j 's and the graph associated to C A2 (6, 5) are given by Figure 1.

τ 0,0 := (s 0,0 s 0,2 )(s 2,0 s 2,2 )(s 4,0 s 4,2 ) τ 0,1 := (s 0,1 s 0,3 )(s 2,1 s 2,3 )(s 4,1 s 4,3 ) τ 0,2 := (s 0,4 )(s 2,4 )(s 4,4 ) τ 1,0 := (s 1,0 s 1,2 )(s 3,0 s 3,2 )(s 5,0 s 5,2 ), τ 1,1 := (s 1,1 s 1,3 )(s 3,1 s 3,3 )(s 5,1 s 5,3 ), τ 1,2 := (s 1,4 )(s 3,4 )(s 5,4 ).

Figure 1: Matrix C A2 (6, 5) and corresponding graph.

The B series

In this section, we study the binary groups of SL 3 C. We give a general formula for the types BDa, BT a, BO and BI. In all these cases, the group Γ contains two normal subgroups Γ 1 and Γ 2 such that Γ 1 ∩ Γ 2 = {id}, and

|Γ 1 | • |Γ 2 | = |Γ|, so that Γ = Γ 2 Γ 1 and Γ ≃ Γ 2 × Γ 1 .
The group Γ 1 is isomorphic to a binary group of SL 2 C and Γ 2 is isomorphic to Z/mZ. So, we can deduce the results for Γ of the results obtained for the group Γ 1 .

If we denote by T k the character table of the group Γ k , the character table of the direct product Γ = Γ 2 ×Γ 1 is the Kronecker product T Γ = T 2 ⊗T 1 . The matrix T 2 is given in Section 2.2.1 (Type A -Cyclic groups), and the matrix T 1 is given in the section dealing with the corresponding binary group of SL 2 C. We also have

T -1 Γ = T -1 2 ⊗ T -1 1 = 1 m (T 2 Q σ2 ) ⊗ T -1 1
, where σ 2 is the permutation matrix defined by σ 2 (0) = 0, and

∀ i ∈ [[0, m -1]], σ 2 (i) = m -i.
Let us denote by h the number of conjugacy classes of Γ 1 . The columns of T Γ give a basis of eigenvectors and the eigenvalues of A (1) are the numbers χ i,j , (i,

j) ∈ [[0, m -1]] × [[0, h -1]]
where χ i,j is the value of the natural character of Γ on the (i, j)-th conjugacy class. Let us also denote by Λ (1) the diagonal block-matrix with m × m blocks of size h × h defined by

Λ (1) := Diag Λ (1) 0 , . . . , Λ (1) m-1 , (Λ (1) i ) jj = χ i, j .
According to Formula 6,

P Γ (t, u) = T Γ ∆(t, u)T -1 Γ v 0,0 = 1 m (T 2 ⊗ T 1 )∆(t, u)((T 2 Q σ2 ) ⊗ (T 1 T -2 1 ))v 0,0 ,
where ∆(t, u) is the diagonal block-matrix defined by

∆(t, u) = Diag ∆(t, u) (0) , . . . , ∆(t, u) (m-1) ,
with ∆(t, u)

(i) jj = f (χ i,j , χ i,j ) = 1 -tu (1 -tχ i,j + t 2 χ i,j -t 3 )(1 -uχ i,j + u 2 χ i,j -u 3 ) =: f ij .
The decomposition of the matrix A (1) and the description of the associated graph are made in the same way for the binary tetrahedral, octahedral and icosahedral groups: the results are collected in 3.3.5.

3.3.1

The BDa subseries -Binary dihedral groups

• For (q, m) ∈ N 2 , let ψ 2q , τ and φ 2m be the following elements of SL 3 C:

ψ 2q =   1 0 0 0 ζ 2q 0 0 0 ζ -1 2q   , τ =   1 0 0 0 0 i 0 i 0   , φ 2m =   ζ -2 2m 0 0 0 ζ 2m 0 0 0 ζ 2m   .
In this section 5 , we assume that 1 < q < n, n ∧ q = 1, and m := n -q ≡ 1 mod 2, and we consider the subgroup Γ := ψ 2q , τ, φ 2m of SL 3 C. Note that

φ 2m = ψ q 2q φ -m-1 2 m
, so that Γ = ψ 2q , τ, φ m . Set Γ 1 := ψ 2q , τ and Γ 2 := φ m ≃ Z/mZ. Then Γ ≃ Γ 2 × Γ 1 . With the notations used for SL 2 C, ψ 2q (resp. τ ) represents a q (resp. b), where a q , b is the binary dihedral subgroup of SL 2 C.

The natural character of Γ is given by χ = (χ i ) i=0...m-1 , with

χ i = [χ i,0 , χ i,1 , χ i,2 , χ i,3 , χ i,4 , χ i,5 , χ i,6 , . . . , χ i,q+2 ] = tr(φ i m ), tr(φ i m ψ 2q τ ), tr(φ i m τ ), tr(φ i m ψ q 2q ), tr(φ i m ψ 2q ), tr(φ i m ψ 2 2q ), tr(φ i m ψ 3 2q ), . . . , tr(φ i m ψ q-1 2q ) = ζ -2i m + 2ζ i m , ζ -2i m , ζ -2i m , ζ -2i m -2ζ i m , ζ -2i m + ζ i m (ζ 2q + ζ -1 2q ), ζ -2i m + ζ i m (ζ 2 2q + ζ -2 2q ), ζ -2i m + ζ i m (ζ 3 2q + ζ -3 2q ), . . . , ζ -2i m + ζ i m (ζ q-1 2q + ζ -(q-1) 2q
) .

We deduce the following formula for the series P Γ (t, u):

P Γ (t, u) i1(q+3) = 1 m m-1 r=0 ζ i1r m 3q -1 8q 2 d (r) 1 + d (r) 2 + d (r) 3 + d (r) 4 + 2 q-1 k=1 δ (r) k +(-1) q q -1 8q 2 d (r) 1 + (-1) q (d (r) 2 + d (r) 3 ) + d (r) 4 + 2 q-1 k=1 (-1) k δ (r) k + q-1 l=1 (-1) l q -1 4q 2 d (r) 1 + (-1) l (d (r) 2 + d (r) 3 ) + d (r) 4 + q-1 k=1 (ζ lk 2q + ζ -lk 2q )δ (r) k , 5
The other case -the type BDb -is 1 < q < n, n ∧ q = 1, and m := nq ≡ 0 mod 2. This group is not a direct product, the general expression for groupes in this subseries is unclear.

P Γ (t, u) i1(q+3)+1 = 1 m m-1 r=0 ζ i1r m 3q -1 8q 2 d (r) 1 + i n d (r) 2 -i n d (r) 3 + (-1) n d (r) 4 + 2 q-1 k=1 (-1) k δ (r) k +(-1) q q -1 8q 2 d (r) 1 + (-1) q (i n d (r) 2 -i n d (r) 3 + d (r) 4 ) + 2 q-1 k=1 δ (r) k + q-1 l=1 (-1) l q -1 4q 2 d (r) 1 + (-1) l (i n d (r) 2 -i n d (r) 3 ) + (-1) n d (r) 4 + q-1 k=1 (-1) k (ζ lk 2q + ζ -lk 2q )δ (r) k
, and P Γ (t) i1(q+3)+3 (resp. P Γ (t) i1(q+3)+2 ) is obtained by replacing in P Γ (t) i1(q+3) (resp. P Γ (t) i1(q+3)+1 ) d 2 by -d 2 and d 3 by -d 3 . Finally, for i 2 ∈ [[1, q -1]], we have

P Γ (t, u) i1(q+3)+i2+3 = 1 m m-1 r=0 ζ i1r m 3q -1 8q 2 2d (r) 1 + 2(-1) i2 d (r) 4 + 2 q-1 k=1 (ζ ki2 2q + ζ -ki2 2q )δ (r) k +(-1) q q -1 8q 2 2d (r) 1 + 2(-1) i2 d (r) 4 + 2 q-1 k=1 (-1) k (ζ i2k 2q + ζ -i2k 2q )δ (r) k + q-1 l=1 (-1) l q -1 4q 2 2d (r) 1 + 2(-1) i2 d (r) 4 + q-1 k=1 (ζ ki2 2q + ζ -ki2 2q )δ (r) k (ζ kl 2q + ζ -kl 2q
) .

• We now make the matrix A (1) explicit: A (1) is a block-matrix with m × m blocks of size (q + 3) × (q + 3). ⊲ If m ≥ 5, then the matrices A (1) and C Γ := 2I -A (1) -t A (1) are defined by

A (1) =          0 I B B . . . . . . . . . . . . I I . . . . . . I B 0          , C Γ =              2I -B -I -I -B -B 2I -B -I -I -I -B 2I -B -I -I -B . . . . . . . . . -I . . . . . . -B -I -I . . . -B 2I -B -B -I -I -B 2I              , with B =       0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0       , if q = 2, and B =              
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 . . . 0 0 0 0 0 0 . . . . . . 1

0 1 1 0 0 0 1 0               if q ≥ 3 ⊲ If m = 3
, then the matrices A (1) and C Γ := 2I -A (1) -t A (1) are defined by

A (1) =   0 0 B B 0 0 0 B 0   , C Γ =   0 B B B 0 B B B 0   , with B =       1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 1       , if q = 2, and B =              
1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 . . . 0 0 0 0 0 0 . . . . . . 1

0 1 1 0 0 0 1 1               if q ≥ 3.
⊲ If m = 1, then the matrices A (1) and C Γ are A (1) = B and C Γ := 2I -A (1) -t A (1) + 2 Diag(A (1) ), with B defined as in the case m = 3.

• For i 1 ∈ [[0, m -1]] and i 2 ∈ [[0, q + 2
]], let s i1,i2 be the reflection associated to the (i 1 (q + 3) + i 2 )-th root. Then the set S may be decomposed in p sets where p ∈ {2, 3, 4, 5}:

-If m ≥ 3, then:

⊲ If m ≡ 0 mod 3, set S l := {s 3k+l,i2 / (k, i 2 ) ∈ [[0, m 3 -1]]×[[0, q+2]]} for l ∈ [[0, 2]]. Then S = S 0 ⊔S 1 ⊔S 2 , p = 3, and C Γ = 3I -τ 0 -τ 1 -τ 2 . ⊲ If m ≡ 1 mod 3, set S l := {s 3k+l,i2 / (k, i 2 ) ∈ [[0, m-1 3 -1]] × [[0, q + 2]]} for l ∈ [[0, 2]],
and

S 3 := {s m-1,i2 / i 2 ∈ [[0, q + 2]]}. Then S = S 0 ⊔ S 1 ⊔ S 2 ⊔ S 3 , p = 4, and C Γ = 4I -τ 0 -τ 1 -τ 2 -τ 3 . ⊲ If m ≡ 2 mod 3, set S l := {s 3k+l,i2 / (k, i 2 ) ∈ [[0, m-2 3 -1]] × [[0, q + 2]]} for l ∈ [[0, 2]],
and

S 3 := {s m-2,i2 / i 2 ∈ [[0, q + 2]]}, S 4 := {s m-1,i2 / i 2 ∈ [[0, q + 2]]}. Then S = S 0 ⊔ S 1 ⊔ S 2 ⊔ S 3 ⊔ S 4 , p = 5, and C Γ = 5I -τ 0 -τ 1 -τ 2 -τ 3 -τ 4 .
-If m = 1, then: ⊲ If q = 2, set S 0 := {s 0,0 , s 0,1 , s 0,2 s 0,3 } and S 1 := {s 0,4 }. Then S = S 0 ⊔S 1 , p = 2, and C Γ = 2I -τ 0 -τ 1 . ⊲ If q ≥ 3 and q is even, set S 0 := {s 0,0 , s 0,1 , s 0,2 } and S 1 := {s 0,3 , . . . , s 0, q+1 }, S 2 := {s 0,4 , . . . , s 0, q+2 }. Then S = S 0 ⊔ S 1 ⊔ S 2 , p = 3, and C Γ = 2I -τ 0 -τ 1 -τ 1 . ⊲ If q ≥ 3 and q is odd, set S 0 := {s 0,0 , s 0,1 , s 0,2 } and S 1 := {s 0,3 , . . . , s 0, q+2 }, S 2 := {s 0,4 , . . . , s 0, q+1 }. Then S = S 0 ⊔ S 1 ⊔ S 2 , p = 3, and C Γ = 2I -τ 0 -τ 1 -τ 1 .

• If m = 1, the graph associated to Γ is the following:

1 0 q + 2 • • • 4 2 3
If m ≥ 2, the graph associated to Γ consists in q + 3 m-gones that are linked together.

The BT a subseries -Binary tetrahedral groups

Let ψ 4 , τ, η, φ 2m be the elements

ψ 4 :=   1 0 0 0 ζ 4 0 0 0 ζ -1 4   , τ :=   1 0 0 0 0 i 0 i 0   , η := 1 √ 2   √ 2 0 0 0 ζ 7 8 ζ 7 8 0 ζ 5 8 ζ 8   , φ 2m :=   ζ -2 2m 0 0 0 ζ 2m 0 0 0 ζ 2m   .
In this section 6 , we assume that m ≡ 1 or 5 mod 6, and we consider the subgroup Γ := ψ 4 , τ, η, φ

2m of SL 3 C. Note that φ 2m = ψ 2 4 φ -m-1 2 m
, so that Γ := ψ 4 , τ, η, φ m . Set Γ 1 := ψ 4 , τ, η and Γ 2 := 6 The other case -the type BT b -is m ≡ 3 mod 6. This group is not a direct product.

φ m ≃ Z/mZ. Then we have Γ ≃ Γ 2 × Γ 1 .
With the notations of the binary tetrahedral subgroup of SL 2 C, ψ 4 (resp. τ , η) represents a 2 (resp. b, c). Γ 1 ≃ a 2 , b, c is the binary tetrahedral subgroup of SL 2 C. Representatives to its 7 conjugacy classes are {id, a 4 = -id, b, c, c 2 , -c, -c 2 }, Finally, we obtain the series P Γ (t, u): for p ∈ [[0, m -1]], we get

P Γ (t, u) 7p = 1 24 m m-1 k=0 ζ pk m (f k,0 + f k,1 + 6 f k,2 + 4 f k,3 + 4 f k,4 + 4 f k,5 + 4 f k,6 ), P Γ (t, u) 7p+1 = 1 24 m m-1 k=0 ζ pk m (f k,0 + f k,1 + 6 f k,2 + (4 f k,3 + 4 f k,5 ) j + (4 f k,4 + 4 f k,6 ) j 2 ), P Γ (t, u) 7p+3 = 1 24 m m-1 k=0 ζ pk m (2 f k,0 -2 f k,1 + 4 f k,3 -4 f k,4 -4 f k,5 + 4 f k,6 ), P Γ (t, u) 7p+4 = 1 24 m m-1 k=0 ζ pk m (2 f k,0 -2 f k,1 + (4 f k,3 -4 f k,5 ) j + (-4 f k,4 + 4 f k,6 ) j 2 ), P Γ (t, u) 7p+6 = 1 24 m m-1 k=0 ζ pk m (3 f k,0 + 3 f k,1 -6 f k,2 ),
and P Γ (t, u) 7p+2 (resp. P Γ (t, u) 7p+5 ) is obtained by exchanging j and j 2 in P Γ (t, u) 7p+1 (resp. P Γ (t, u) 7p+4 ).

The BO subseries -Binary octahedral groups

For m ∈ N such that m ∧ 6 = 1, let ψ 8 , τ, η, φ 2m be the elements

ψ 8 :=   1 0 0 0 ζ 8 0 0 0 ζ 7 8   , τ :=   1 0 0 0 0 i 0 i 0   , η := 1 √ 2   √ 2 0 0 0 ζ 7 8 ζ 7 8 0 ζ 5 8 ζ 8   , φ 2m :=   ζ -2 2m 0 0 0 ζ 2m 0 0 0 ζ 2m   ,
and consider the subgroup

Γ = ψ 8 , τ, η, φ 2m of SL 3 C. Note that φ 2m = ψ 4 8 φ -m-1 2 m , so that Γ := ψ 8 , τ, η, φ m . Set Γ 1 := ψ 8 , τ, η and Γ 2 := φ m ≃ Z/mZ. Then Γ ≃ Γ 2 × Γ 1 .
With the notations of the binary octahedral subgroup of SL 2 C, ψ 8 (resp. τ , η) represents a (resp. b, c). Γ 1 ≃ a, b, c is the binary octahedral subgroup of SL 2 C. Reperesentatives of its 8 conjugacy classes are {id, a 4 = -id, ab, b, c2 , c, a, a 3 }, and its character table is the matrix given in Section 2.3.2 (Type E 7 -Binary octahedral group). The natural character of Γ is given by Finally, we obtain the series P Γ (t, u): for p ∈ [[0, m -1]], we have

P Γ (t, u) 8p = 1 48 m m-1 k=0 ζ pk m (f k,0 + f k,1 + 12 f k,2 + 6 f k,3 + 8 f k,4 + 8 f k,5 + 6 f k,6 + 6 f k,7 ), P Γ (t, u) 8p+2 = 1 48 m m-1 k=0 ζ pk m (2 f k,0 + 2 f k,1 + 12 f k,3 -8 f k,4 -8 f k,5 ), P Γ (t, u) 8p+3 = 1 48 m m-1 k=0 ζ pk m (2 f k,0 -2 f k,1 -8 f k,4 + 8 f k,5 + 6 √ 2f k,6 -6 √ 2f k,7 ), P Γ (t, u) 8p+5 = 1 48 m m-1 k=0 ζ pk m (3 f k,0 + 3 f k,1 -12 f k,2 -6 f k,3 + 6 f k,6 + 6 f k,7 ), P Γ (t, u) 8p+7 = 1 48 m m-1 k=0 ζ pk m (4 f k,0 -4 f k,1 + 8 f k,4 -8 f k,5 ),
and P Γ (t, u) 8p+1 (resp. P Γ (t, u) 8p+4 , P Γ (t, u) 8p+6 ) is obtained by replacing f k,2 , f k,6 , f k,7 by their opposite in P Γ (t, u) 8p (resp. P Γ (t, u) 8p+3 , P Γ (t, u) 8p+5 ).

The BI subseries -Binary icosahedral groups

For m ∈ N such that m ∧ 30 = 1, let µ, τ, η be the elements

µ :=   1 0 0 0 -ζ 3 5 0 0 0 -ζ 2 5   , τ :=   1 0 0 0 0 1 0 -1 0   , η := 1 ζ 2 5 -ζ -2 5   ζ 2 5 -ζ -2 5 0 0 0 ζ 5 -ζ -1 5 1 0 1 -ζ 5 -ζ -1 5   ,
and consider the subgroup Γ = µ, τ, η, φ 2m of SL 3 C.

Note that

φ 2m = η 2 φ -m-1 ζ pk m (2 f k,0 -2 f k,1 + (6 -6 √ 5)f k,2 + (-6 -6 √ 5)f k,3 + (6 + 6 √ 5)f k,4 +(-6 + 6 √ 5)f k,5 + 20 f k,6 -20 f k,7 ), P Γ (t, u) 9p+3 = 1 120 m m-1 k=0 ζ pk m (3 f k,0 + 3 f k,1 + (6 + 6 √ 5)f k,2 + (6 -6 √ 5)f k,3 + (6 -6 √ 5)f k,4 +(6 + 6 √ 5)f k,5 -30 f k,8 ), P Γ (t, u) 9p+5 = 1 120 m m-1 k=0 ζ pk m (4 f k,0 + 4 f k,1 -12 f k,2 -12 f k,3 -12 f k,4 -12 f k,5 + 20 f k,6 + 20 f k,7 ) P Γ (t, u) 9p+7 = 1 120 m m-1 k=0 ζ pk m (5 f k,0 + 5 f k,1 -20 f k,6 -20 f k,7 + 30 f k,8 ) P Γ (t, u) 9p+8 = 1 120 m m-1 k=0 ζ pk m (6 f k,0 -6 f k,1 -12 f k,2 + 12 f k,3 -12 f k,4 + 12 f k,5 ),
and P Γ (t, u) 9p+4 (resp. P Γ (t, u) 9p+2 ) is obtained by replacing 6 √ 5 by its opposite in P Γ (t, u) 9p+3 (resp. P Γ (t, u) 9p+1 ), and P Γ (t, u) 9p+6 is obtained by replacing f k,1 , f k,2 , f k,4 , f k,6 by their opposite in P Γ (t, u) 9p+5 .

Decomposition of C Γ for the subseries BT a, BO, BI

• For the subseries BT a (resp. BO, BI), we set n = 7 (resp. n = 8, n = 9). We now make the matrix A (1) explicit: A (1) is a block-matrix with m × m blocks of size n × n. ⊲ If m ≥ 5, then the matrices A (1) and C Γ := 2I -A (1) -t A (1) are defined by

A (1) = 0 B B B B B B B B @ 0 I Bn Bn . . . . . . . . . . . . I I . . . . . . I Bn 0 1 C C C C C C C C A , CΓ = 0 B B B B B B B B B B B B @ 2I -Bn -I -I -Bn -Bn 2I -Bn -I -I -I -Bn 2I -Bn -I -I -Bn . . . . . . . . . -I . . . . . . -Bn -I -I . . . -Bn 2I -Bn -Bn -I -I -Bn 2I 1 C C C C C C C C C C C C A , with B7 = 0 B B B B B B B B @
0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 1 0

1 C C C C C C C C A , B8 = 0 B B B B B B B B B B B B B B B B B B @
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1

0 0 1 0 0 1 1 0 1 C C C C C C C C C C C C C C C C C C A , B9 = 0 B B B B B B B B B B B B B B B B B B B B B @
0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0

0 0 0 0 1 1 0 1 1 C C C C C C C C C C C C C C C C C C C C C A . ⊲ If m = 1, then the matrix A (1) is A (1) := B n + I, and C Γ is defined by C Γ := 2I -A (1) -t A (1) + 2 Diag(A (1) ), i.e. C Γ = 2I -2B n .
• The decomposition of C Γ is the following:

for i 1 ∈ [[0, m -1]] and i 2 ∈ [[0, n -1]]
, let s i1,i2 be the reflection associated to the (ni 1 +i 2 )-th root. Then the set S may be decomposed in p sets where p ∈ {2, 4, 5}.

-If m ≥ 5, then:

⊲ If m ≡ 1 mod 3, set S l := {s 3k+l,i2 / (k, i 2 ) ∈ [[0, m-1 3 -1]] × [[0, n -1]]} for l ∈ [[0, 2]],
and

S 3 := {s m-1,i2 / i 2 ∈ [[0, n -1]]}. Then S = S 0 ⊔ S 1 ⊔ S 2 ⊔ S 3 , p = 4, and C Γ = 4I -τ 0 -τ 1 -τ 2 -τ 3 . ⊲ If m ≡ 2 mod 3, set S l := {s 3k+l,i2 / (k, i 2 ) ∈ [[0, m-2 3 -1]] × [[0, n -1]]} for l ∈ [[0, 2]],
and

S 3 := {s m-2,i2 / i 2 ∈ [[0, n -1]]}, S 4 := {s m-1,i2 / i 2 ∈ [[0, n -1]]}. Then S = S 0 ⊔ S 1 ⊔ S 2 ⊔ S 3 ⊔ S 4 , p = 5, and C Γ = 5I -τ 0 -τ 1 -τ 2 -τ 3 -τ 4 .
-If m = 1, then S = S 0 ⊔ S 1 , p = 2, and C Γ = 2I -τ 0 -τ 1 , with ⊲ If n = 7, S 0 := {s 0,0 , s 0,1 , s 0,2 , s 0,6 } and S 1 := {s 0,3 , s 0,4 , s 0,5 }. ⊲ If n = 8, S 0 := {s 0,0 , s 0,1 , s 0,2 , s 0,5 , s 0,6 } and S 1 := {s 0,3 , s 0,4 , s 0,7 }. ⊲ If n = 9, S 0 := {s 0,0 , s 0,3 , s 0,4 , s 0,5 , s 0,7 } and S 1 := {s 0,1 , s 0,2 , s 0,6 , s 0,8 }.

• If m = 1, the graph associated to Γ is the following: If m ≥ 5, the graph associated to Γ is a graph of type BT a (resp. BO, BI), m = 1, such that every vertex is a m-gone.

The C series

Let H ≃ Z/j 1 Z × Z/j 2 Z be a group of the series A, with eventually j 1 = 1 or j 2 = 1, and consider the matrix T :=   0 1 0 0 0 1 1 0 0   , that is the matrix of the permutation (1, 2, 3) of S 3 . In this section, we study Γ := H, T , the finite subgroup of SL 3 C generated by H and T . The subgroup N of Γ which consists of all the diagonal matrices of Γ is a normal subgroup of Γ. By using the Bezout theorem,

N =    g k1,k2 :=   ζ k1 m 0 0 0 ζ k2 m 0 0 0 ζ -k1-k2 m   / (k 1 , k 2 ) ∈ [[0, m -1]] 2    . (7)
Moreover, we have

N ∩ T = {id} and |N T | = |N || T | |N ∩ T | = 3m 2 = |Γ|. So, Γ is the semi-direct product Γ ≃ N ⋊ T ≃ (Z/mZ) 2 ⋊ T .
We will obtain all the irreducible characters of Γ by induction; we distinguish two cases corresponding to the two following subsections.

3.4.1

Series Cm non divisible by 3

• Set n ′ := m 2 -1 3 , so that |N | = 3n ′ + 1 and |G| = 3m 2 = 3(3n ′ + 1)
. The conjugacy classes of Γ are:

Class id T T -1 g ∈ N \{id}(n ′ classes) Cardinality 1 m 2 m 2 3
For each element g k1,k2 ∈ N \{id}, the conjugacy class of g is the set {g k1,k2 , g k2,-k1-k2 , g -k1-k2,k1 }. In order to obtain a transversal of N \{id}, i.e. a set containing exactly one representant of each conjugacy class of N \{id}, we represent the elements of N \{id} by points (k

1 , k 2 ) of [[0, m -1]] 2 .
So, we search a transversal for the set of elements of the form (k

1 , k 2 ), (k 2 , -k 1 , -k 2 mod m) and (-k 1 , -k 2 mod m, k 1 ), with (k 1 , k 2 ) ∈ [[0, m -1]] 2 .
A solution is the following: for a given conjugacy class, its three elements are on the edges of a triangle (see Figure 2), with exactly one element on each edge of the triangle. Therefore we may take as transversal the set of all points that belong to the vertical edges minus the nearest point of the diagonal. More precisely, a transversal for N \{id} is the set E cc defined by We choose the usual lexicographic order on E cc , so that we can number its elements:

{(0, k 2 ) / k 2 ∈ [[1, m -1]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[1, m 3 ]], k 2 ∈ [[k 1 , m -1 -2k 1 ]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[m -⌊ m 3 ⌋, m -1]], k 2 ∈ [[2(m -k 1 ) + 1, k 1 ]]}.
E cc = {c 1 , c 2 , . . . , c m 2 -1 3 }, with c i = (c (1)
i , c

(2)

i ) ∈ [[0, m -1]] 2 .
• The group Γ is generated by R := g 1,0 and T , which verify the relations R m = (RT ) 3 = T 3 = id. As m is not divisible by 3, the irreducible characters of degree 1 are χ 0,l : R → 1, T → j l for l ∈ [[0, 2]]. We have [G : N ] = 3 with N abelian, so the possible degrees of the irreducible characters are 1, 2, 3. The irreducible characters χ l1, l2 induced by the irreducible characters of N are given by

Class [id] [T ] [T -1 ] [g], g ∈ N \{id} Value 3 0 0 ζ k1l1+k2l2 m + ζ (-k1-k2)l1+k1l2 m + ζ k2l1+(-k1-k2)l2 m
The characters χ l1, l2 with (l 1 , l 2 ) = (0, 0) are represented by points (l 1 , l 2 ) of [[0, m -1]] 2 . The points that are associated to the same character are on a triangle or on a "trident ", with exactly one point on each edge (see Figure 2). So the set of irreducible characters χ l1, l2 with (l 1 , l 2 ) = (0, 0) is obtained by taking the following set E ic of indexes:

{(0, k 2 ) / k 2 ∈ [[1, m -1]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[1, m 3 ]], k 2 ∈ [[2k 1 + 1, m -k 1 ]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[m -⌊ m 3 ⌋, m -1]], k 2 ∈ [[m -k 1 , 2k 1 -m -1]]}.
We choose the usual lexicographic order on E ic , so that we can number its elements:

E ic = {d 1 , d 2 , . . . , d m 2 -1 3 with d i = (d (1) i , d (2)
i ) ∈ [[0, m -1]] 2
, and the character table T Γ is

T Γ =          1 1 1 1 • • • 1 1 j j 2 1 • • • 1 1 j 2 j 1 • • • 1 3 0 0 . . . . . . . . . C 3 0 0          , where the general term of C ∈ M m 2 -1 3 C is c i, j := ζ c (1) j d (1) i +c (2) j d (2) i m + ζ (-c (1) j -c (2) j )d (1) i +c (1) j d (2) i m + ζ c (2) j d (1) i +(-c (1) j -c (2) j )d (2) i m , (i, j) ∈ [[1, m 2 -1 3 ]] 2 .
• The values of the natural character χ of Γ are

Class [id] [T ] [T -1 ] [g], g ∈ N \{id} Value 3 0 0 ζ k1 m + ζ k2 m + ζ -k1-k2 m Therefore the diagonal matrix ∆(t, u) is Diag ε 1 , ε 2 , ε 3 , ∆(t, u) , with ε 1 := f (3, 3), ε 2 = ε 3 := f (0, 0), and the general term of ∆(t, u) ∈ M m 2 -1 3 C is γ j := f ζ -c (1) j m + ζ -c (2) j m + ζ c (1) j +c (2) j m , ζ c (1) j m + ζ c (2) j m + ζ -c (1) j -c (2) j m
, j ∈ [[1, m 2 -1 3 ]].
Then, by setting

Σ := m 2 -1 3 p=1 γ p , Σ i := m 2 -1 3 j=1 γ j ζ c (1) j d (1)
i +c

(2)

j d (2) i m + ζ (-c (1) j -c
(2)

j )d (1)
i +c

(1)

j d (2) i m + ζ c (2) j d (1)
i +(-c

(1)

j -c (2) j )d
(2) i m , we obtain the formula for P Γ (t, u):

P Γ (t, u) 0 = 5(m + 1)(m -1) + 1 9m 4 (3Σ + ε 1 + ε 2 + ε 3 ) - 2(m + 1)(m -1) 9m 2 (ε 1 -ε 2 -ε 3 ) + 2(m + 1)(m -1) 3m 4 m 2 -1 3 q=1   ε1 + ε 2 + ε 3 + m 2 -1 3 p=1 γ p c p, q    , P Γ (t, u) 1 = 5(m + 1)(m -1) + 1 9m 4 (3Σ + ε 1 + jε 2 + j 2 ε 3 ) - 2(m + 1)(m -1) 9m 2 (ε 1 -jε 2 -j 2 ε 3 ) + 2(m + 1)(m -1) 3m 4 m 2 -1 3 q=1   ε1 + jε 2 + j 2 ε 3 + m 2 -1 3 p=1 γ p c p, q    ,
and P Γ (t, u) 2 is obtained by exchanging j and j 2 in P Γ (t, u) 1 , and

for i ∈ [[1, m 2 -1 3 ]], P Γ (t, u) i+2 = 5(m + 1)(m -1) + 1 9m 4 (3Σ i + 3ε 1 ) - 6(m + 1)(m -1) 9m 2 ε 1 + 2(m + 1)(m -1) 3m 4 m 2 -1 3 q=1   3ε1 + m 2 -1 3 p=1 γ p c i, p c p, q    . 3.4.2 Series C -m divisible by 3 • Set n ′ := m 2 3 , so that |N | = 3n ′ and |G| = 3m 2 = 3(3n ′ ). Set a := R m 3 = Diag(j, j, j). The conjugacy classes of Γ are: Class id a a 2 T T -1 RT RT -1 R 2 T R 2 T -1 g ∈ N \{id, a, a 2 }(n ′ -1 classes) Cardinality 1 1 1 m 2 3 m 2 3 m 2 3 m 2 3 m 2 3 m 2 3 3
For each element g k1,k2 ∈ N \{id, a, a 2 }, the conjugacy class of g is the set {g k1,k2 , g k2,-k1-k2 , g -k1-k2,k1 }. Its three elements are on the edges of a triangle, with exactly one element of each edge of the triangle. So, a transversal of N \{id, a, a 2 } has the same form as in the case where 3 does not divide m, i.e. a transversal for N \{id, a, a 2 } is the set E cc defined by

{(0, k 2 ) / k 2 ∈ [[1, m -1]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[1, m 3 -1]], k 2 ∈ [[k 1 , m -1 -2k 1 ]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[m -m 3 + 1, m -1]], k 2 ∈ [[2(m -k 1 ) + 1, k 1 ]]}.
• As m is divisible by 3, the irreducible characters of degree 1 of Γ are, for (k, l) ∈ [[0, 2]] 2 , the nine elements χ k,l : R → j k T → j l .

As for the case where m is divisible by 3, the set of irreducible characters χ l1, l2 with (l 1 , l 2 ) = (0, 0) is obtained by taking the following set E ic of indexes:

{(0, k 2 ) / k 2 ∈ [[1, m -1]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[1, m 3 -1]], k 2 ∈ [[2k 1 + 1, m -k 1 ]]} ⊔ {(k 1 , k 2 ) / k 1 ∈ [[m -m 3 + 1, m -1]], k 2 ∈ [[m -k 1 , 2k 1 -m -1]]}.
We choose the usual lexicographic order on E cc and E ic , so that we can number its elements:

E cc = {c 1 , c 2 , . . . , c m 2 3 -1 }, E ic = {d 1 , d 2 , . . . , d m 2 3 -1 }, with c j = (c (1) j , c
(2)

j ) ∈ [[0, m -1]] 2 , and d i = (d (1) i , d (2)
i ) ∈ [[0, m -1]] 2 , and we deduce the character table T Γ of Γ:

T Γ =                        1 1 1 1 1 1 1 1 1 1 1 1 j j 2 j j 2 j j 2 1 1 1 1 j 2 j j 2 j j 2 j 1 1 1 1 1 j j j 2 j 2 1 1 1 j j 2 j 2 1 1 j j c (1) i -c (2) i 1 1 1 j 2 j 1 j 2 j 1 1 1 1 1 1 j 2 j 2 j j 1 1 1 j j 2 1 j j 2 1 j 2(c (1) i -c (2) i ) 1 1 1 j 2 j j 1 1 j 2 3 J (1) 1 J (2) 1 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . C 3 J (1) m 2 3 -1 J (2) m 2 3 -1 0 0 0 0 0 0                        , where C ∈ M m 2
3 -1 C is a block-matrix with general term

c i, j := ζ c (1) j d (1)
i +c

(2)

j d (2) i m + ζ (-c
(1) j -c

(2)

j )d (1)
i +c

(1)

j d (2) i m + ζ c (2) j d (1)
i +(-c

(1) j -c

(2)

j )d (2) i m , (i, j) ∈ [[1, m 2 3 -1]] 2 , J (1) i
:= j d (1) i +d (2) i +j -2d (1) i +d (2) i +j d (1)
i -2d

(2) i , J

(2) i := j 2d (1) i +2d

(2) i +j -d (1) i +2d

(2)

i +j 2d (1) i -d (2) i , i ∈ [[1, m 2 3 -1]] 2 .
• The values of the natural character χ of Γ are

Class id a a 2 T T -1 RT RT -1 R 2 T R 2 T -1 g ∈ N \{id, a, a 2 } Value 3 3j 3j 2 0 0 0 0 0 0 ζ k1 m + ζ k2 m + ζ -k1-k2
m Therefore the diagonal matrix ∆(t, u) is Diag( ∆(t, u), ∆(t, u)), with

∆(t, u) = Diag(β 1 , β 2 , β 3 , β 4 , . . . , β 4 6 terms ) = Diag   f (3, 3), f (3j 2 , 3j), f (3j, 3j
2 ), f (0, 0), . . . , f (0, 0)

6 terms   ,
and the general term of ∆(t, u) ∈ M m 2 3 -1 C is

γ j := f ζ -c (1) j m + ζ -c (2) j m + ζ c (1) j +c (2) j m , ζ c (1) j m + ζ c (2) j m + ζ -c
(1)

j -c (2) j m , j ∈ [[1, m 2 3 -1]].
For (i, r) ∈ {1, 2} × {1, 2, 3}, and (s,

q) ∈ {1, 2} × [[1, m 2 3 -1]], let us define Σ (r) := m 2 3 -1 p=1 γ p j (r-1)(c (1) p -c (1) p ) , Φ (i,r) := m 2 3 -1 p=1 J (i) p γ p j (r-1)(c (1) p -c (1) p ) , Σ (s) q := m 2 3 -1 p=1 J (s) p γ p ζ c (1) p d (1) q +c (2) p d (2) q m + ζ (-c (1) p -c (2) p )d (1) q +c (1) p d (2) q m + ζ c (2) p d (1) q +(-c (1) p -c (2) p )d (2) q m , ξ := (-1, -1, 2 1 , -1, -1, 2 2 , -1, -1, 2 3 , . . . , -1, -1, 2 m 3 ( m 3 +1) 2 -1
, -1, -1, -1, -1, 2 , -1, -1, 2 , . . . , -1, -1, 2 ).

Then, we may give the expression of the series P Γ (t, u):

PΓ(t, u)0 = 3(5( m 3 ) 2 -1) + 1 3m 4 (3Σ (1) + β1 + β2 + β3 + 6β4) + 6( m 3 ) 2 -2 3m 4 (Φ (11) + Φ (21) + 2β1 + 2β2 + 2β3 + 12β4) + 3( m 3 ) 2 -1 m 4 m 2 3 -1 X q=1 ξq 0 B @β1 + β2 + β3 + 3j c (1) q -c
(2)

q β4 + 3j 2(c (1) q -c (2) q ) β4 + m 2 3 -1 X p=1 γpcp,q 1 C A , PΓ(t, u)1 = 3(5( m 3 ) 2 -1) + 1 3m 4 (3Σ (1) + β1 + β2 + β3 -3β4) + 6( m 3 ) 2 -2 3m 4 (Φ (11) + Φ (21) + 2β1 + 2β2 + 2β3 -6β4) + 3( m 3 ) 2 -1 m 4 m 2 3 -1 X q=1 ξq 0 B @β1 + β2 + β3 + (2j + j 2 )j c (1) q -c
(2)

q β4 + (2j 2 + j)j 2(c (1) q -c (2) q ) β4 + m 2 3 -1 X p=1 γpcp,q 1 C A , PΓ(t, u)3 = 3(5( m 3 ) 2 -1) + 1 3m 4 (3Σ (2) + β1 + β2 + β3) + 6( m 3 ) 2 -2 3m 4 (Φ (12) + Φ (22) + 2β1 + 2β2 + 2β3) + 3( m 3 ) 2 -1 m 4 m 2 3 -1 X q=1 ξq 0 B @β1 + β2 + β3 + (2 + j)j c (1) q -c
(2)

q β4 + (2j 2 + j)j 2(c (1) q -c (2) q ) β4 + m 2 3 -1 X p=1 γpj c (1)
p -c

(2)

p cp,q 1 C A , PΓ(t, u)5 = 3(5( m 3 ) 2 -1) + 1 3m 4 (3Σ (2) + β1 + β2 + β3) + 6( m 3 ) 2 -2 3m 4 (Φ (12) + Φ (22) + 2β1 + 2β2 + 2β3) + 3( m 3 ) 2 -1 m 4 m 2 3 -1 X q=1 ξq 0 B @β1 + β2 + β3 + m 2 3 -1 X p=1 γpj c (1)
p -c

(2)

p cp,q 1 C A , PΓ(t, u)6 = 3(5( m 3 ) 2 -1) + 1 3m 4 (3Σ (3) + β1 + β2 + β3) + 6( m 3 ) 2 -2 3m 4 (Φ (13) + Φ (23) + 2β1 + 2β2 + 2β3) + 3( m 3 ) 2 -1 m 4 m 2 3 -1 X q=1 ξq 0 B @β1 + β2 + β3 + (2 + j 2 )j c (1) q -c
(2)

q β4 + (2j + j 2 )j 2(c (1) q -c (2) q ) β4 + m 2 3 -1 X p=1 γpj 2(c (1) p -c (2) p ) cp,q 1 C A ,
and P Γ (t, u) 2 (resp. P Γ (t, u) 4 , P Γ (t, u) 8 ) is obtained by exchanging the coefficients 2j + j 2 and 2j 2 + j (resp. 2 + j and 2j 2 + j, 2 + j 2 and 2j + j 2 ) in P Γ (t, u) 1 (resp. P Γ (t, u) 3 , P Γ (t, u) 6 ); P Γ (t, u) 7 is obtained by replacing Φ (1i) by Φ (2i) and j c (1)

p -c (2) p by j 2(c (1) p -c (2) p ) in P Γ (t, u) 5 . Finally, for i ∈ [[1, m 2 3 -1]], we have PΓ(t, u)i+8 = 3(5( m 3 ) 2 -1) + 1 3m 4 (3β1 + J (1)
i β2 + J

(2)

i β3 + 3Σi) + 6( m 3 ) 2 -2 3m 4 (6β1 + 2J (1)
i β2 + 2J

(2)

i β3 + g Σ (1) i + g Σ (2) i ) + 3( m 3 ) 2 -1 m 4 m 2 3 -1 X q=1 ξq 0 B @3β1 + J (1) i β2 + J (2) i β3 + m 2 3 -1 X p=1 γpci,pcp,q 1 C A .

Decomposition of C Γ

We now make the matrix A (1) explicit: the form of the matrix A (1) is nearly the same in the case where m is divisible by 3 as in the other case. The main difference between these two cases is due to the fact that in the case where m is divisible by 3, there are 9 irreducible characters of degree 1 instead of 3. Set

• If m = 2, then A (1) = 0 3,3 1 3,1 1 1,3 2
, and C Γ = 3I -τ 0 -τ 1 -τ 2 , with τ 0 := s 0 s 2 , τ 1 := s 1 , τ 2 := s 3 .

• If m = 3, then A (1) = 0 9,9 1 9,2 1 2,9 A , A = 0 0 3 0 , and C Γ = 2I -(s 0 s 1 . . . s 9 ) -(s 10 s 11 ).

• Now, we assume that m ≥ 4. For i

1 ∈ {0} ⊔ [[1, κ m ]] ⊔ [[m -κ m , m -1]],
we define the set S i1 by:

S i1 :=    {s 0,i2 / i 2 ∈ [[1, m -1]]} if i 1 = 0, {s i1,i2 / i 2 ∈ [[i 1 , m -1 -2i 1 ]]} if i 1 ∈ [[1, κ m ]], {s i1,i2 / i 2 ∈ [[2(m -i 1 ) + 1, i 1 ]]} if i 1 ∈ [[m -κ m , m -1]].
Then, we distinguish two cases: ⊲ If κ m is odd, we set I 0 := {0, 2, 4, . . . , κ m -1, m -κ m , m -κ m + 2, . . . , m -3},

I 1 := {1, 3, 5, . . . , κ m , m -κ m + 1, m -κ m + 3, . . . , m -2}, I 2 := {m -1}.
⊲ If κ m is even, we set I 0 := {0, 2, 4, . . . , κ m , m -κ m + 1, m -κ m + 3, . . . , m -3},

I 1 := {1, 3, 5, . . . , κ m -1, m -κ m , m -κ m + 2, . . . , m -2}, I 2 := {m -1}.
Then, the roots associated to the reflections of distinct S i1 's for i 1 belonging to a same I k are orthogonal. Now, we decompose each S i1 , i.e. S i1 = S i1,0 ⊔ • • • ⊔ S i1,q-1 , such that q ∈ {1, 2, 3} and for every k ∈ [[0, q -1]], the roots associated to the reflections belonging to S i1,k are orthogonal:

⊲ If i 1 = 0, then ⋄ If m -1 is odd, then S 0 = S 0,0 ⊔ S 0,1 ⊔ S 0,2 , with
S 0,0 = {s 0,1 , s 0,3 , . . . , s 0,m-3 }, S 0,1 = {s 0,2 , s 0,4 , . . . , s 0,m-2 }, S 0,2 = {s 0,m-1 }.

⋄ If m -1 is even, then S 0 = S 0,0 ⊔ S 0,1 , with S 0,0 = {s 0,1 , s 0,3 , . . . , s 0,m-2 }, S 0,1 = {s 0,2 , s 0,4 , . . . , s 0,m-1 }.

⊲ If i 1 ∈ [[1, κ m ]], then we have S i1 = S i1,0 ⊔ S i1,1 ⊔ S i1,2 , with: ⋄ If m -3i 1 is odd, then S i1,0 = {s i1,i1 , s i1,i1+2 , . . . , s i1,m-2i1-3 }, S i1,1 = {s i1,i1+1 , s i1,i1+3 , . . . , s i1,m-2i1-2 }, S i1,2 = {s i1,m-2i1-1 }. ⋄ If m -3i 1 is even, then S i1,0 = {s i1,i1 , s i1,i1+2 , . . . , s i1,m-2i1-4 }, S i1,1 = {s i1,i1+1 , s i1,i1+3 , . . . , s i1,m-2i1-1 }, S i1,2 = {s i1,m-2i1-2 }. ⊲ If i 1 ∈ [[m -κ m , m -1]], then we have S i1 = S i1,0 ⊔ S i1,1 ⊔ S i1,2 , with: ⋄ If 3i 1 -2m is odd, then S i1,0 = {s i1,2(m-i1)+1 , s i1,2(m-i1)+3 , . . . , s i1,i1-2 }, S i1,1 = {s i1,2(m-i1)+2 , s i1,2(m-i1)+4 , . . . , s i1,i1-1 }, S i1,2 = {s i1,i1 }. ⋄ If 3i 1 -2m is even, then S i1,0 = {s i1,2(m-i1)+1 , s i1,2(m-i1)+3 , . . . , s i1,i1-1 }, S i1,1 = {s i1,2(m-i1)+2 , s i1,2(m-i1)+4 , . . . , s i1,i1-2 }, S i1,2 = {s i1,i1 }.
Note that some sets S i1,k can be empty for k ∈ {1, 2}.

Example 11

For m = 16, we have the following decomposition: τ 0,0 = (s 0,1 s 0,3 . . . s 0,13 )(s 2,2 s 2,4 . . . s 2,8 )(s 4,4 )(s 11,11 )(s 13,7 s 13,9 s 13,11 ) τ 0,2 = (s 0,15 )(s 2,10 )(s 4,6 )(s 13,13 ) τ 0,1 = (s 0,2 s 0,4 . . . s 0,14 )(s 2,3 s 2,5 . . . s 2,11 )(s 4,5 s 4,7 )(s 13,8 s 13,10 s 13,12 ) τ 1,2 = (s 1,13 )(s 3,9 )(s 12,12 )(s 14,14 ) τ 1,0 = (s 1,1 s 1,3 . . . s 1,11 )(s 3,3 s 3,5 s 3,7 )(s 5,5 )(s 12,9 s 12,11 )(s 14,5 s 14,7 . . . s 14,13 ) τ 2,0 = (s 15,3 s 15,5 . . . s 15,13 ) τ 1,1 = (s 1,2 s 1,4 . . . s 1,12 )(s 3,4 s 3,6 s 3,8 )(s 12,10 )(s 14,6 s 14,8 . . . s 14,12 ) τ 2,1 = (s 15,4 s 15,6 . . . s 15,14 ) τ 2,2 = (s 15,15 )(s -1,0 s -1,1 s -1,2 )

The D series

A group of type D is generated by a group of type C and a matrix

Q :=   a 0 0 0 0 b 0 c 0   ,
with abc = -1. This group is not a direct product. We can't give a general formula for this group and we only give a simple example.

Example 12

Consider the group

Γ :=   1 0 0 0 -1 0 0 0 -1   ,   0 1 0 0 0 1 1 0 0   ,   -1 0 0 0 0 1 0 1 0   .
This group is isomorphic to the symmetric group S 4 , so Γ has 5 conjugacy classes. The series P Γ (t, u) verifies

∀ i ∈ [[0, 4]], P Γ (t, u) i = (1 -tu) N (t, u) i D(t)D(u) ,
with D(t) = (t -1) 3 (t 2 + t + 1)(t 2 + 1)(t + 1) 2 , and

N (t, u) 0 = t 6 u 6 + t 5 u 5 -t 6 u 3 -t 3 u 6 + t 5 u 3 + 2 t 4 u 4 + t 3 u 5 + t 4 u 3 + t 3 u 4 + t 6 + t 5 u + 2 t 4 u 2 + 4 t 3 u 3 +2 t 2 u 4 + tu 5 + u 6 + t 3 u 2 + t 2 u 3 + t 3 u + 2 t 2 u 2 + tu 3 -t 3 -u 3 + tu + 1, N (t, u) 1 = t 6 u 3 + t 5 u 4 + t 4 u 5 + t 3 u 6 + t 5 u 3 + t 4 u 4 + t 3 u 5 + t 5 u 2 + t 4 u 3 + t 3 u 4 + t 2 u 5 + t 4 u 2 + t 2 u 4 +t 4 u + t 3 u 2 + t 2 u 3 + tu 4 + t 3 u + t 2 u 2 + tu 3 + t 3 + t 2 u + tu 2 + u 3 , N (t, u) 2 = t 4 u 2 + t 3 u 3 + t 2 u 4 + t 3 u 2 + t 2 u 3 + t 3 u + tu 3 + t 2 u + tu 2 + t 2 + tu + u 2 t 2 + 1 u 2 + 1 , N (t, u) 3 = t 4 u 2 + t 3 u 3 + t 2 u 4 + t 3 u + tu 3 + t 2 + tu + u 2 t 2 + t + 1 u 2 + u + 1 , N (t, u) 4 = (t 4 u 3 + t 3 u 4 -t 4 u 2 -t 3 u 3 -t 2 u 4 + t 4 u + 2 t 3 u 2 + 2 t 2 u 3 + tu 4 -t 3 u -tu 3 + t 3 + 2 t 2 u +2 tu 2 + u 3 -t 2 -tu -u 2 + t + u) t 2 + t + 1 u 2 + u + 1 .
4 Exceptional subgroups of SL 3 C -Types E, F, G, H, I, J, K, L

For every exceptional subgroup of SL 3 C, we begin by making the matrix A (1) explicit. Then we give a decomposition of C Γ := 2 I -A (1) -A (2) + 2 Diag(A (1) ) as a sum of p elements, with p ∈ {3, 4}, so that C Γ = p I -(τ 0 + • • • + τ p-1 ), and we give the graph associated to C Γ . We also write the list Θ of eigenvalues of A (1) .

Finally, we compute the sum of the series P Γ (t, u) = N Γ (t, u) D Γ (t, u) . In all the cases, the denominator is of the form D Γ (t, u) i = D Γ (t) i D Γ (u) i . Moreover, we will take the lowest common multiple D Γ (t) of the D Γ (t) i 's in order that all the denominators are the same and have the form D Γ (t)D Γ (u), i.e.

∀ i ∈ [[0, l]], P Γ (t, u) i = (1 -tu) M Γ (t, u) i D Γ (t)D Γ (u) .

Because of the to big size of the numerators, only the denominator and the relations between the numerators are given in the text: all the numerators may be found on the web. We also give the Poincaré series of the invariant ring P Γ (t) := P Γ (t, 0) 0 = P Γ (0, t) 0 .

P G (t) = -t 18 -t 36 -1 (t -1)3 (t 6 -t 3 + 1) (t2 + t + 1) 3 (t + 1) 2 (t 2 + 1) (t4 -t 2 + 1) (t 2 -t + 1) 2 (t 6 + t 3 + 1) 2

Type H

The group of type H, isomorphic to the alternating group A 5 , is the group S, U, T , with M J (t, u) 2 = M J (u, t) 1 M J (t, u) 6 = M J (u, t) 5 M J (t, u) 8 = M J (u, t) 7 M J (t, u) 11 = M J (u, t) 10 M J (t, u) 14 = M J (u, t) 13 D J (t) = t 4 + t 3 + t 2 + t + 1 t 8 -t 7 + t 5 -t 4 + t 3 -t + 1 (t + 1) 2 t 2 -t + 1 2 (t -1) 3 t 2 + t + 1 3 P J (t) = -t 24 -t 12 -1 (t 4 + t 3 + t 2 + t + 1) (t 8 -t 7 + t 5 -t 4 + t 3 -t + 1) (t + 1) 2 (t 2 -t + 1) 2 (t -1) 3 (t 2 + t + 1) 3

Type K

The group of type K is the group S, T, R, W , with S, T, R as for the type I, and W := Diag(j, j, j). It is the direct product of the group of type I and the center of SL 3 C. Here l + 1 = 18, rank(A (1) ) = 15, Θ = (3, 0, 0, 0, 3ζ 3 , 3ζ ), p = 3, and τ 0 := s 1 s 5 s 6 s 11 s 14 s 17 , τ 1 := s 0 s 3 s 4 s 9 s 12 s 15 , τ 2 := s 2 s 7 s 8 s 10 s 13 s 16 . M K (t, u) 2 = M K (u, t) 1 M K (t, u) 4 = M K (u, t) 3 M K (t, u) 7 = M K (u, t) 6 M K (t, u) 8 = M K (u, t) 5 M K (t, u) 11 = M K (u, t) M K (t, u) 14 = M K (u, t) M K (t, u) 17 = M K (u, t) D K (t) = (t -1) 3 t 2 + t + 1 3 t 2 + 1 t 4 -t 2 + 1 t 6 + t 5 + t 4 + t 3 + t 2 + t + 1 t 12 -t 11 + t 9 -t 8 + t 6 -t 4 + t 3 -t + 1 (t + 1) 2 t 2 -t + 1 2 P K (t) = -t 36 -t 18 -1 (t-1) 3 (t 2 +t+1) 3 (t 2 +1)(t 4 -t 2 +1)(t 6 +t 5 +t 4 +t 3 +t 2 +t+1)(t 12 -t 11 +t 9 -t 8 +t 6 -t 4 +t 3 -t+1)(t+1) 2 (t 2 -t+1) 2

Type L

The group of type L is the group S, U, T, V , with S, U, T as for the type H, and . Here l + 1 = 17, rank(A (1) ) = 15, p = 3, and τ 0 := s 0 s 5 s 6 s 9 s 10 s 11 s 14 , τ 1 := s 1 s 2 s 7 s 12 s 15 , τ 2 := s 3 s 4 s 8 s 13 s 16 . D L (t) = (t -1) 3 t 2 + t + 1 3 t 2 + 1 t 4 -t 2 + 1 t 4 + t 3 + t 2 + t + 1 t 8 -t 7 + t 5 -t 4 + t 3 -t + 1 (t + 1) 2 t 2 -t + 1 2 P L (t) = -t 30 +t 15 -1 (t-1) 3 (t 2 +t+1) 3 (t 2 +1)(t 4 -t 2 +1)(t 4 +t 3 +t 2 +t+1)(t 8 -t 7 +t 5 -t 4 +t 3 -t+1)(t+1) 2 (t 2 -t+1) 2

Figure 2 :

 2 Figure 2: Conjugacy classes and irreducible characters for m = 11.

  κ m := m 3 -1 if 3 divides m, and κ m := m 3 otherwise. The matrix A (1) is a block-matrix with (2 + 2κ m ) × (2 + 2κ m ) blocks: for example, if m = 16, the matrices A (1) and C Γ are matrices of size 88.

Figure 3 :

 3 Figure 3: The matrices A (1) and C Γ for m = 16.

Finally, we

  set S k,l := i1∈ b I k S i1,l , for (k, l) ∈ {0, 1, 2} 2 \{(2, 2)},and S 2,2 := i1∈ b I2 S i1,2 ⊔{s -1,0 , . . . , s -1,r }, with r = 8 if 3 divides m, and r = 2 otherwise. We denote by p ∈ [[1, 9]] the number of non-empty sets S k,l , and by τ k,l the commutative product of the reflections of S k,l . Then, C Γ = p I -(k,l)∈{0, 1, 2} 2 τ k,l .

DD

  Here l + 1 = 5, rank(A (1) ) = 4, A(1) is symmetric, Θ = (3, -1, -ζ 2 5 -ζ 3 5 , -ζ 5 -ζ 4 5 , 0), p = 3, and τ 0 := s 0 s 3 , τ 1 := s 1 s 2 , τ 2 := s 4 . H (t) = (t -1) 3 t 2 + t + 1 t 4 + t 3 + t 2 + t + 1 (t + 1)4.5 Type IThe group of type I is the group S, T, R , withHere l + 1 = 6, rank(A (1) ) = 5, Θ = (3, 0, 1,ζ 7 + ζ 2 7 + ζ 4 7 , ζ 3 7 + ζ 5 7 + ζ 6 7, -1), p = 4, and τ 0 := s 5 s 0 , τ 1 := s 1 s 4 , τ 2 := s 2 , τ 3 := s 3 . I (t) = (t -1)

24. 6

 6 Type JThe group of type J is the group S, U, T, W , with S, U, T as for the type H, and W := Diag(j, j, j). It is the direct product of the group of type H and the center ofSL 3 C. Here l + 1 = 15, rank(A (1) ) = 12, Θ = (3, -1, -ζ 2 5 -ζ 3 5 , -ζ 5 -ζ 4 5 , -ζ 2 3 , -ζ 3 , 3ζ 3 , -ζ 11 15 -ζ 14 15 , -ζ 2 15 -ζ 8 15 , 3ζ 2 3 , -ζ 7 15 -ζ 13 15 , -ζ 15 -ζ 415 , 0, 0, 0), p = 3, and τ 0 := s 2 s 5 s 7 s 10 s 13 , τ 1 := s 1 s 6 s 8 s 11 s 14 , τ 2 := s 0 s 3 s 4 s 9 s 12 .

  s := ζ 2 5 + ζ 3 5 , t := ζ 5 + ζ 4 5 , λ 1 := -1+i √ 15 4 , and λ 2 := -1-i √ 15 4

M

  Θ = (3, -ζ3, -ζ 2 3 , 3ζ 2 3 , 3ζ3, -1, ζ 2 3 , ζ3, 1, -ζ 2 15 -ζ 8 15 , -ζ15 -ζ 4 15 , -ζ 11 15 -ζ 14 15 , -ζ 7 15 -ζ 13 15 , L (t, u) 3= M L (u, t) 2 M L (t, u) 5 = M L (t, u) 6 M L (t, u) 8 = M L (u, t) 7 M L (t, u) 13 = M L (u, t) 12 M L (t, u) 16 = M L (u, t) 15

  and then P Γ (t) 3 (resp. P Γ (t) 2 ) is obtained by replacing in P Γ (t) 0 (resp. P Γ (t) 1 ) d 2 by -d 2 and d 3 by -d 3 .

Finally, for i ∈ [[1, n -1]], we have

  ) 4 = t 13 + t 11 + t 7 + t 5 , N Γ (t) 1 = t 12 + t 6 , N Γ (t) 5 = t 16 + t 12 + t 10 + t 8 + t 6 + t 2 , N Γ (t) 2 = t 14 + t 10 + t 8 + t 4 , N Γ (t) 6 = t 14 + t 12 + t 10 + t 8 + t 6 + t 4 , N Γ (t) 3 = t 17 + t 11 + t 7 + t, N Γ (t) 7 = t 15 + t 13 + t 11 + 2t 9 + t 7 + t 5 + t 3 .

  ) 4 = t 24 + t 20 + t 16 + t 14 + t 10 + t 6 , N Γ (t) 1 = t 23 + t 17 + t 13 + t 7 , N Γ (t) 5 = t 24 + t 22 + t 18 + t 16 + t 14 + t 12 + t 8 + t 6 , N Γ (t) 2 = t 29 + t 19 + t 11 + t, N Γ (t) 6 = t 27 + t 21 + T 19 + t 17 + t 13 + t 11 + t 9 + t 3 , N Γ (t) 3 = t 28 + t 20 + t 18 + t 12 + t 10 + t 2 , N Γ (t) 7 = t 26 + t 22 + t 20 + t 18 + t 16 + t 14 + t 12 + t 10 + t 8 + t 4 , N Γ (t) 8 = t 25 + t 23 + t 21 + t 19 + t 17 + 2t 15 + t 13 + t 11 + t 9 + t 7 + t 5 .

  3 t 2 + t + 1 t 2 + 1 (t + 1) 2 t 6 + t 5 + t 4 + t 3 + t 2 + t + 1 M I (t, u) 2 = M I (u, t) 1 P I (t) =-t 12 -t 11 + t 9 + t 8 -t 6 + t 4 + t

3 -t -1 (t -1)

3 (t 2 + t + 1) (t 6 + t 5 + t 4 + t 3 + t 2 + t + 1) (t 2 + 1) (t + 1)

f : C 2 → C(t, u) (d 1 , d 2 ) → 1 -tu (1 -td 1 + t 2 d 2 -t 3 )(1 -ud 2 + u 2 d 1 -u 3 ).The complete classification up to conjugation of all finite subgroups of SL 3 C is given in[START_REF] Yau | Gorenstein Quotient Singularities in Dimension Three[END_REF]. It consists in four infinite series (types A, B, C, D) and eight exceptional cases (types E, F, G, H, I, J, K, L). In all the cases, the character table T Γ is invertible, and Λ (1) := T -1 Γ A (1) T Γ and Λ (2) := T -1 Γ A (2) T Γ are diagonal matrices, with Λ(1) jj = χ(g j ) and Λ(2) jj = χ(g j ). According to Proposition 5, we may writeT Γ 1 -tΛ (1) + t 2 Λ (2) -t 3 1 -uΛ (2) + u 2 Λ (1) -u 3 T -1Γ P Γ (t, u) = (1 -tu)v 0,0 .

Recall that the Kronecker product of two matrices A ∈ MmC and B ∈ MnC is the block-matrix A ⊗ B ∈ MmnC defined by the formula:∀ (i, j) ∈ [[1, m]], (A ⊗ B) ij = a ij B.An important property of the Kronecker product is the relation tr(A ⊗ B) = tr(A)tr(B).The equality T Γ = T 1 ⊗ T 2 is implied by this relation

m

P H (t) = -t 8 -t 7 +t 5 +t 4 +t

-t-1 (t-1) 3 (t 2 +t+1)(t

+t 3 +t 2 +t+1)(t+1) 2

Type E

The group of type E is the group S, T, V , with

Here l + 1 = 14, rank(A

), p = 3, and τ 0 := s 0 s 1 s 2 s 3 s 12 s 13 , τ 1 := s 5 s 7 s 10 s 11 , τ 2 := s 4 s 6 s 8 s 9 .

Type F

The group of type F is the group S, T, V, P , with S, T, V as for the type E, and

Here l + 1 = 16, rank(A

), p = 3, and τ 0 := s 0 s 1 s 2 s 3 s 4 s 15 , τ 1 := s 5 s 7 s 9 s 11 s 13 , τ 2 := s 6 s 8 s 10 s 12 s 14 .

Type G

The group of type G is the group S, T, V, U , with S, T, V as for the type E, and

Here l + 1 = 24, rank(A (1) ) = 21,

), p = 3, and τ 0 := s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 19 s 20 s 21 , τ 1 := s 7 s 8 s 9 s 13 s 14 s 15 s 22 , τ 2 := s 10 s 11 s 12 s 16 s 17 s 18 s 23 .