On Coxeter Diagrams of complex reflection groups

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices, namely, the $\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Brou\'{e}, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an"affine diagram"with $\ZZ/7\ZZ$ symmetry.

1. Introduction 1.1. Background on unitary reflection groups: A unitary reflection group is a finite subgroup of the unitary group of a positive definite hermitian vector space V , generated by "complex reflections". They were classified by Shepherd and Todd in [12]. For a selfcontained proof of the classification (which is also similar in spirit to part of our work) see [6]. A convenient table of all these groups and their properties may be found in [5]. There is an infinite series, denoted by G(de, e, r), and 34 others, denoted by G 4 , G 5 , · · · , G 37 . Unitary reflection groups have many invariant theoretic properties that are similar to those of the orthogonal reflection groups. Most of these properties were initially established for the unitary groups, via case by case verification through Shepherd and Todd's list. Recently, there has been a lot of progress in trying to find unified and more conceptual proofs. (For example, see [4] and the references therein.) But a coherent theory, like that of the classical Coxeter groups and Weyl groups, is still not in place and lot of mysteries still remain. One of these mysteries involve the diagrams for unitary reflection groups.
Coxeter-Dynkin diagrams for orthogonal reflection groups encode presentations of these groups. For each unitary reflection group G, there is a diagram D G , that encodes a presentation of G (see [5]). Most of these diagrams were first introduced by Coxeter in [9]. The vertices of D G correspond to complex reflections that form a minimal set of generators for G. Other than that, the definition of D G is ad-hoc and case by case. It is curious that even though these diagrams do not have any uniform definition, in most cases they contain a lot of non-trivial information about the group G. We quote two sample results.
In section 4 we describe "affine diagrams" for unitary reflection groups defined over E. The affine diagrams are obtained from the unitary diagrams by adding an extra node. They encode presentations for the corresponding affine complex reflection groups. For each affine diagram, we describe a "balanced numbering" on its vertices, like the 3 2 4 6 5 4 3 2 1 numbering on the affine E 8 diagram. The existence of a balanced numbering on a diagram implies that the corresponding reflection group is not finite. So an affine diagram cannot occur as a full sub-graph of a diagram for a unitary reflection group. These facts, coupled with a combinatorial argument, complete the classification of E-root lattices. The affine diagrams are often more symmetric compared to the unitary diagrams, like in the real case. For example, in the case of K E 12 , we get an affine diagram with rotational Z/7Z symmetry. The complex root systems that we use in computer experiments are described in the appendix (see A.1 to A.5). If the diagram found by our algorithm is new, then a presentation of the corresponding reflection group, encoded by this diagram, is given in appendix A. 6. We finish this section by introducing some basic definitions and notations to be used.

Reflection groups and root systems:
Let V be a complex vector space with an hermitian form (always assumed to be linear in the second variable). If x ∈ V , then |x| 2 = x, x is called the norm of x. Given a vector x of non-zero norm and a root of unity u = 1, let φ u x (y) = y − (1 − u) x, y |x| −2 x. Then φ u x is an automorphism of the hermitian space V , called an u-reflection in x, or simply, a complex reflection. The hyperplane x ⊥ (or its image in the projective space P(V )), fixed by φ u x , is called the mirror of reflection. A complex reflection group G is a discrete subgroup of Aut(V, , ), generated by complex reflections. A mirror of G is a hyperplane fixed by a reflection in G. A complex reflection (resp. complex reflection group) is called a unitary reflection (resp. unitary reflection group) if the hermitian form on V is positive definite. We shall omit the words "complex" or "unitary", if they are clear from context. A unitary reflection group G acting on V is reducible (resp. imprimitive) if V is a direct sum V = V 1 ⊕ · · · ⊕ V l such that 0 = V 1 = V and each V j is fixed by G (resp. {V j : 1 ≤ j ≤ l} is fixed by G ) as a set. Otherwise it is irreducible (resp. primitive). Unless otherwise stated, we always assume that G is irreducible.
Let G be a unitary reflection group acting on C k with the standard hermitian form. Let F be the field of definition of G and O be the ring of integers of F . Let O * be the group of units of O. A vector r in an O-module K is primitive if m −1 r ∈ K and m ∈ O implies m ∈ O * . Let Φ be a set of primitive vectors in O k such that: • Φ is stable under the action of G, • {r ⊥ : r ∈ Φ} is equal to the set of mirrors of G, and • given r ∈ Φ, ur ∈ Φ if and only if u is an unit of O. Such a set of vectors will be called a (unitary) root system for G, defined over O. The group of units O * , acts on Φ by multiplication. An orbit is called a projective root. A set of projective roots for G is denoted by Φ * or Φ * (G).
1.4. Lattices and their reflection groups: Let F be a number field. Let O be the ring of integers of F . For simplicity, we assume that O is a unique factorization domain. We shall fix an embedding of F in C and identify F and O as subsets of C via this embedding. The three cases when O forms a discrete set in C are the integers, the Gaussian integers G = Z[i] and the Eisenstein integers E = Z[e 2πi /3 ]. These are the rings that will be important for us. To fix ideas, one may take O = E, since we will mostly be dealing with this case.
A lattice K, defined over O, is a free O-module of finite rank with an O-valued hermitian form. Let V = C ⊗ O K be the complex vector space underlying K. The dual lattice of K, denoted by K ′ , is the set of vectors y ∈ V such that y, x ∈ O for all x ∈ K.
A root of K is a primitive vector r ∈ K of non-zero norm such that φ u r ∈ Aut(K) for some root of unity u = 1. The reflection group of K, denoted by R(K), is the subgroup of Aut(K) generated by reflections in the roots of K. The projective roots of K are in bijection with the mirrors of R(K). If K is positive definite, then the roots of K form a unitary root system, denoted by Φ K , for the unitary reflection group R(K).
1.5. Diagrams: Consider the permutation matrices acting on k × k hermitian matrices by conjugation. An orbit D of this action is called a diagram. If M = ((m ij )) is a representative of an orbit D, then we say that M is a gram matrix of D. Let ∆ = {r 1 , · · · , r k } be a subset of a hermitian space V . The matrix (( r i , r j )) is called a gram matrix of ∆ and the corresponding diagram is denoted by D(∆). Let Φ be a root system for a unitary reflection group G. If ∆ is a subset of Φ such that {φ ur r : r ∈ ∆} is a minimal generating set for G (for some units u r ), then we say that D(∆) is a diagram for G.
Pictorially, a diagram D is conveniently represented by drawing a directed graph D with labeling of vertices and edges, as follows: Let v(D) = {x 1 , · · · , x k } be the set of vertices of D. We remember the entry g ii by labeling the vertex x i with g ii . We remember the entry g ij by drawing a directed edge from x j to x i labeled with g ij or equivalently, by drawing a directed edge from x i to x j labeled withḡ ij (but not both).
Let D be a diagram with gram matrix ((m ij )). Assume that m ij ∈ O for all i and j. Define L(D) to be the O-lattice generated by linearly independent vectors {x 1 , · · · , x k } with x i , x j = m ij . Let L be an O-lattice having a set of roots ∆ which form a minimal spanning set for L as an O-module. Then the diagram D(∆) is called a root diagram or simply a diagram for L. A diagram for L will be denoted by D L . We shall usually denote the vertices of D L and the corresponding vectors of L by the same symbol. If D is a root diagram of L, then L(D) surjects onto L preserving the hermitian form. If, further, the gram matrix of D is positive definite, then L(D) ≃ L.
Two diagrams D and D ′ are equivalent if L(D) ≃ L(D ′ ). In this case, we write D ≃ D ′ . Let L = L(D), v(D) = {x 1 , · · · , x k } and u 1 , · · · , u k be units. Then there is a diagram for L whose vertices correspond to the generators {u 1 x 1 , · · · , u k x k }. These two diagrams are equivalent. The only difference between them is in the edge labeling, which may differ by units.
Acknowledgments: I would like to thank Prof. Daniel Allcock, Prof. Jon Alperin, Prof. Michel Broue and Prof. George Glauberman for useful discussions and my advisor Prof. Richard Borcherds for his help and encouragement in the early stages of this work.

2.1.
Definition. An Eisenstein root lattice or E-root lattice is a positive definite E-lattice K, generated by vectors of norm 3, such that K ⊆ θK ′ . A root lattice is indecomposable if it is not a direct sum of two proper non-zero root lattices.
Let D be a diagram with gram matrix ((m ij )). The following assumptions about D will remain in force for the rest of this section. We assume that m ij ∈ E for all i and j. We assume that m ii = 3 for all i. So we omit the labels on the vertices. If m ij = −p, we omit the label on the edge going from j to i. If m ij = 0, we omit the edge {i, j}. The connectedness of a diagram is considered with these conventions. Each E-root lattice has at-least one diagram. Any diagram for an indecomposable E-root lattice is connected.

2.2.
Remark. Let K be an E-lattice satisfying K ⊆ θK ′ . The following observations are immediate: If r ∈ K has norm 3, then r is a root of K. The order 3 reflections in r, denoted by φ r = φ ω r and φ −1 r = φω r , belong to the reflection group of K. Let x and y be two linearly independent vectors of K of norm 3. If K is positive definite, then one must have | x, y | ≤ √ 3. So either x, y = 0, which imply φ x φ y = φ y φ x , or x, y ∈ θE * , which imply and E E 8 , with the bilinear form 2 3 Re x, y , are the lattices A 2 , D 4 , E 6 and E 8 respectively. We use the complex diagrams to give a new proof of the following theorem (theorem 2. Thus it suffices to classify the equivalence classes of root diagrams of indecomposable E-root lattices and show that there are only four classes. The proof of this classification, given below, is like the well-known classification of A-D-E root systems. 2.5. Definition. Let D be a connected root diagram for an E-lattice L. A balanced numbering on D is a function from v(D) to E, denoted by x → n x , such that n x = 1 for some x ∈ v(D) and . So |y| 2 = 0. A diagram that admits a balanced numbering and is minimal with this property, is called an affine diagram. Figure 1 shows a few affine diagrams, each with a balanced numbering. The number shown next to a vertex x is n x . Given a ∈ v(D), if b 1 , · · · , b m are the vertices connected to a and c j is the label on the directed edge going from b j to a, that is, a, b j = c j , then to verify equation (1), one needs to check that −3n a = m j=1 c j n b j . This is easily verified. We say that a connected diagram D is indefinite if D cannot appear as a full sub-graph of a diagram of an Eisenstein root lattice. Otherwise, we say that D is definite. Figure 1. A few affine diagrams with balanced numbering. Choose any ver- x is a norm zero vector of L(D) and n x j = 1 for some One checks that the vector y = k i=1 n i,ν x i has norm zero in L(Circ k,e 2πiν/6 ), for k = 3, 4, 5 and ν = 1, · · · , 6, except for the three cases considered in part (a) and (b). Part (c) now follows from lemma 2.6. Part (d) also follows from lemma 2.6.
Proof of the theorem 2.4. Let D be a root diagram for an indecomposable E-root lattice K. We shall repeatedly use lemma 2.8 in two ways. First, it implies that D cannot contain the diagrams mentioned in part (c) and (d) of the lemma. Secondly, from the proof of lemma 2.8, we observe the following: If Circ 3,u or Circ 4,u is a sub-graph of D, then we are in one of the cases considered in part (a) or (b) of the lemma and we can change one of the vertices to get an equivalent diagram, where one of the edges has been removed. However this may introduce new edges elsewhere in the graph. Since In the latter case, D ≃ Circ 4,ω , since all other circuits of length 4 are indefinite, by lemma 2.8(c).
Since D cannot be the affine diagram ∆ 4 , there must be at-least two edges joining x 5 with D ′ . Since the diagrams of the form Circ 5,u are not definite, D must contain a circuit of length 3 or 4. If {x 1 , x 5 } is an edge, it is part of a circuit of length 3 or 4 and as before, we can remove it by shifting to an equivalent diagram. The remaining possibilities are shown in figure 2.
(The arrows on edges are not important here, so they have been omitted). In cases (iii) and (iv), we may add a multiple of x 3 to x 5 and disconnect x 4 from x 5 (note that this does not introduce an edge between x 1 and x 5 ). So we are reduced to the first two cases. But (i) is affine (either ∆ 6 or ∆ 7 ) and (ii) contains the affine diagram ∆ 3 (See figure 1).

An attempt to characterize the diagrams for unitary reflection groups
In this section we want to address a question that was raised in [5]: How to characterize the diagrams for unitary reflection groups? We maintain the definitions and notations introduced in 1.3, 1.4 and 1.5. In 3.8-3.9, we shall describe an algorithm which, given a unitary reflection group, picks out a set of reflections. For the Weyl groups and the primitive unitary reflection groups defined over Q(ω) and Q(i), the reflections selected by the algorithm form a minimal set of generators. There are ten primitive unitary reflection groups defined over Q(ω) and Q(i). Most of these can be viewed as sets of automorphisms of some special complex lattices defined over E or G, namely, and K E 12 , see example 11b, 13b and 10a respectively, in chapter 7, section 8 of [8]). The lattice K E 10 is the orthogonal complement of any root in K E 12 . We take cue from the fact that for a Weyl group, vertices of the Dynkin diagram correspond to the simple roots, which are the positive roots having minimal inner product with a Weyl vector. Our algorithm is based on a generalization of the notion of Weyl vector.
3.1. Definition. Let G be an irreducible unitary reflection group with a root system Φ, defined over O. Let Φ * = Φ/O * be the set of projective roots. Given a projective root r, let o(r) denote the order of the subgroup of G generated by reflections in r. Let K be the O-lattice spanned by Φ. Let V be the vector space underlying K and M = ∪ r∈Φ * r ⊥ be the union of the mirrors. Define a function α : V \ M → V , by 3.2. Remark.
(1) Note that the quantity o(r) −2 r,w | r,w | r |r| does not change if we change r by a scalar. So the function α is well defined and only depends on the reflection group G and not on the choice of the roots. The function α descends to a function α : P(V \ M) → P(V ). Also note that α is G-equivariant, that is, α(gw) = gα(w) for all g ∈ G. So α induces a function from P(V \ M)/G to P(V )/G. Proof. For this argument, assume that the factor o(r) −2 = 1 4 is absent in the definition of α. Let Φ + (w) be the set of roots having strictly positive inner product with w. Then Φ + (w) can be chosen as a set of representatives for the projective roots, so α(w) = r∈Φ + (w) r/|r|. Now, w and w ′ are in the same Weyl chamber if and only if Φ + (w) = Φ + (w ′ ), so α(w) = α(w ′ ).
To check that w and α(w) belong to the same Weyl chamber, first, let Φ be a root system of type A n , D n , E 6 , E 7 or E 8 . Let w ∈ V R \ M R . Then ρ = 1 2 r∈Φ + (w) r is a Weyl vector which belong to the same Weyl chamber as w. Note that α(w) = √ 2ρ and α( √ 2ρ) = √ 2ρ. Now consider the non-simply laced case. Since α is G-equivariant, it is enough to check that w and α(w) are in the same Weyl chamber, for a single chamber. We show the calculation for B n . The Weyl group of type C n is isomorphic to the group of type B n . Calculations for type G 2 and F 4 are only little more complicated and will be omitted.
Let e j be the j-th unit vector in Z n . The roots of B n are {±e j , ±e i ± e j : j < i}. Choose w = (w 1 , · · · , w n ) such that 0 < w 1 < · · · < w n . Then Φ + (w) = {e j , e i ± e j : j < i}. So Observe that ρ (n) and w are in the same Weyl chamber. So α(ρ (n) ) = α(w) = ρ (n) .

3.4.
Definition. Let Φ be a unitary root system defined over O. In view of 3.3, a fixed point of α will be called a Weyl vector, when the root system is not defined over Z. One 8 may try to find a fixed point of α by iterating the function. Our method for selecting a set of generating reflections for G, is based on this notion of Weyl vector. Before describing it we show that Weyl vectors exist.
3.5. Theorem. Let Φ be a root system for a unitary reflection group G. Then the function α, defined in (2), has a fixed point.
For notational simplicity, let µ r = |r| −1 .o(r) −2 , so that, The argument given below actually shows, that for any sequence of non-zero positive numbers µ r , a function of the form (4) has a fixed point. We need a lemma, which converts the problem of finding a fixed point of α to a maximization problem.

Lemma. Let Φ be a unitary root system and let M be the union of mirrors. Consider the function S : P(V ) → R, defined by
For w / ∈ M, one has ∂ w (S) = 0 if and only if α(w) = |w| −1 S(w)w. (here ∂ w denotes the holomorphic derivative with respect to w).
Proof. Let n(w) = Φ * µ r | r, w |, so that S(w) = n(w)/|w|. We fix a basis for the vector space V and write r, w = r * Mw, where r * is the conjugate transpose of r and M is the matrix of the hermitian form. Differentiating, we get ∂ w (| r, w | 2 ) = w, r r * M. So ∂ w (µ r | r, w |) = µ r (2| r, w |) −1 w, r r * M.

It follows that
Note that the third term can be made non-negative by choosing ξ suitably, and the second term is positive, since r 0 ∈ Ψ 0 . This proves the claim.
Let k be the minimum number of reflections needed to generate G. Define ∆(w) = {r 1 , · · · , r k }. In other words, ∆(w) consists of k projective roots, whose mirrors are closest to w.

3.8.
Method to obtain a set of "simple reflections": We now describe a computational procedure in which, the input is a unitary reflection group G, (or equivalently, a projective root system Φ * for G and the numbers {o(r) : r ∈ Φ * }) and the output is either the empty set or a non-empty set of projective roots, to be called the simple roots. The reflections in the simple roots are called simple reflections. A set of simple reflections form a simple system.
(4) If r 1 , · · · , r k are linearly independent, then return ∆(w) (the simple roots). In practice, for each group G to be studied, we execute the following algorithm many times.
3.9. Algorithm: Start with a random vector w 0 ∈ V . Generate a sequence w n by w n = α(w n−1 ). If the sequence stabilizes, then say that the algorithm converges and let w = lim w n . Note down ∆(w) and S(w).
For computer calculation, we assume that w n stabilizes if |w n+1 − w n | 2 /|w n | 2 becomes small, say less than 10 −8 , and remains small and decreasing for many successive values of n. Note that if α(w) = w, then S(w) = |w|. From the values of S(w), the maximum, denoted by S G max , can be found. (In all the examples that we have studied, S(w) takes atmost two values on the set of fixed points of α found experimentally, so finding the maximum is not difficult). Each instance of the algorithm, that produced a vector w with S(w) = S G max , now yields a simple system ∆(w) (provided that ∆(w) is a linearly independent set).

3.10.
Observations for Weyl groups: First, suppose that Φ is a root system for a Weyl group W . We maintain the notations of 3.3 and ignore the factor o(r) −2 = 1 4 in the definition of α. Given w ∈ V R , let δ w (r) = | r, w |/|r|. Arranging the roots according to increasing distance from w (according to the spherical metric or the Fubini-Study metric) is equivalent to arranging them according to increasing order of δ w (r). Claim: Let W be a Weyl group, w 0 ∈ V R \ M R and w = α(w 0 ). Then the algorithm 3.9 converges in one iteration and yields a simple system ∆(w). For Weyl groups, the definition of a set of simple roots given in 3.8 agrees with the classical notion. Further, for each simple root r ∈ ∆(w), we have δ w (r) = 1. So the simple roots are equidistant from w.
Suppose Φ is a root system of type A, D or E. Then w = α(w 0 ) = √ 2ρ, where ρ is a Weyl vector. So ∆(w) = ∆(ρ) consists of a set of simple roots and for each r ∈ ∆(w), one has r, ρ = 1, so δ w (r) = 1.
3.11. Observations for complex root systems: The primitive unitary reflection groups defined over Q(ω) (resp. Q(i)) are {G 4 , G 5 , G 25 , G 26 , G 32 , G 33 , G 34 } (resp. {G 8 , G 29 , G 31 }). In the following discussion, let G be one of these groups. In each case, a projective root system Φ * (G) for G is chosen. These are described in the appendix (see A.1, A.2, A.3). For each of these groups G, we have run algorithm 3.9 atleast one thousand times and obtained many simple systems by method 3.8. The calculations were performed using the GP/PARI calculator. The computer codes are contained in the files finite.gp and ei-linear-alg.gp available online at http://www.math.uchicago.edu/˜tathagat/codes/index.html. The main observations made from the computer experiments are the following: The sequence {w n } stabilizes in each trial for each G mentioned above. The simple reflections generate G and the simple roots form the same diagram every time. For G 4 ,G 5 , G 8 , G 25 , G 26 , G 32 , these are Coxeter's diagrams, which can be found, for example, in [5]. In the exceptional cases G 29 , G 31 , G 33 and G 34 new diagrams are obtained. These diagrams and the presentations of the groups obtained from these diagrams are given in appendix A.6. The "affine" diagrams of type G 33 and G 34 are shown in figure 3. The diagrams for the unitary groups are obtained by omitting the extending vertex (the vertex joined by dotted lines).
Further observations from the computer experiments are summarized below.
(1) If G is not G 29 , G 31 or G 32 , then for all w such that α(w) = w, the function S(w) attains the same value. So each trial of the algorithm yields a maxima w for S. For G 29 , G 31 and G 32 , the function S(w) attains two values on the fixed point set of α.
In these three cases, the Z-span of the roots form the E 8 lattice. (2) If G = G 33 , then in each trial of the algorithm, we find that the vectors {r 1 , · · · , r k } are linearly independent. So each choice of a maxima w for the function S, yields a set of simple roots ∆(w). For G 33 = R(K E 10 ), in most of the trials, we find that {r 1 , · · · , r 5 } are linearly dependent. So most trials do not yield a set of simple roots ∆(w). In an experiment with 5000 trials, only 297 yielded simple systems.
(3) Let G be G 33 = R(K E 10 ) or G 34 = R(K E 12 ) or G 29 . Let g 1 , · · · , g k be a set of simple reflections of G obtained by method 3.8. For these three groups, we have verified the following result. (almost a re-statement of sample theorem (1), quoted in 1.1): Fix a permutation π so that the order of the product p = g π 1 · · · g π k is maximum (over all permutations). Then, the order of p is equal to the Coxeter number of G (denoted by h) and the eigenvalues of either p or p −1 are e 2πi(d 1 −1)/h , · · · , e 2πi(d k −1)/h , where d 1 , · · · , d k are the invariant degrees of G.
For G 29 , G 33 and G 34 the invariant degrees are (4,8,12,20), (4, 6, 10, 12, 18) and (6,12,18,24,30,42) respectively. In all three cases, h is equal to the maximum degree. These three are the well generated groups for which the diagrams obtained by method 3.8 are different from the ones in the literature. The remaining group G 31 is not well generated. (4) Start with w 0 ∈ V and consider the sequence defined by w n = α(w n−1 ). Roughly speaking, each iteration of the function α makes the vector w n more symmetric with respect to the set of projective roots Φ * . The function S measures this symmetry. So the fixed points w of α are often the vectors that are most symmetrically located with respect to Φ * . Based on this discussion, an alternative definition of a Weyl vector may be suggested, namely, a vector w ∈ V , such that d(w, M) is maximum. (This was suggested to me by Daniel Allcock). It seems harder to compute these vectors, so we have not experimented much with this alternative definition. However, we would like to remark, that for some complex and quaternionic Lorentzian lattices, similar analogs of Weyl vectors and simple roots, are useful. (One such example is studied in [3]; other examples are studied in [2]. In these examples of complex and quaternionic Lorentzian lattices, the simple roots are again defined as those whose mirrors are closest to the "Weyl vector".) (5) The method 3.8 fails for the primitive unitary reflection groups that are not defined over Q, Q(i) or Q(ω) and for the imprimitive groups G(de, e, n), except when they are defined over Q, that is, for the cases G(1, 1, n + 1) ≃ A n , G(2, 1, n) ≃ BC n and G(2, 2, n) ≃ D n . It fails in the sense that the set ∆(w) does not in general form a minimal set of generators for the group. This was found by experimenting with G(de, e, n) for small values of (d, e, n) and also with G 6 and G 9 (see appendix A.4). Although method 3.8 fails, the following observation holds for G(de, e, n): Let k be the minimum number of reflections needed to generate G(de, e, n) (k = n or n+ 1). There exists a vector w ∈ V such that, if r ⊥ 1 , · · · , r ⊥ k are the mirrors closest to w, then reflections in {r 1 , · · · , r k } generate G(de, e, n).
For G(de, e, n), one can take w = ρ (n) (the vector we obtained for the Weyl group B n ; see equation (3)). It is easy to check that ∆(ρ (n) ) forms the known diagrams for G(de, e, n). We found ρ (n) by using an algorithm that tries to find a point in V whose distance from M is at a local maximum. For small values of d, e and n, we find that this algorithm always converge to ρ (n) .

the affine reflection groups
4.1. In this section we shall describe affine diagrams for the primitive unitary reflection groups defined over E, except for G 5 . (Including G 5 would further complicate notations). An affine diagram is obtained by adding an extra node to the corresponding "unitary diagram" and each admit a balanced numbering. A unitary diagram can be extended to an affine diagram in many ways. We have chosen one that makes the diagram more symmetric. The Weyl vector, that yielded the unitary diagram, is often fixed by the affine diagram automorphisms.
The discussion below and the lemma following it are direct analogs of the corresponding results for Euclidean root lattices. We have included a proof since we could not find a convenient reference.
Let Φ be a unitary root system defined over E, for an unitary reflection group G. Let K be the E-lattice spanned by Φ. Assume that the subgroup of Aut(K) generated by reflections in Φ is equal to G. Let E 0 be the one dimensional free module over E with zero hermitian form. LetK = K ⊕ E 0 . Let us write the elements ofK in the form (y, m) with y ∈ K and m ∈ E. LetG be the subgroup of Aut(K) generated by the reflections iñ Consider the semi-direct product K ⋊ G, in which the product is defined by (x, g).(y, h) = (x + gy, gh). The faithful action of K ⋊ G on K via affine transformations is given by (x, g)y = x + gy.
(a) The affine reflection groupG is isomorphic to the semi-direct product K ⋊ G.
The automorphisms t x ofK are called translations. The subgroup of Aut(K) generated by translations is isomorphic to the additive group of K. For any root (x, m) ∈Φ, one has, φ ω (x,m) (y, n) = (φ ω x (y), n − mp −1 x, y ) = φ ω (x,0) • tm x (y, n). Let us write φ (x,0) = φ x . From the above equation, one has, in particular, From equation (7), it follows that φ a t x φ −1 a = t φa(x) . So (x, g) → t x • g is an isomorphism from K ⋊ G ontoG.
(b) Let G 1 be the subgroup ofG generated by φ r 1 , · · · , φ r k and φ (r 0 ,1) . Then G ⊆ G 1 . Let x = gr 0 be a root in the G-orbit of r 0 . Then t Since the roots in the G-orbit of r 0 span K as a Z-module, it follows that t x ∈ G 1 for all x ∈ K. The translations, together with G, generateG. SoG = G 1 .

4.3.
Method to get an affine diagram: Lemma 4.2 applies to the root systems Φ(G 4 ), Φ(G 25 ) and Φ(G 32 ) and the corresponding lattices D E 4 , E E 6 and E E 8 . Similar result holds for the root systems Φ(G 33 ) and Φ(G 34 ) and the corresponding lattices K E 10 and K E 12 , if one replaces order three reflections by two reflections and p by −ω. In each of these cases, any root can be chosen as r 0 in part (b) of 4.2.
Further modifications are necessary for G 26 . In this case p −1 K is not a subset of K ′ but there is a sub-lattice E E 6 ⊆ K such that p −1 E E 6 ⊆ K ′ . Accordingly, the translations t x ,   . Affine diagrams with balanced numbering (shown next to the vertices). The number shown inside the vertex is the norm of the root as well as the order of a reflection in that root. An edge (resp. a double edge) between x and y implies the Coxeter relation φ x φ y φ x = φ y φ x φ y (resp. φ x φ y φ x φ y = φ y φ x φ y φ x ). An unmarked directed edge or double edge from y to x means that x, y = −p for G 4 , G 25 , G 26 , G 32 and it means x, y = ω for G 33 , G 34 .
given in (6), define automorphisms ofK only for x ∈ E E 6 . The conclusion in part (a) is that G ≃ E E 6 ⋊ G. In part (b), any root of an order 3 reflection can be chosen as r 0 . The details are omitted.
Let G ∈ {G 4 , G 25 , G 26 , G 32 , G 33 , G 34 }. We take the diagram for G obtained by method 3.8 and extend it by adding an extra node corresponding to a suitable root, thus obtaining the corresponding affine diagram. These are shown in Figure 3. The extending node is joined with dotted lines. The vertices of an affine diagram of type G correspond to a minimal set of generators for the affine reflection groupG. The edges indicate the Coxeter relations among the generators. Additional relations may be needed to obtain a presentation ofG (like those given in appendix A.6). We describe a root system for each unitary reflection group considered in section 3. Notation: a set of co-ordinates marked with a line (resp. an arrow) above, means that these co-ordinates can be permuted (resp. cyclically permuted).
A.1. G 4 , G 5 , G 25 , G 26 , G 32 : For each of these groups, a set of projective roots and the lattices spanned by these roots are given in table 1. The groups G 5 , G 25 and G 32 are reflection groups of the E-root lattices D E 4 , E E 6 and E E 8 respectively. A.2. G 33 , G 34 : These are reflection groups of the E-lattices K E 10 and K E 12 respectively, where K E 12 is a complex form of the Coxeter-Todd lattice (see example 10a in chapter 7, section 8 of [8]) and K E 10 is the orthogonal complement of any vector of minimal norm in K E 12 . The minimal norm vectors of K E 10 and K E 12 form root systems of type G 33 and G 34 respectively. A.3. G 8 , G 29 , G 31 : These are the primitive unitary reflection groups defined over G = Z[i]. Let q = 1 + i.
The reflection group of the two dimensional G-lattice D G 4 = {(x, y) ∈ G 2 : x+y ≡ 0 mod q} is G 8 . The six projective roots are (q, 0), (0, q) and (1, i j ). The reflection group contains order 4 and order 2 reflections in these roots.
There are 12 projective roots of norm 4 and six of norm 4 + 2 √ 2. The group G 9 contains order 2 reflections in all the roots and order 4 reflections in the roots of norm (4 + 2 √ 2).
A.5. G(de, e, n): Let ζ m = e 2πi/m and let e j be the j-th unit vector in C n . The projective roots can be chosen to be {e j , e j − ζ t de e k : 1 ≤ j < k ≤ n, 1 ≤ t ≤ de}. For a detailed study of these groups, see [5].
A.6. The diagrams and presentations for G 29 , G 31 , G 33 and G 34 . The presentations given here were found by coset enumeration for which we used MAGMA. One reason for including these presentations is to observe the following fact: each extra relation needed beyond Coxeter's relations correspond to "deflating" a triangle or a square in the diagram in the sense of Conway [7]. If x, y, z are three vertices in a diagram forming a triangle, then let def(x, y, z) be the "deflation" relation xyzy = yzyx. The Coxeter group of a triangle is affine. Adding the deflation relation def(x, y, z) reduces it to a finite group.
Let q = 1 + i. Then a gram matrix for the diagram of G 31 (obtained according to our method described in 3.8) is given by 2.
The Coxeter relations implied by this gram matrix are φ x φ y φ x = φ y φ x φ y if x, y = 2 and φ x φ y φ x φ y = φ y φ x φ y φ x if | x, y | = 2 √ 2. Let a 1 , · · · , a 5 be the simple reflections forming this diagram for G 31 . Adding the relations def(a 2 , a 4 , a 1 ), def(a 1 , a 5 , a 2 ), def(a 1 , a 3 , a 4 ), def(a 2 , a 4 , a 5 ), a 3 a 4 a 5 = a 4 a 5 a 3 and a 1 a 2 a 3 a 2 a 1 a 2 = a 2 a 3 a 2 a 1 a 2 a 3 to the Coxeter relations, yields a presentation of G 31 . Omitting the generator a 5 one obtains the diagram for G 29 (found by method 3.8). Omitting a 5 and the relations involving a 5 , one obtains the corresponding presentation for G 29 .
The diagrams for G 33 and G 34 are described in figure 3. Starting from top left and going counter-clockwise, let us denote the simple reflections of the diagram for G 34 by a 1 , · · · , a 6 . A presentation of G 34 is obtained by adding the relations def(a i , a i+1 , a i+2 ) for i = 1, 2, 3, 4 and a 1 a 3 a 5 a 6 a 5 a 3 a 1 a 3 a 5 = a 3 a 5 a 6 a 5 a 3 a 1 a 3 a 5 a 6 to the Coxeter relations. A presentation for G 33 is obtained by deleting the generator a 2 and the relations involving a 2 .