The set of dominance-minimal roots

Brigitte Brink

School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia.

Abstract

If α and β are positive roots in the root system of a Coxeter group W, we say that α dominates β if w · β is negative whenever w · α is negative for w ∈ W. We say that α is elementary or dominanceminimal, if it does not dominate any β ≠ α. It is shown in [BH] that the set E of dominance-minimal roots is finite if W has finite rank, and this is used to show that W is automatic. To limit the size of the relevant automata, and possibly facilitate other Coxeter group algorithms, we give an explicit description of the set of elementary roots.

§1 Introduction

The proof of Theorem (2.8) of [BH] yields that |E| is bounded by

∑_{d=1}^{c^{|R|}(|R|+1)+1} |R|^{d} = \frac{|R|}{|R| − 1}(|R|^{c^{|R|}(|R|+1)+1} − 1),

where c equals the cardinality of the set

{ cos(nπ/m_{rs}) | r, s ∈ R, m_{rs} < ∞ and n ∈ {1, …, m_{rs} − 1} }.

This bound, however, is rather large. For example, if

W = ⟨ r, s | r^{2} = s^{2} = (rs)^{3} = 1 ⟩,

then |E| = 3, but |R| = 2 and c = 3, and thus

\frac{|R|}{|R| − 1}(|R|^{c^{|R|}(|R|+1)+1} − 1) = 2(2^{28} − 1).

Moreover, the algorithm given in [BH] to calculate the set of elementary roots is not very practical. In [C] (p.104, ll.41-43) Casselman asks for a more realistic algorithm and a better bound for |E|. We will explicitly determine E, thus making it possible to calculate |E| precisely.

§2 Summary of background material

Let (W, R) be a Coxeter system, as defined in [B] or [H], and for r, s ∈ R let m_{rs} be the order of rs. Let Π = {a_{r} | r ∈ R} be the set of simple roots, and for a ∈ Π denote the element of R corresponding to a by r_{a}. Further, let V be the ℝ-vector space with basis Π, and let ⟨ , ⟩: V ×V → ℝ be a symmetrical bilinear form which satisfies ⟨a_{r}, a_{s}⟩ = − cos(π/m_{rs}) for all r, s ∈ R such that m_{rs} is finite, and ⟨a_{r}, a_{s}⟩ ≤ −1 if m_{rs} is infinite. Then r·v = v−2⟨a_{r}, v⟩a_{r} (for all r ∈ R and v ∈ V) determines a faithful action of W on V which preserves ⟨ , ⟩. The set Φ = {w · a_{r} | r ∈ R, w ∈ W } is the root system of W in V. For S ⊆ Π, we denote the Coxeter group generated by r_{a} with a ∈ S by W_{S}, and the root system of W_{S} on the subspace of V spanned by S by Φ_{S}.

The Coxeter graph of W has vertex set Π, and two vertices a_{r} and a_{s} are adjoined by an edge or bond of weight m_{rs} if m_{rs} ≥ 3 or m_{rs} = ∞; if m_{rs} = 3, the bond adjoining a_{r} and a_{s} is said to be simple, and if m_{rs} ≥ 4 or m_{rs} = ∞, the bond adjoining a_{r} and a_{s} is labelled by m_{rs}. We frequently identify S ⊆ Π with the subgraph of the Coxeter graph consisting of the vertices in S and the edges adjoining them.

Each root α can be written uniquely as ∑_{a∈Π} λ_{a}a; we say that λ_{a} is the coefficient of a in α, and define the support supp(α) of α to be the set of a ∈ Π such that λ_{a} ≠ 0. It can be easily seen that supp(α) is a connected subgraph of the Coxeter graph. Further, α is said to be positive (negative), if all λ_{a} ≥ 0 (all λ_{a} ≤ 0), and the sets of positive and negative roots are denoted by Φ^{+} and Φ^{−} respectively. It is a well known fact that w · a_{r} is positive for w ∈ W and r ∈ R if l(wr) ≥ l(w) (where l(w) is the minimal integer n such that w equals r_{1} ⋯ r_{n} for some r_{i} ∈ R). This yields that each root is either positive or negative. An extension of the proof of this fact yields the next proposition.

(2.1) Proposition Let λ be the coefficient of a simple root in α ∈ Φ.

(i) If λ > 0, then λ ≥ 1. (ii) If ⟨a, b⟩ ≥ −1 for all a, b ∈ Π, then λ is a polynomial in

C = \{\frac{sin(lπ/m)}{sin(π/m)} | 4 ≤ m = m_{rs} < ∞ for a_{r}, a_{s} ∈ Π and l ∈ ℕ with l ≤ \frac{m}{2}\}

with coefficients in ℕ_{0} (the non-negative integers), or coefficients in −ℕ_{0}. In particular, if λ ∈ (0, 2), then λ = 1, or λ = 2 cos(π/m_{st}) for some s, t ∈ R with m_{st} ≥ 4.

Proof. Suppose that λ is the coefficient of x in α, and let w ∈ W and r ∈ R with α = w·a_{r}. If l(w) = 0, the assertion is trivially true; so suppose that l(w) > 0, and let s ∈ R and w′ ∈ W such that w = w′s and l(w′) = l(w) − 1. Choose w′′ ∈ wW_{{a_{r},a_{s}}} of minimal length; then l(w′′) < l(w) as l(ws) < l(w), and l(w′′r), l(w′′s) ≥ l(w′′) by minimality of w′′. Hence the inductive hypothesis applies to w′′ · a_{r} and w′′ · a_{s}, and the roots w′′ · a_{r} and w′′ · a_{s} are positive.

Next, let u ∈ W_{{a_{r},a_{s}}} with w = w′′u. Then u · a_{r} = μa_{r} + νa_{s} for some μ, ν ∈ ℝ; moreover, a straightforward calculation of the rank 2 case yields that μ, ν ≤ 0 or μ, ν ≥ 0. Let μ_{x}, ν_{x} ≥ 0 be the coefficients of x in w′′·a_{r} and w′′·a_{s} respectively, so that λ = μμ_{x}+νν_{x}.

If λ > 0, then μ, μ_{x} > 0 (and thus ν ≥ 0), or ν, ν_{x} > 0 (and thus μ ≥ 0). If μ > 0, it can be easily verified that μ ≥ 1; further, if μ > 0, then μ_{x} ≥ 1 by the inductive hypothesis. Hence λ ≥ μμ_{x} ≥ 1 if μ, μ_{x} > 0; symmetrical arguments apply if ν, ν_{x} > 0, and this proves (i).

Suppose now that ⟨a, b⟩ ≥ −1 for all a, b ∈ Π. By the inductive hypothesis, μ_{x} and ν_{x} are polynomials in C with coefficients in ℕ_{0}. Further, either ⟨a_{r}, a_{s}⟩ = −1, or m_{rs} is finite, and in both cases it can be easily verified that μ and ν are both polynomials in C with coefficients in ℕ_{0}, or both polynomials in C with coefficients in −ℕ_{0}, and (ii) follows.

Following [S], we define for α ∈ Φ^{+} the depth of α to be the minimal integer l such that w · α is negative for some w ∈ W of length l; it follows that α equals u · a_{s} for some s ∈ R and u ∈ W of length dp(α) − 1. In [BH], it is shown that if r ∈ R and α ∈ Φ^{+} \ {a_{r}}, then

dp(r · α) = \{ \begin{matrix}dp(α) − 1 & if ⟨α, a_{r}⟩ > 0, \\ dp(α) & if ⟨α, a_{r}⟩ = 0, \\ dp(α) + 1 & if ⟨α, a_{r}⟩ < 0. \end{matrix}  (∗)

For α, β ∈ Φ^{+} we say that β precedes α (we write β ⪯ α) if and only if there exists a w ∈ W such that α = w · β and dp(α) − dp(β) = l(w); we write β ≺ α if β ⪯ α and β ≠ α. It is shown in [BH] that ⪯ defines a partial order on the set of positive roots. Note that (∗) yields for r ∈ R and α ∈ Φ^{+} \ {a_{r}} that r · α ≺ α if and only if ⟨α, a_{r}⟩ > 0. It follows that each positive root is preceded by a simple root. If α ⪯ β, an easy induction on dp(β) − dp(α) yields that the coefficients of simple roots in α must be less than or equal to the corresponding coefficients in β.

For α, β ∈ Φ^{+} we say that α dominates β with respect to W (we write α dom β) if and only if w · β ∈ Φ^{−} whenever w · α ∈ Φ^{−} for w ∈ W. It is shown in [BH] that dom also defines a partial order on the set of positive roots. The dominance-minimal roots are called elementary, and the set of elementary roots is denoted by E; that is, E is the set of positive roots that do not dominate any root other than themselves. From here on, when we refer to a minimal element of a set of roots, we mean precedence-minimal rather than dominance-minimal.

The next proposition summarizes some basic properties of dominance from [BH].

(2.2) Proposition Let α, β ∈ Φ^{+} and r ∈ R. Then (i) α dom β if and only if ⟨α, β⟩ ≥ 1 and dp(α) ≥ dp(β); (ii) if α dom β and w · β ∈ Φ^{+} for some w ∈ W, then (w · α) dom (w · β); (iii) if β ∉ E and β ⪯ α, then α ∉ E; (iv) if α ∈ E, then r · α ∈ E if and only if ⟨α, a_{r}⟩ ∈ (−1, 1).

§3 Initial reduction

For S ⊆ Π, let Φ_{S}^{>0} denote the set of positive roots with support S, and define E_{S} = Φ_{S}^{>0} ∩E. Then Φ is the disjoint union of all Φ_{S}^{>0} with S ⊆ Π. It is clear that Φ_{S}^{>0} is empty if S is infinite, or if the Coxeter graph corresponding to S has more than one connected component. So E is the disjoint union of all E_{S} with S ⊆ Π finite and connected.

It follows readily from (2.2)(i) that a root α ∈ Φ_{S}^{+} is elementary in Φ_{S}^{+} (with respect to W_{S}) if and only if it is elementary in Φ^{+}. Hence we may restrict our attention to Φ_{S} and W_{S} when investigating E_{S}. The root system Φ_{S} is well known if W_{S} is finite. Furthermore, since the subspace spanned by Φ_{S} is Euclidean in this case, it follows from (2.2)(i) that every root in Φ_{S}^{+} is elementary (since ⟨α, β⟩ = cos(θ) for some θ ∈ (0, π) if α ≠ β ∈ Φ_{S}); that is, E_{S} = Φ_{S}^{>0}. It remains to determine E_{S} for S ⊆ Π finite and connected such that W_{S} infinite.

(3.1) Proposition E_{S} is empty if S contains a circuit or an infinite bond.

Proof. Let α_{0} be a root whose support contains a circuit, and let α ⪯ α_{0} be minimal such that supp(α) contains a circuit; we will show that α ∉ E, and thus α_{0} ∉ E by (2.2)(iii).

Let r ∈ R with r · α ≺ α; by minimality of α it follows that the support of r · α does not contain any circuits. If α = ∑_{x∈Π} λ_{x}x, then

r · α = ∑_{x≠a_{r}} λ_{x}x + (λ_{a_{r}} − 2⟨β, a_{r}⟩)a_{r},

and we deduce that the coefficient of a_{r} in r · α equals 0, and that a_{r} is part of a circuit in supp(α). Hence there exist at least two elements b, c of supp(α) \ {a_{r}} adjoined to a_{r}. By definition of ⟨ , ⟩, it follows that ⟨a_{r}, b⟩ and ⟨a_{r}, c⟩ are both at most − cos(\frac{π}{3} ) = −\frac{1}{2}, while ⟨a_{r}, x⟩ ≤ 0 for all other x ∈ Π \ {a_{r}}. Furthermore, λ_{b}, λ_{c} ≥ 1 by (2.1)(i), and thus

⟨r · α, a_{r}⟩ = ∑_{x≠a_{r}} λ_{x}⟨x, a_{r}⟩ ≤ λ_{b}⟨b, a_{r}⟩ + λ_{c}⟨c, a_{r}⟩ ≤ −1.

So ⟨α, a_{r}⟩ = −⟨r·α, a_{r}⟩ ≥ 1, and thus α dom a_{r} by (2.2)(i) (as α is certainly of depth greater than dp(a_{r}) = 1). Hence α ∉ E, and α_{0} ∉ E, as required. A similar argument applies when supp(α) contains an infinite bond.

We will now determine E_{S} for S ⊆ Π finite and connected, and not containing any circuits (that is, S is a tree); moreover, we can assume that S does not contain any infinite bonds and that W_{S} is infinite. To facilitate the relevant statements, we make the following definition.

(3.2) Definition Let x ∈ S and T ⊆ S. We say that x is a disconnector of S if the subgraph of the Coxeter graph corresponding to S \ {x} is not connected. Define D(S) to be the set of disconnectors of S. A T -component of S is a maximal connected subset C of S such that T ∩ D(C) = ∅.

For example, if S is the graph shown below, and the subset T corresponds to the white vertices, then the T -components of S are as indicated:

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Note that in this example, each y ∈ S \ T is contained in exactly one T -component C(y) of S; moreover, C(y) is the union of the connected component of S \ T containing y, and the set of x ∈ T which are adjoined to some element of this connected component. Each x ∈ T is contained in as many T -components of S as there are z ∈ S adjoined to x, and we denote this number by val_{S}(x). Finally, observe that each T -component C of S equals ⋂_{x∈T} C_{x}, where C_{x} is the {x}-component of S containing C. It can be easily shown that the above observations are true in general (given that S is a tree).

(3.3) Definition For α ∈ Φ, let C_{1}(α) be the set of a ∈ Π with coefficient 1 in α. We say that α is indecomposable if C_{1}(α)∩D(supp(α)) = ∅, otherwise α is said to be decomposable. For S ⊆ Π, we denote the set of indecomposable elementary roots with support S by E_{S}^{indec}.

If α is a decomposable root and x ∈ C_{1}(α) ∩ D(supp(α)), the following assertion yields that α can be written as the ‘sum’ (minus a correction factor) of roots, each of which has support an {x}-component of supp(α) (and is thus of cardinality less than | supp(α)|). Moreover, α is elementary if and only if each of the summands is elementary.

(3.4) Proposition Let x ∈ S and denote the {x}-components of S by S_{1}, …, S_{n}. Then

ϕ: (β_{1}, …, β_{n}) ↦ β_{1} + ⋯ + β_{n} − (n − 1)x

defines a one-one correspondence between the set of n-tuples in Φ_{S_{1}}^{+} × ⋯ × Φ_{S_{n}}^{+} such that the coefficient of x in each component equals 1, and the set of roots in Φ_{S}^{+} with coefficient 1 for x. Moreover, this map restricts to a one-one correspondence between the set of n-tuples in E_{S_{1}} × ⋯ × E_{S_{n}} such that x has coefficient 1 in each component, and the set of roots in E_{S} with coefficient 1 for x.

Before we prove (3.4), we establish the following lemma.

(3.5) Lemma Let α ∈ Φ^{+} and suppose that x ∈ Π has coefficient 1 in α. Then x ⪯ α, and there exists a w ∈ W_{supp(α)\{x}} such that α = w · x and l(w) = dp(α) − 1.

Proof. If α is of depth 1, the assertion is certainly true. Suppose now that dp(α) > 1, and let y ∈ S with r_{y} · α ≺ α. If y ≠ x, the assertion follows by induction, so suppose that y = x. Then the coefficient of x in r_{x} · α is less than 1, and thus equal to 0 by (2.1)(i). Hence ⟨α, x⟩ = \frac{1}{2}, and it follows that x is adjacent to exactly one element z of supp(r_{x} · α); and, moreover, that x and z are adjoined by a simple bond, and the coefficient of z in r_{x} · α equals 1. By induction, there exists a u ∈ W_{supp(r_{x}·α)\{z}} of length dp(r_{x} · α) − 1 such that r_{x} · α = u · z. Since x is not adjoined to supp(r_{x} · α) \ {z}, we deduce that r_{x}u = ur_{x}, and thus

α = r_{x} · (u · z) = u · (r_{x} · z) = u · (z + x) = u · (r_{z} · x) = (ur_{z}) · x.

As l(ur_{z}) ≤ l(u) + 1 = dp(r_{x} · α) = dp(α) − 1, we find that α is preceded by x, as required.

Proof of (3.4). It can be easily verified using Lemma (3.5) that ϕ is well defined and onto; moreover, ϕ(w_{1} · x, …, w_{n} · x) = (w_{1} ⋯ w_{n}) · x for w_{i} ∈ W_{S_{i}\{x}}. Since ϕ is certainly one-one, it follows that ϕ is a one-one correspondence.

In order to show that ϕ(β_{1}, …, β_{n}) ∈ E_{S} if and only if β_{i} ∈ E_{S_{i}} for all i, it suffices to show that if w = w_{1} ⋯ w_{n} with w_{i} ∈ W_{S_{i}\{x}}, then w · x ∉ E if and only if w_{i} · x ∉ E for some i ∈ {1, …, n}. Since it can be easily seen that w_{i} · x ⪯ w · x for i ∈ {1, …, n}, Proposition (2.2)(iii) implies that w · x ∉ E if w_{i} · x ∉ E for some i; hence it remains to show the converse.

Suppose that w · x dominates some γ ∈ Φ^{+} \ {w · x}. Since x ∈ E, (2.2)(ii) yields that w^{−1} · γ ∈ Φ^{−}. As w ∈ W_{S\{x}}, in particular x ∉ supp(γ); since the support of γ has to be connected, we may assume without loss of generality that γ ∈ Φ_{S_{1}\{x}}. Then (w_{2} ⋯ w_{n})^{−1} · γ is positive, and hence (w_{1} · x) dom (w_{2} ⋯ w_{n})^{−1} · γ by (2.2)(ii). Since w · x ≠ γ, it follows that w_{1} · x ≠ (w_{2} ⋯ w_{n})^{−1} · γ, and thus w_{1} · x ∉ E, as required.

(3.6) Corollary Let α ∈ Φ, and suppose that supp(α) is a tree which does not contain any infinite bonds. Further, let x and y be in the support of α, and suppose that x has coefficient 1 in α, and y has coefficient μ in α.

(i) If x and y are adjoined by a simple bond, then μ = 1, or μ ≥ 2. (ii) If x and y are adjoined by a non-simple bond, then μ > 1.

Proof. Suppose that α = β_{1} + ⋯ + β_{n} − nx in accordance with (3.4), where β_{i} ∈ Φ_{S_{i}} and S_{i} is an {x}-component of supp(α) for all i. If y ∈ S_{i}, the coefficient of x in r_{x} · β_{i} equals −1 − 2⟨x, y⟩μ. By (2.1)(i), we deduce that this equals 0, or is greater than or equal to 1, and the assertion follows.

For S ⊆ Π and T ⊆ S, we define Φ_{S}^{T} to be the set of roots α ∈ Φ_{S}^{>0} such that C_{1}(α) = T; that is, Φ_{S}^{T} is the set of roots in with support S and coefficient 1 for x ∈ S if and only if x ∈ T. Define E_{S}^{T} to be Φ_{S}^{T} ∩ E.

It is clear that Φ_{S}^{>0} is the disjoint union of all Φ_{S}^{T} with T ⊆ S. If T ∩ D(S) = ∅, every element in E_{S}^{T} is indecomposable. If T ∩ D(S) ≠ ∅ and x is an element of this intersection, let S_{1}, …, S_{n} be the {x}-compoments of S. By (3.4), each element of E_{S}^{T} can be written as

α_{1} + ⋯ + α_{n} − (n − 1)x,

where α_{i} ∈ E_{S_{i}} with coefficient 1 for x. Furthermore, if y ∈ S_{i}, the coefficient of y in α_{i} equals 1 if and only if y ∈ T; that is, α_{i} ∈ E_{S_{i}}^{S_{i}∩T}. Conversely, it is clear that β_{1} + ⋯ + β_{n} − (n − 1)x is in E_{S}^{T} if β_{i} ∈ E_{S_{i}}^{S_{i}∩T} for all i.

An iteration of this yields the following assertion, which reduces our quest to finding E_{S}^{indec} for S ⊆ Π.

(3.7) Proposition Suppose that S ⊆ Π is a tree, and let T ⊆ S and S_{1}, …, S_{n} be the T -components of S. Then the elements of E_{S}^{T} are:

∑_{i=1}^{n} α_{i} − ∑_{x∈T}(val_{S}(x) − 1)x,

where α_{i} ∈ E_{S_{i}}^{S_{i}∩T} for i ∈ {1, …, n}. (Note that since T ∩D(S_{i}) = ∅, clearly E_{S_{i}}^{S_{i}∩T} ⊆ E_{S_{i}}^{indec}).

(3.8) Proposition E_{S}^{indec} is empty if S contains two or more non-simple bonds.

Proof. As in the proof of (3.1), we assume that α ∈ Φ_{S} is minimal with coefficients greater than 1 for x_{1}, …, x_{l}, and x_{0}, x_{l} ∈ supp(α), where the graph of {x_{0}, …, x_{l+1}} is as follows:

•  •  •  •  •  • x_{0}  x_{1} x_{2}  x_{l−1} x_{l}  x_{l+1} m_{1} ≥ 4  m_{2} ≥ 4

Let x ∈ Π with r_{x} · α ≺ α. Again, it suffices to show that ⟨r_{x} · α, x⟩ ≤ −1; for then α and all the roots preceded by α are non-elementary. By (3.1), we may assume without loss of generality that supp(α) does not contain any circuits.

Denote the coefficient of y ∈ S in α by λ_{y}; then λ_{x_{i}} ≥ \sqrt{2} for i = 1, …, l. Minimality of α forces x = x_{i} for some i ∈ {0, …, l + 1}. If i = 0, minimality of α yields further that x_{0} ∉ supp(r_{x_{0}} · α), and thus

⟨r_{x_{0}} · α, x_{0}⟩ = ∑_{y∈S} λ_{y}⟨y, x_{0}⟩ = 0⟨x_{0}, x_{0}⟩ + ∑_{y≠x_{0}} λ_{y}⟨y, x_{0}⟩ ≤ λ_{x_{1}}⟨x_{1}, x_{0}⟩.

Since λ_{x_{1}} ≥ \sqrt{2} and ⟨x_{1}, x_{0}⟩ = cos(π/m_{1}) ≤ − cos(\frac{π}{4} ) = −\frac{1}{\sqrt{2}}, this is less than or equal to −1, as required. Similar arguments apply if i = l + 1.

Suppose next that i ∈ {1, …, l}. Then the coefficient of x_{i} in r_{x_{i}} · α is less than or equal to 1 by minimality of α. Since supp(α) does not contain any circuits, and the support of r_{x_{i}} · α must be connected, Proposition (2.1)(i) forces the coefficient of x_{i} in r_{x_{i}} · α to be equal to 1. If i = 1, then λ_{x_{0}} ≥ \sqrt{2} by Corollary (3.6)(ii) together with (2.1)(ii); hence

λ_{x_{i−1}}⟨x_{i−1}, x_{i}⟩ = λ_{x_{0}}⟨x_{0}, x_{1}⟩ ≤ \sqrt{2}(− cos(π/m_{1})) ≤ −1,

as cos(π/m_{1}) ≥ \frac{1}{\sqrt{2}}. If i > 1, Corollary (3.6)(i) implies that λ_{x_{i−1}} ≥ 2, and thus again λ_{x_{i−1}}⟨x_{i−1}, x_{i}⟩ ≤ −1. Symmetrical arguments also yield λ_{x_{i+1}}⟨x_{i+1}, x_{i}⟩ ≤ −1 and therefore

⟨r_{x_{i}} · α, x_{i}⟩ ≤ 1 + λ_{x_{i−1}}⟨x_{i−1}, x_{i}⟩ + λ_{x_{i+1}}⟨x_{i+1}, x_{i}⟩ ≤ −1,

as required.

This leaves us to determine E_{S}^{indec} for S ⊆ Π a tree, containing at most one non-simple bond (which has finite weight) such that W_{S} is infinite.

§4 Simple bonds only *

For the duration of this section suppose that S ⊆ Π is a tree and contains only simple bonds. Note that if α ∈ Φ_{S}, all the coefficients in α are integers; hence it is clear that ⟨α, x⟩ is an integer multiple of \frac{1}{2} for x ∈ S. In particular, if α is elementary and α ≠ x, then ⟨α, x⟩ ≤ \frac{1}{2}; moreover, if r_{x} · α ≺ α, then ⟨α, x⟩ = \frac{1}{2}.

* A description of the set of elementary roots in this case has been obtained in [He]

(4.1) Proposition E_{S}^{indec} is empty if S contains

•  •  • •  • • • or •  •  • • • •  x_{1} x_{2}  x_{l−1} x_{l}  y_{3} y_{4} y_{1} y_{2}  y_{2} y_{1}  y_{3} y_{4} x_{1} ........................................................................ ......................................................................... ........................................................................ .........................................................................  ............................................................................................................................................................. .............................................................................................................................................................

(Of course, the right hand case is just a special case of the left hand one).

Proof. As in the proof of (3.1), we let α be minimal with y_{1}, y_{2}, y_{3}, y_{4} ∈ supp(α) and coefficient λ_{i} > 1 for x_{i} for all i ∈ {1, …, n}. Note that since λ_{i} is an integer, it follows that λ_{i} ≥ 2. Further, let z ∈ S with r_{z} · α ≺ α; it suffices to show that ⟨r_{z} · α, z⟩ ≤ −1.

Minimality of α forces z = x_{i} for some i ∈ {1, …, l} or z = y_{j} for some j ∈ {1, 2, 3, 4}. If z = x_{i}, minimality of α yields that the coefficient of x_{i} in r_{x_{i}} · α is less than or equal to 1. If 1 < i < l, we deduce that

⟨r_{x_{i}} · α, x_{i}⟩ ≤ 1 + λ_{i−1}⟨x_{i−1}, x_{i}⟩ + λ_{i+1}⟨x_{i+1}, x_{i}⟩ ≤ 1 + 2(−\frac{1}{2}) + 2(−\frac{1}{2}) ≤ −1,

as required. If 1 = i < l, the coefficients of y_{1} and y_{2} in α are greater than or equal to 1, while λ_{i+1} ≥ 2, and thus again ⟨r_{x_{i}} · α, x_{i}⟩ ≤ −1. Similarly, ⟨r_{x_{i}} · α, x_{i}⟩ ≤ −1 if 1 < i = l. Finally, if 1 = i = l, the coefficients of y_{1}, y_{2}, y_{3} and y_{4} in α are greater than or equal to 1, and again ⟨r_{x_{i}} · α, x_{i}⟩ ≤ −1.

If z = y_{1}, minimality of α yields that y_{1} ∉ supp(r_{y_{1}} · α). Since λ_{1} ≥ 2, we deduce that ⟨r_{y_{1}} · α, y_{1}⟩ ≤ −1. Symmetrical arguments apply if z ∈ {y_{2}, y_{3}, y_{4}}.

This leaves us to determine E_{S}^{indec} for S of the shape

•  •  •  •  •  •  • • • • y_{m} y_{m−1}  y_{1} x_{0} z_{1}  z_{n−1} z_{n} x_{1} x_{l−1} x_{l} (0)

with l, m, n ≥ 1. We show now that each root in E_{S}^{indec} is preceded by

x_{l} + 2(x_{l−1} + ⋯ + x_{1}) + y_{m} + 2(y_{m−1} + ⋯ + y_{1}) + 2x_{0} + 2(z_{1} + ⋯ + z_{n−1}) + z_{n},

by proving that if α = ∑_{x∈S} λ_{x}x and β = ∑_{x∈S} μ_{x}x are elementary roots with μ_{x} ≤ λ_{x} for all x ∈ S, then β ⪯ α.

If dp(α) = 1, this is certainly true. Suppose next that dp(α) > 1, and let y ∈ S with r_{y} · α ≺ α; then ⟨α, y⟩ = \frac{1}{2}, and the coefficient of y in r_{y} · α equals λ_{y} − 1. If μ_{y} ≤ λ_{y} − 1, induction yields that β ⪯ r_{y} · α, and thus β ⪯ α by transitivity of ⪯. Suppose next that μ_{y} > λ_{y} − 1; that is, μ_{y} = λ_{y}. If β = y, then λ_{y} = 1, and thus y ⪯ α by (3.5), as required. If β ≠ y, then ⟨β, y⟩ ≤ \frac{1}{2}. Further,

⟨β, y⟩ = ∑_{x∈S} μ_{x}⟨x, y⟩ = μ_{y}⟨y, y⟩ + ∑_{x≠y} μ_{x}⟨x, y⟩ ≥ λ_{y}⟨y, y⟩ + ∑_{x≠y} λ_{x}⟨x, y⟩ = ⟨α, y⟩ = \frac{1}{2},

and so we must have equality everywhere. Hence ⟨β, y⟩ = \frac{1}{2}, and if x ∈ S with ⟨x, y⟩ ≠ 0, then λ_{x} = μ_{x}. Since the coefficient of y in r_{y} ·β equals μ_{y} −1 = λ_{y} −1, induction yields that r_{y} · β ⪯ r_{y} · α. Let w ∈ W_{S} be of length dp(r_{y} · α) − dp(r_{y} · β) with r_{y} · α = w · (r_{y} · β). Since for x ∈ S with ⟨x, y⟩ ≠ 0, the coefficients of x in r_{y} · β and r_{y} · α coincide, we deduce that w ∈ W_{S_{0}}, where S_{0} consists of all x ∈ S with ⟨x, y⟩ = 0. Therefore r_{y}w = wr_{y}, and thus β = r_{y} ·(r_{y} ·β) = r_{y} ·(w·(r_{y} ·α)) = w·α. Since dp(α)−dp(β) = dp(r_{y} ·α)−dp(r_{y} ·β) = l(w), we conclude that β ⪯ α, as required.

To shorten the notation, we write [a··b] for the sum of the simple roots on the minimal path in S joining a ∈ S to b ∈ S. A straightforward induction on depth yields the following assertion.

(4.2) Proposition Suppose that S is of shape (0), and set x_{0} = y_{0} = z_{0}.

(1) If l, m, n ≥ 2, then E_{S}^{indec} has exactly one element, namely x_{l} + 2(x_{l−1} + ⋯ + x_{1}) + y_{m} + 2(y_{m−1} + ⋯ + y_{1}) + 2x_{0} + 2(z_{1} + ⋯ + z_{n−1}) + z_{n}.

(2) If l = 1 and m, n ≥ 3, the elements of E_{S}^{indec} are as follows:

(i) x_{1} + y_{m} + 2[y_{m−1} ·· y_{j(3)}] + 3[y_{j(3)−1} ·· y_{j(4)}] + ⋯ + (M − 1)[y_{j(M−1)−1} ·· y_{j(M)}]

+ M[y_{j(M)−1} ·· z_{k(M)−1}]

+ (M − 1)[z_{k(M)} ·· z_{k(M−1)−1}] + ⋯ + 3[z_{k(4)} ·· z_{k(3)−1}] + 2[z_{k(3)} ·· z_{n−1}] + z_{n}, with M ∈ {2, …, min(m, n)+1}, m > j(3) > j(4) > … > j(M −1) > j(M) > 0

and 0 < k(M) < … < k(4) < k(3) < n,

(ii) 2x_{1} + y_{m} + 2[y_{m−1} ·· y_{j}] + 3[y_{j−1} ·· z_{k−1}] + 2[z_{k} ·· z_{n−1}] + z_{n},

where m > j > 0 and 0 < k < n,

(iii) 2x_{1} + y_{m} + 2[y_{m−1} ·· y_{1}] + 4[x_{0} ·· z_{k(2)−1}] + 3[z_{k(2)} ·· z_{k(1)−1}] + 2[z_{k(1)} ·· z_{n−1}] + z_{n},

where 0 < k(2) < k(1) < n, and

(iv) 2x_{1} + y_{m} + 2[y_{m−1} ·· y_{j(1)}] + 3[y_{j(1)−1} ·· y_{j(2)}] + 4[y_{j(2)−1} ·· x_{0}] + 2[z_{1} ·· z_{n−1}] + z_{n},

where m > j(1) > j(2) > 0.

(3) If l = 1, m = 2 and n ≥ 5, the elements of E_{S}^{indec} are as follows:

(i) x_{1} + y_{2} + 2[y_{1} ·· z_{n−1}] + z_{n}, (ii) λx_{1} +y_{2} +2y_{1} +3[x_{0} ··z_{k−1}]+2[z_{k} ··z_{n−1}]+z_{n} where k ∈ {1, …, n−1}, λ ∈ {1, 2},

(iii) λx_{1} + y_{2} + μy_{1} + M[x_{0} ·· z_{k(M−1)−1}]

+ (M − 1)[z_{k(M−1)} ·· z_{k(M−2)−1}] + ⋯ + 3[z_{k(3)} ·· z_{k(2)−1}] + 2[z_{k(2)} ·· z_{n−1}] + z_{n}, where M ∈ {4, …, n + 1}, 0 < k(M − 1) < k(M − 2) < … < k(3) < k(2) < n and λ ∈ \{\frac{M−1}{2}, \frac{M+1}{2}\}, μ = \frac{M+1}{2} if M is odd, and λ = \frac{M}{2} , μ ∈ \{\frac{M}{2} , \frac{M}{2} + 1\} if M is even,

(iv) 2x_{1} + 2y_{2} + 3y_{1} + 4[x_{0} ·· z_{k(3)−1}] + 3[z_{k(3)} ·· z_{k(2)−1}] + 2[z_{k(2)} ·· z_{n−1}] + z_{n},

where 0 < k(3) < k(2) < n,

(v) λx_{1} + 2y_{2} + μy_{1} + 5[x_{0} ·· z_{k(4)−1}] + 4[z_{k(4)} ·· z_{k(3)−1}] + ⋯ + 2[z_{k(2)} ·· z_{n−1}] + z_{n}, where 0 < k(4) < k(3) < k(2) < n and λ ∈ {2, 3}, μ ∈ {3, 4},

(vi) λx_{1} + 2y_{2} + μy_{1} + 6[x_{0} ·· z_{k(5)−1}] + 5[z_{k(5)} ·· z_{k(4)−1}] + ⋯ + 2[z_{k(2)} ·· z_{n−1}] + z_{n},

where 0 < k(5) < k(4) < k(3) < k(2) < n and (λ, μ) ∈ {(3, 3), (2, 4)}, (vii) 3x_{1} + 2y_{2} + 4y_{1} + 6x_{0} + 4[z_{1} ·· z_{k(2)−1}] + 3[z_{k(3)} ·· z_{k(2)−1}] + 2[z_{k(2)} ·· z_{n−1}] + z_{n},

where 1 < k(3) < k(2) < n, (viii) 3x_{1} + 2y_{2} + 4y_{1} + 6x_{0} + 5z_{1} + 3[z_{2} ·· z_{k−1}] + 2[z_{k} ·· z_{n−1}] + z_{n}, where 2 < k < n,

(ix) 3x_{1} + 2y_{2} + 4y_{1} + 6x_{0} + 5z_{1} + 4z_{2} + 2[z_{3} ·· z_{n−1}] + z_{n}.

§5 One non-simple bond of weight m ≥ 6

For the duration of this section assume that S ⊆ Π is a tree and contains exactly one nonsimple bond which has weight m ≥ 6. Let a and b be the vertices of this non-simple bond, and denote 2 cos(\frac{π}{m}) by c_{m}; then c_{m} ≥ \sqrt{3}.

We show now that each α ∈ E_{S} is preceded by a + c_{m}b or c_{m}a + b (this is true for all m ≥ 4). Let β ⪯ α be maximal such that {a, b} ⊈ supp(β); we may assume without loss of generality that b ∉ supp(β). By maximality of β, we find that β ≺ r_{b} · β ⪯ α and a ∈ supp(β). So r_{b} · β ∈ E by (2.2)(iii), and thus ⟨β, b⟩ > −1 by (2.2)(iv). We deduce that the coefficient of a in β is less than \frac{2}{c_{m}} ≤ \sqrt{2}, and thus must be equal to 1 by (2.1)(ii). Lemma (3.5) yields that a ⪯ β, and that there exists a w ∈ W_{supp(β)\{a}} of length dp(β)−1 such that β = w · a. As b ∉ supp(β), and since S does not contain any circuits, we know that b cannot be adjacent to supp(β) \ {a}, and so r_{b}w = wr_{b}. Now r_{b} · β = w · (a + c_{m}b), and since dp(r_{b} · β) − dp(a + c_{m}b) = dp(β) − dp(a) = l(w), we find that r_{b} · β is preceded by a + c_{m}b. Transitivity of ⪯ yields that α is preceded by a + c_{m}b, as required.

(5.1) Lemma Suppose that S contains the following subgraph

•  •  • a  b  c m ≥ 6

and let α ∈ Φ_{S} be preceded by c_{m}a + b, with a, c ∈ supp(α), and coefficient for b in α greater than 1. Then α ∉ E.

Proof. Let β be maximal with c_{m}a+b ⪯ β ⪯ α such that the coefficient of b in β equals 1; then β ≺ r_{b} · β ⪯ α by maximality of β. If c ∈ supp(β), we find that

⟨β, b⟩ ≤ 1 + (−\frac{c_{m}}{2} )c_{m} + (−\frac{1}{2})1 ≤ −1,

and thus r_{b} · β ∉ E by (2.2)(iv); hence α ∉ E, as required. Assume now that c ∉ supp(β), and let γ be of maximal depth with r_{b} ·β ⪯ γ ⪯ α such that c ∉ supp(γ); then γ ≺ r_{c} ·γ ⪯ α by maximality of γ. The coefficient of b in γ must be greater than or equal to the coefficient of b in r_{b} · β, and this is greater than or equal to

−1 + 2(−\frac{c_{m}}{2} )c_{m} ≥ 2.

Hence clearly ⟨γ, c⟩ ≤ −1, and thus α ∉ E by (2.3)(iii), (iv).

Since in an indecomposable root the coefficient of each disconnector must be greater than 1, the previous lemma has the following immediate consequence.

(5.2) Proposition E_{S}^{indec} is empty if S contains the following subgraph:

•  •  •  • m ≥ 6

This leaves us to determine E_{S}^{indec} for S of the following two shapes:

•  •  •  •  • a  b_{1}  b_{2}  b_{n−1} b_{n} m ≥ 6 (1)  (n ≥ 2) ........ ........ ........ ........ ........ ........ ........

•  •  •  •  • a  b_{1}  b_{2}  b_{N−1} b_{N} m ≥ 6  B (2)  ........................................................................ ........................................................................ ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ..... ........ ........ ........ .... .............................. ...

where B is a tree which contains only simple bonds such that b_{N} is adjoined to two or more elements of B (that is, val_{B}(b_{N}) ≥ 2).

Note that if S is of shape (1) or (2), Lemma (5.1) yields that each root in E_{S}^{indec} is preceded by a + c_{m}b. If S is of shape (1), it follows readily that each root in E_{S}^{indec} is preceded by a + c_{m}(b_{1} + ⋯ + b_{n}), and the next result follows:

(5.3) Proposition Suppose that S is of shape (1) with m ≥ 7. Then E_{S}^{indec} = {a + c_{m}(b_{1} + ⋯ + b_{n}), (c_{m}^{2} − 1)a + c_{m}(b_{1} + ⋯ + b_{n})}.

(5.4) Proposition Suppose that S is of shape (1) and m = 6. Then the elements of E_{S}^{indec} are as follows:

(i) a + c_{m}(b_{1} + ⋯ + b_{n}), and (ii) 2a + \sqrt{3}(2(b_{1} + ⋯ + b_{j−1}) + (b_{j} + ⋯ + b_{n})), where j ∈ \{1, …, n\}.

It remains to determine E_{S}^{indec} for S of shape (2). In order to do so, we prove the following two assertions, which are valid for all m ≥ 4, and since these statements will be used for the cases m = 4, 5, we prove the results in general.

(5.5) Proposition Let x ∈ S, and let S_{0}, …, S_{k} be the {x}-components of S such that S_{0} contains the bond of weight m; that is, {a, b} ⊆ S_{0}. Then

ϕ: (β_{0}, …, β_{k}) ↦ β_{0} + c_{m}(β_{1} + ⋯ + β_{k} − kx)

defines a one-one correspondence between the set of (k + 1)-tuples in Φ_{S_{0}}^{+} × ⋯ × Φ_{S_{k}}^{+} such that the coefficient of x in the first component equals c_{m}, while for all other components the coefficient of x equals 1, and the set of roots in Φ_{S}^{+} with coefficient c_{m} for x. Moreover, ϕ restricts to a one-one correspondence between the set of (k + 1)-tuples in E_{S_{0}} × ⋯ × E_{S_{k}} such that x has coefficient c_{m} in the first component, and 1 in the others, and the set of roots in E_{S} with coefficient c_{m} for x.

Proof. We show first that ϕ is well defined. So let (β_{0}, …, β_{k}) ∈ Φ_{S_{0}}^{+} ×⋯×Φ_{S_{k}}^{+} such that the coefficient of x in β_{0} equals c_{m}, while the coefficient of x in β_{j} equals 1 for j ∈ {1, …, k}. Lemma (3.5) implies that x ⪯ β_{j} for j ∈ {1, …, k}, and hence there exist w_{j} ∈ W_{S_{j}\{x}} of length dp(β_{j}) − 1 such that β_{j} = w_{j} · x. Define α = (w_{1} ⋯ w_{k}) · β_{0}; then the coefficient of x in α equals c_{m}, and it can be easily seen that α equals ϕ(β_{0}, …, β_{k}). Therefore ϕ is well defined; it is clear that ϕ is one-one, and we show now that ϕ is onto.

Suppose that α ∈ Φ_{S}^{+} has coefficient c_{m} for x, and let γ ⪯ α be minimal such that the coefficient of x in α equals c_{m}. We deduce from (2.1)(ii) that {a, b} ⊆ supp(γ). Since {a, b} ⊆ S_{0}, in particular | supp(γ) ∩ S_{0}| ≥ 2, and thus

(supp(γ) ∩ S_{0}) \ {x} ≠ ∅.  (∗)

Now r_{x} · γ ≺ γ by minimality of γ, and it follows by (2.1)(i) that the coefficient of x in r_{x} · γ equals 0 or 1. If x ∉ supp(r_{x} · γ), then supp(r_{x} · γ) ⊆ S_{i} \ {x} for some i by connectedness of the support of r_{x} · γ, and thus supp(r_{x} · γ) ⊆ S_{0} \ {x} by (∗).

Assume now that the coefficient of x in r_{x} · γ equals 1. Let z ∈ supp(γ) be adjoined to x, and denote the coefficient of z in γ by ν. If x and z are adjoined by a simple bond, it follows by (3.6)(i) applied to γ that ν ≠ 1; therefore (3.6)(i) applied to r_{x} · γ yields that ν ≥ 2. If x and z are adjoined by the bond of weight m, then ν ≥ c_{m} by (3.6)(ii). Hence in any case, ν⟨z, x⟩ ≤ −1; since ⟨γ, x⟩ > 0 as r_{x} · γ ≺ γ, we deduce that x is only adjoined to one z ∈ supp(γ). By (∗) we conclude that supp(γ) ⊆ S_{0}.

Since γ has coefficient c_{m} for x, there exists a w ∈ W_{S\{x}} such that α = w · γ and dp(α)−dp(γ) = l(w). As W_{S\{x}} is the direct product of W_{S_{0}\{x}}, …, W_{S_{k}\{x}}, there exist w_{j} ∈ W_{S_{j}\{x}} for all j ∈ {0, …, k} such that w equals w_{0} ⋯ w_{k} with length adding. Define β = w_{0} · γ; this is an element of Φ_{S_{0}}^{+} with coefficient c_{m} for x, and clearly

α = (w_{1} ⋯ w_{k}) · β = ϕ(β, w_{1} · x, …, w_{k} · x).

Hence ϕ is also onto. We show now for w_{i} ∈ W_{S_{i}\{x}} and β ∈ E_{S_{0}} with coefficient c_{m} for x that α = (w_{1} ⋯ w_{k}) · β ∉ E if and only if β ∉ E or w_{i} · x ∉ E for some i ≥ 1. Note that a straightforward calculation yields for i ≥ 1 and γ ∈ Φ_{S_{i}\{x}}, that

⟨γ, α⟩ = c_{m}⟨γ, w_{i} · x⟩.  (∗∗)

It can be easily seen that α is preceded by β; thus α ∉ E by (2.2)(iii) if β ∉ E. Next, suppose that w_{i} · x dominates γ ∈ Φ^{+} \ {w_{i} · x}. Then γ ∈ Φ_{S_{i}\{x}} as x ∉ E. Since ⟨γ, w_{i} · x⟩ ≥ 1 by (2.2)(i), we deduce from (∗∗) that ⟨α, γ⟩ ≥ 1, and it follows readily that α ∉ E, as required.

Conversely, if α dominates some γ ∈ Φ^{+} \ {α}, (2.2)(ii) implies that either β dominates ((w_{1} ⋯ w_{k})^{−1}) · γ, or ((w_{1} ⋯ w_{k})^{−1}) · γ is negative. So β ∉ E (as required), or γ ∈ Φ_{S_{i}\{x}}^{+}

for some i ≥ 1. If γ ∈ Φ_{S_{i}\\{x\}}^{+}, then (∗∗) implies that ⟨γ, w_{i} · x⟩ ≥ \frac{1}{c_{m}} > \frac{1}{2}. As S_{i} contains only simple bonds, this forces ⟨γ, w_{i} · x⟩ ≥ 1; hence w_{i} · x dominates γ, and thus cannot be an element of E, as required.

(5.6) Lemma Suppose that S contains the following subgraph

•  •  •  •  •  • • a  b_{1}  b_{2}  b_{N−1} b_{N}  y_{1} y_{2} m  ............................................................................ ............................................................................

and let α_{0} ∈ Φ_{S} with coefficient for a greater than 1, coefficients for b_{1}, …, b_{N} greater than or equal to 2, and y_{1}, y_{2} ∈ supp(α_{0}). Then α_{0} ∉ E.

Proof. As in the proof of (3.1), we let α ⪯ α_{0} be minimal such that y_{1}, y_{2} ∈ supp(α), the coefficient of a in α is greater than 1, and the coefficients of b_{1}, …, b_{N} in α greater than or equal to 2. Further, let x ∈ S with r_{x} · α ≺ α; again, it suffices to show that ⟨r_{x} · α, x⟩ ≤ −1.

Minimality of α forces x = a, or x = y_{1}, y_{1}, or x = b_{j} for some j ∈ {1, …, N}. If x = a, minimality of α together with (2.1)(ii) yield that the coefficient of a in r_{a} · α equals 0 or 1. In the former case, it follows easily that ⟨r_{a} · α, a⟩ ≤ −1. If the coefficient of a in r_{a} · α equals 1, an easy induction on depth yields that the coefficient of b_{1} in α is an integer multiple of c_{m}, and thus greater than or equal to 2c_{m}; whence again, ⟨r_{a} · α, a⟩ ≤ −1.

If x = y_{1}, minimality of α yields that y_{1} ∉ supp(r_{y_{1}} · α). Since the coefficient of b_{N} in r_{y_{1}} · α is greater than or equal to 2, we deduce that ⟨r_{y_{1}} · α, y_{1}⟩ ≤ −1. Symmetrical arguments apply if x = y_{2}.

Finally, if x = b_{j} for some j ∈ {1, …, N}, minimality of α together with (2.1)(ii) yield that the coefficient μ_{j} of b_{j} in r_{b_{j}} · α equals 0, 1 or c_{m}. If μ_{j} ≤ 1, it can be easily seen that ⟨r_{b_{j}} · α, b_{j}⟩ ≤ −1, so suppose that μ_{j} = c_{m}. If j < N, the coefficient of b_{j+1} in r_{b_{j}} · α is an integer multiple of c_{m} by (5.5), and thus greater than or equal to 2c_{m}; if j = N, the coefficients of y_{1} and y_{2} are integer multiples of c_{m} by (5.5), and we can deduce in both cases that ⟨r_{b_{j}} · α, b_{j}⟩ ≤ −1.

The above shows that for S of shape (2) each element of E_{S} has coefficient 1 for a, or coefficient c_{m} for some b_{j}. Propositions (5.3), (5.4) and (5.5) enable us to describe the roots with coefficient c_{m} for some b_{j}, and we will now describe the set of roots in Φ_{S} with coefficient 1 for a. Note that {z} ∪ (S \ {x}) defined below contains only simple bonds, therefore E_{{z}∪(S\{x})}^{indec} has already been determined.

(5.7) Proposition Since E_{S} does not depend on Π as long as S ⊆ Π, we may assume without loss of generality that there exists a z ∈ Π \ S such that for a_{r} ∈ S,

m_{r_{z}r} = m_{rr_{z}} = \{ \begin{matrix}3 & if r = r_{b_{1}}, \\ 2 & if r ≠ r_{b_{1}}. \end{matrix}

Then ϕ: β ↦ a + c_{m}(β − z) defines a bijection from the set of roots in Φ_{{z}∪(S\{a})}^{+} with

coefficient 1 for z, to the set of roots in Φ_{S}^{+} with coefficient 1 for a. Moreover, ϕ restricts to a bijection from the set of roots in E_{{z}∪(S\{a})} with coefficient 1 for z, to the set of roots in E_{S} with coefficient 1 for a.

Proof. An easy calculation yields for β ∈ Φ and y ∈ S \ {a} that

r_{y} · (a + c_{m}(β − z)) = a + c_{m}(r_{y} · β − z).  (∗)

Now let β ∈ Φ_{{z}∪(S\{a}}^{+} with coefficient 1 for z, and in accordance with (3.5) let w ∈ W_{S\{a}} be of length dp(β) − 1 such that β = w · z. The coefficient of a in w · a equals 1, and since a = a + c_{m}(z − z), a straightforward induction on l(w) using (∗) yields that w · a = a + c_{m}(w · z − z); hence ϕ is well defined. By (3.5), every element of Φ_{S}^{+} with coefficient 1 for a can be written as w · a for some w ∈ W_{S\{a}}, so the above also shows that ϕ is surjective. Since ϕ is certainly injective, ϕ is a bijection. As in the proof of (5.5), it follows readily that ϕ restricts to a bijection from the set of roots in E_{{z}∪(S\{a})} with coefficient 1 for z to the set of roots in E_{S} with coefficient 1 for a.

(5.8) Proposition Suppose that S is of shape (2) with m ≥ 7. Then the elements of E_{S}^{indec} are as follows:

(i) a + c_{m}(β − z), where β ∈ E_{{z}∪(S\{a})} with coefficient 1 for z (and z as in (5.7)), and

(ii) (c_{m}^{2} − 1)a + c_{m}γ, where γ ∈ E_{S\{a}} with coefficient 1 for b_{1}.

(5.9) Proposition Suppose that S is of shape (2) with m = 6. Then the elements of E_{S}^{indec} are as follows:

(i) a + c_{m}(β − z), where β ∈ E_{{z}∪(S\{a})} with coefficient 1 for z (and z as in (5.7)), and

(ii) 2a + \sqrt{3}(2(b_{1} + ⋯ + b_{n−1}) + β_{n}), where n ∈ {1, …, N} and β_{n} ∈ E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n}.

§6 One non-simple bond of weight m ∈ {4, 5}

Addressing now the final case, we assume for the duration of this section that S ⊆ Π is a tree, and contains exactly one non-simple bond which has weight m ∈ {4, 5}. Let a and b be the vertices of the non-simple bond, and denote 2 cos(\frac{π}{m}) by c_{m}.

(6.1) Definition For α ∈ Φ_{S}, define C_{c}(α) to be the set of x ∈ Π with coefficient c_{m} in α. If C_{c}(α) ∩ D(supp(α)) is empty, α is said to be skew-indecomposable, otherwise α is said to be skew-decomposable. The set of skew-indecomposable roots in E_{S}^{indec} is denoted by E_{S}^{skew-ind}.

Proposition (5.5) together with Lemma (5.6) yield the next three propositions, which reduce the problem to the determination of E_{S}^{skew-ind} for S of the following shape:

•  •  •  •  •  •  •  • a_{l}  a_{l−1}  a_{2} a_{1} b_{1}  b_{2}  b_{n−1} b_{n} m (3)  ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ........ ........ ........ ........

with m ∈ {4, 5}. In fact, (6.2), (6.3) and (6.4) are valid for all m ≥ 4; we chose to treat the case m ≥ 6 separately because the statements simplify significantly in that case.

(6.2) Proposition Suppose that S is of the following shape

•  •  •  •  •  •  •  • a_{L} a_{L−1}  a_{2} a_{1} = a b = b_{1} b_{2}  b_{N−1} b_{N} A  B m  ........................................................................ ........................................................................ ........................................................................ ........................................................................  ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ..... ........ ........ ........ .... ............................. ........ ........ ........ .....  ........ ........ ........ ....  .............................  . ... . . ..

where A and B are trees which contain only simple bonds, such that val_{A}(a_{L}), val_{B}(b_{N}) ≥ 2. Then the elements of E_{S}^{indec} are

c_{m}α_{l} − c_{m}a_{l} + γ_{l,n} + c_{m}β_{n} − c_{m}b_{n},

where l ∈ {1, …, L} and α_{l} ∈ E_{A∪{a_{L},...,a_{l}}} with coefficient 1 for a_{l}, while n ∈ {1, …, N} and β_{n} ∈ E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n}, and further, γ_{l,n} ∈ E_{{a_{l},...,a_{1},b_{1},...,b_{n}}}^{skew-ind} with coefficient c_{m} for a_{l} and b_{n}.

(6.3) Proposition Suppose that S is of the following shape:

•  •  •  •  •  •  •  • a_{L} a_{L−1}  a_{2} a_{1} = a b = b_{1} b_{2}  b_{N−1} b_{N} m  B ........................................................................ ........................................................................ ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ..... ........ ........ ........ .... .............................. ...

where B is a tree which only contains simple bonds, such that val_{B}(b_{N}) ≥ 2.

(1) If L = 1, the elements of E_{S}^{indec} are

(i) γ_{1,n} + c_{m}β_{n} − c_{m}b_{n}, where n ∈ {1, …, N} and β_{n} ∈ E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n},while

γ_{1,n} ∈ E_{{a,b_{1},...,b_{n}}}^{skew-indec} with coefficient c_{m} for b_{n}, and

(ii) a+c_{m}(δ −z), where δ ∈ E_{{z}∪(S\{a})} with coefficient 1 for z (where z as in (5.7)).

(2) If L ≥ 2, the elements of E_{S}^{indec} are

c_{m}(a_{L} + ⋯ + a_{l+1}) + γ_{l,n} + c_{m}β_{n} − c_{m}b_{n},

where l ∈ {1, …, L}, n ∈ {1, …, N} and β_{n} ∈ E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n}, while γ_{l,n} ∈ E_{{a_{l},...,a_{1},b_{1},...,b_{n}}}^{skew-ind} with coefficient c_{m} for b_{n} and if l < L, coefficient c_{m} for a_{l}.

(6.4) Proposition Suppose that S is of the following shape:

•  •  •  •  •  •  •  • a^{L} a^{L−1}  a^{2} a^{1} b^{1}  b^{2}  b^{N−1} b^{N} m ........ ........ ........ ........ ........ ........ ........  ........ ........ ........ ........ ........ ........ ........

Then the elements of E_{S}^{indec} are

c_{m}(a_{L} + ⋯ + a_{l+1}) + γ_{l,n} + c_{m}(b_{n+1} + ⋯ + b_{N}),

where l ∈ {1, …, L} and n ∈ {1, …, N}, while γ_{l,n} ∈ E_{{a_{l},...,a_{1},b_{1},...,b_{n}}}^{skew-ind} with coefficient c_{m} for a_{l} if l < L, and coefficient c_{m} for b_{n} if n < N.

Suppose first that S is of shape (3) with m = 4. We proved in §5 that each element of E_{S} is preceded by a_{1} + \sqrt{2}b_{1} or \sqrt{2}a_{1} + b_{1}. If α ∈ Φ_{S} is preceded by a_{1} + \sqrt{2}b_{1}, an easy induction on depth yields that the coefficients of the a_{i} in α are integers, and the coefficients of the b_{j} are integer multiples of \sqrt{2}. As in §3, it follows readily from (2.2)(iii),(iv) that if α = ∑_{x∈S} λ_{x}x and β = ∑_{x∈S} μ_{x}x are elementary roots preceded by a_{1} + \sqrt{2}b_{1} with μ_{x} ≤ λ_{x} for all x ∈ S, then β ⪯ α. In particular, each element of E_{S}^{skew-ind} which is preceded by a_{1} + \sqrt{2}b_{1} is preceded by a_{l} + 2(a_{l−1} + ⋯ + a_{1}) + \sqrt{2}(2(b_{1} + ⋯ + b_{n−1}) + b_{n}). A straightforward induction on depth now yields the following result:

(6.5) Proposition Suppose that S is of shape (3) with m = 4.

(1) If l, n ≥ 3, the elements of E_{S}^{skew-ind} are a_{l} + 2(a_{l−1} + ⋯ + a_{1}) + \sqrt{2}(2(b_{1} + ⋯ + b_{n−1}) + b_{n}) and

\sqrt{2}(a_{l} + 2(a_{l−1} + ⋯ + a_{1})) + 2(b_{1} + ⋯ + b_{n−1}) + b_{n}.

(2) If l = 2 and n ≥ 3, the elements of E_{S}^{skew-ind} are as follows:

(i) a_{2} +Ma+\sqrt{2}(M(b_{1} +⋯+b_{j(M−1)−1})+(M −1)(b_{j(M−1)} +⋯+b_{j(M−2)−1})+⋯

⋯ + 3(b_{j(3)} + ⋯ + b_{j(2)−1}) + 2(b_{j(2)} + ⋯ + b_{n−1}) + b_{n})
where M ∈ {2, …, n + 1} and 0 < j(M − 1) < ⋯ < j(2) < n,

(ii) 2a_{2} + 3a + \sqrt{2}(3(b_{1} + ⋯ + b_{j−1}] + 2(b_{j} + ⋯ + b_{n−1}) + b_{n}), where 0 < j < n, and

(iii) \sqrt{2}(a_{2} + 2a_{1}) + 2(b_{1} + ⋯ + b_{n−1}) + b_{n}.

(6.6) Corollary

(1) Suppose that S is of the shape described in (6.2) with m = 4. Then E_{S}^{indec} is empty.

(2) Suppose that S is of the shape described in (6.3) with L ≥ 3 and m = 4. Then the

elements of E_{S}^{indec} are

a_{L} + 2(a_{L−1} + ⋯ + a_{1}) + \sqrt{2}(2(b_{1} + ⋯ + b_{n−1}) + β_{n}),

where n ∈ {1, …, N} and β_{n} ∈ E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n}.

(3) Suppose that S is of the shape described in (6.4) with L, N ≥ 3 and m = 4. Then the

elements of E_{S}^{indec} are as follows:

(i) a_{L} + 2(a_{L−1} + ⋯ + a_{1}) + \sqrt{2}(2(b_{1} + ⋯ + b_{n−1}) + (b_{n} + ⋯ + b_{N})),

where n ∈ {1, …, N} and

(ii) \sqrt{2}((a_{L}+⋯+a_{l})+2(a_{l−1}+⋯+a_{1}))+2(b_{1}+⋯+b_{N−1})+b_{N}, where l ∈ \{1, …, L\}.

Finally suppose that S is of shape (3) with m = 5. Proposition (2.1)(ii) yields that for α ∈ Φ_{S}^{+}, each coefficient of α is in ℕ_{0} + c_{5}ℕ_{0}; that is, equals λc_{5} + μ with λ, μ ∈ ℕ_{0}. Note that the smallest possible nonnegative coefficients are: 0, 1, c_{5}, 2, c_{5} + 1.

(6.7) Proposition Suppose that S is of shape (3) with m = 5 and l, n ≥ 2. Then E_{S}^{skew-ind} = ∅.

Proof. As in the proof of (3.1), we assume that α is a minimal root with coefficients for a_{1}, b_{1} greater than or equal to 2, and a_{2}, b_{2} ∈ supp(α). Let x ∈ S such that r_{x} · α ≺ α; we will show that ⟨r_{x} · α, x⟩ ≤ −1.

Denote the coefficient of a_{i} in α by λ_{i}, and the coefficient of b_{j} in α by μ_{j}. By minimality of α, we deduce that x ∈ {a_{2}, a_{1}, b_{1}, b_{2}}. If x = a_{2}, minimality of α yields that the coefficient of a_{2} in r_{a_{2}} · α equals 0, and since λ_{1} ≥ 2, it follows that ⟨r_{a_{2}} · α, a_{2}⟩ ≤ −1. Similarly, ⟨r_{b_{2}} · α, b_{2}⟩ ≤ −1 if x = b_{2}.

So we are left with x ∈ {a_{1}, b_{1}}, and we may assume without loss of generality that x = a_{1}. By minimality of α together with (2.1)(ii), the coefficient of a_{1} in r_{a_{1}} · α equals 0, 1 or c_{5}. If the coefficient of a_{1} in r_{a_{1}} · α is less than or equal to 1, then

⟨r_{a_{1}} · α, a_{1}⟩ ≤ 1 + λ_{2}(−\frac{1}{2}) + μ_{1}(−\frac{c_{5}}{2} ) ≤ 1 + (−\frac{1}{2}) + 2(−\frac{c_{5}}{2} ) ≤ −1,

as required.

Suppose now that the coefficient of a_{1} in r_{a_{1}}·α equals c_{5}, and assume for a contradiction that the coefficient of b in α equals 2. Let β ⪯ r_{a} · α be minimal such that the coefficient of a in β equals c_{5}, and the coefficient of b in β equals 2. Then r_{a} · β ≺ β or r_{b} · β ≺ β by minimality of β; but ⟨β, a⟩ ≤ c_{5} + (−\frac{c_{5}}{2} )2 = 0, and thus r_{b} · β ≺ β. Denote the coefficient of b_{2} in β by μ′_{2}, and the coefficient of b_{1} in r_{b_{1}} · β by μ′_{1}; then

μ′_{1} = −2 − 2(−\frac{c_{5}}{2} )c_{5} − 2(−\frac{1}{2})μ′_{2} = c_{5} − 1 + μ′_{2}.

This has to be less than 2 since r_{b_{1}} · β ≺ β, and so by (2.1)(ii) it must be equal to 0, 1 or c_{5}. Since μ′_{2} ∈ ℕ_{0} + c_{5}ℕ_{0}, we deduce that μ′_{2} = 1 and μ′_{1} = c_{5}, contradicting Corollary (3.6)(i). Therefore the coefficient of b in α is greater than 2, and thus greater than or equal to c_{5} + 1. Hence

⟨r_{a} · α, a⟩ ≤ c_{5} + (−\frac{1}{2})1 + (−\frac{c_{5}}{2} )(c_{5} + 1) ≤ −1,

as required.

This leaves us to determine E_{S}^{skew-ind} for S of shape (3) with l = 1, m = 5 and n ≥ 4.

(6.8) Lemma Suppose that S is of shape (3) with m = 5, l = 1 and n ≥ 3. Then for each α ∈ E_{S}^{skew-ind}, there exists a γ ∈ E_{{a_{1},b_{1},b_{2},b_{3}}}^{skew-ind} with γ ⪯ α. (Note that if n ≥ 4, the coefficient of b_{3} in γ has to be 1 or c_{5}.)

Proof. If n = 3, the assertion is trivially true. Suppose now that n ≥ 4, and let β ⪯ α be minimal such that β ∈ E_{S}^{skew-ind}. Then r_{x} · β ≺ β for some x ∈ S. If x = a, minimality of β yields that a_{1} ∉ supp(r_{a_{1}} · β); since the coefficient of b_{1} in β is greater than or equal to 2, we find that ⟨r_{a} · β, a_{1}⟩ ≤ −1, contradicting (2.2)(iii), (iv). Similarly x ≠ b_{n}.

So x = b_{j} for some j ∈ {1, …, n − 1}. Denote the coefficient of b_{j} in r_{b_{j}} · β by μ_{j}; then μ_{j} ∈ {1, c_{5}} by minimality of β and connectedness of the support of r_{b_{j}} · β. If μ_{j} = 1, Proposition (3.4) yields that the coefficient of b_{j+1} in β equals 1, while if μ_{j} = c_{5}, Proposition (5.5) yields that the coefficient of b_{j+1} in β equals c_{5}. Since β ∈ E_{S}^{skew-ind}, the coefficient of b_{j+1} in β has to be greater than or equal to 2 if j + 1 < n; whence j + 1 = n. Now the coefficients of b_{n−1} and b_{n} in r_{b_{n−1}} · β coincide, and it follows that r_{b_{n}}r_{b_{n−1}} · β is an element of E_{S\{b_{n}}}^{skew-ind}. By the inductive hypothesis, r_{b_{n}}r_{b_{n−1}} · β is preceded by a root γ ∈ E_{{a_{1},b_{1},b_{2},b_{3}}}^{skew-ind}, and therefore γ ⪯ α, as required.

An induction on n similar to the one just demonstrated, together with an induction on depth yield the next assertion.

(6.9) Proposition Suppose that S is of shape (3) with m = 5, l = 1 and n ≥ 4. Then the elements of E_{S}^{skew-ind} are as follows:

(i) c_{5}a_{1} + (c_{5} + 1)(b_{1} + ⋯ + b_{n−1}) + b_{n} and c_{5}a_{1} + (c_{5} + 1)(b_{1} + ⋯ + b_{n−1}) + c_{5}b_{n}, (ii) (c_{5} + 1)a_{1} + (2c_{5} + 1)(b_{1} + ⋯ + b_{j−1}) + (c_{5} + 1)(b_{j} + ⋯ + b_{n−1}) + b_{n},

(c_{5} + 1)a_{1} + (2c_{5} + 1)(b_{1} + ⋯ + b_{j−1}) + (c_{5} + 1)(b_{j} + ⋯ + b_{n−1}) + c_{5}b_{n}, (c_{5} + 1)a_{1} + (2c_{5} + 1)(b_{1} + ⋯ + b_{j−1}) + 2c_{5}(b_{j} + ⋯ + b_{n−1}) + c_{5}b_{n}, where 1 ≤ j ≤ n,

(iii) (2c_{5} + 1)a_{1} + (2c_{5} + 1)(b_{1} + ⋯ + b_{j−1}) + (c_{5} + 1)(b_{j} + ⋯ + b_{n−1}) + b_{n}, (2c_{5} + 1)a_{1} + (2c_{5} + 1)(b_{1} + ⋯ + b_{j−1}) + (c_{5} + 1)(b_{j} + ⋯ + b_{n−1}) + c_{5}b_{n}, (2c_{5} + 1)a_{1} + (2c_{5} + 1)(b_{1} + ⋯ + b_{j−1}) + 2c_{5}(b_{j} + ⋯ + b_{n−1}) + c_{5}b_{n}, where 2 ≤ j ≤ n, and

(iv) (2c_{5} + 1)a_{1} + (2c_{5} + 2)b_{1} + (c_{5} + 1)(b_{2} + ⋯ + b_{n−1}) + b_{n}, (2c_{5} + 1)a_{1} + (2c_{5} + 2)b_{1} + (c_{5} + 1)(b_{2} + ⋯ + b_{n−1}) + c_{5}b_{n}, (2c_{5} + 1)a_{1} + (3c_{5} + 1)b_{1} + 2c_{5}(b_{2} + ⋯ + b_{n−1}) + c_{5}b_{n}, (2c_{5} + 2)a_{1} + (3c_{5} + 1)b_{1} + 2c_{5}(b_{2} + ⋯ + b_{n−1}) + c_{5}b_{n}.

(6.10) Corollary

(1) Suppose that S is of the shape described in (6.2) with m = 5. Then the elements of

E_{S}^{indec} are as follows:

(i) c_{5}α + (c_{5} + 1)(b_{1} + ⋯ + b_{n−1}) + c_{5}β_{n}, where α ∈ E_{A∪{a_{L},...,a_{1}}} with coefficient 1 for a_{1}, while n ∈ {1, …, N} and β_{n} is in E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n}, and

(ii) c_{5}α_{l} + (c_{5} + 1)(a_{l−1} + ⋯ + a_{1}) + c_{5}β, where l ∈ {1, …, L} and α_{l} ∈ E_{A∪{a_{L},...,a_{l}}} with coefficient 1 for a_{l}, while β is in E_{{b_{1},...,b_{N} }∪B} with coefficient 1 for b_{1}.

(2) Suppose that S is of the shape described in (6.3) with m = 5 and L ≥ 2. Then the elements of E_{S}^{indec} are as follows:

(i) c_{m}(a_{L} + ⋯ + a_{1}) + (c_{5} + 1)(b_{1} + ⋯ + b_{n−1}) + c_{m}β_{n},

where n ∈ {1, …, N} and β_{n} ∈ E_{{b_{n},...,b_{N} }∪B} with coefficient 1 for b_{n},

(ii) a_{L} + (c_{5} + 1)(a_{L−1} + ⋯ + a_{1}) + c_{m}β,

where β ∈ E_{{b_{1},...,b_{N} }∪B} with coefficient 1 for b_{1}, and

(iii) c_{m}(a_{L} + ⋯ + a_{l+1}) + (c_{5} + 1)(a_{l} + ⋯ + a_{1}) + c_{m}β,

where l ∈ {1, …, L} and β ∈ E_{{b_{1},...,b_{N} }∪B} with coefficient 1 for b_{1}.

(3) Suppose that S is of the shape described in (6.4) with m = 5 and L, N ≥ 2. Then the

elements of E_{S}^{indec} are as follows: (i) c_{m}(a_{L} + ⋯ + a_{1}) + (c_{5} + 1)(b_{1} + ⋯ + b_{N−1}) + b_{N}, (ii) c_{m}(a_{L}+⋯+a_{1})+(c_{5}+1)(b_{1}+⋯+b_{n−1})+c_{5}(b_{n}+⋯+b_{N}), where n ∈ {1, …, N}, (iii) a_{L} + (c_{5} + 1)(a_{L−1} + ⋯ + a_{1}) + c_{5}(b_{1} + ⋯ + b_{N}), (iv) c_{m}(a_{1} +⋯+a_{l})+(c_{5} +1)(a_{l−1} +⋯+a_{1})+c_{5}(b_{1} +⋯+b_{N}), where l ∈ {1, …, L}.

Acknowledgements

This paper is taken from my dissertation. I would like to thank my supervisor Dr.R. B. Howlett for his support throughout my candidature and the writing of this paper. The work was completed while I was supported by an Overseas Postgraduate Research Award and a University Postgraduate Research Award.

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