February 9

Definition 1 Let W be a group W generated by a finite set of involutions S = {s_{1}, s_{2}, … s_{n}}. The set S is said to satisfy the exchange condition if whenever

ℓ (s_{k_{1}}s_{k_{2}} … s_{k_{m}}) = m
ℓ (s_{k_{0}}s_{k_{1}}s_{k_{2}} … s_{k_{m}}) < m + 1

we have an integer j with

s_{k_{0}}s_{k_{1}} … s_{k_{j−1}} = s_{k_{1}}s_{k_{2}} … s_{k_{j}}.

Definition 2 A finite Coxeter group W is a finite group with presentation

W = ⟨s_{1}, s_{2}, …, s_{n} : (s_{i}s_{j})^{m_{i,j}} = 1⟩

with m_{i,i} = 1 and m_{i,j} = m_{j,i} > 1 for i ≠ j.

Lemma 1 Let W be a finite group generated by a set S = {s_{1}, s_{2}, …, s_{n}} of involutions. Let m_{i,j} = |s_{i}s_{j}|. Then no two distinct reduced words in s_{i} and s_{j} are equal in W except

\begin{cases}(s_{i}s_{j})^{k_{i,j}} = (s_{j}s_{i})^{k_{i,j}} & if m_{i,j} = 2k_{i,j} \\ (s_{i}s_{j})^{k_{i,j}} s_{i} = (s_{j}s_{i})^{k_{i,j}} s_{j} & if m_{i,j} = 2k_{i,j} + 1. \end{cases}

Theorem 1 (Matsumoto) Assume now that S satisfies the exchange condition. Let M be a monoid with identity e and suppose {m_{1}, m_{2}, … m_{n}} are elements of M satisfying the relations (m_{i}m_{j})^{m_{i,j}} = 1. Then there exists a well-defined map f : W → M such that f (1) = e. and if s_{k_{1}}s_{k_{2}} … s_{k_{l}} is reduced, then f (s_{k_{1}}s_{k_{2}} … s_{k_{l}}) = m_{k_{1}}m_{k_{2}} … m_{k_{l}}

Proof. Define f : W → M by setting f (1) = f (empty word) = e and f (s_{k_{1}}s_{k_{2}} … s_{k_{l}}) = m_{k_{1}}m_{k_{2}} … m_{k_{l}} if s_{k_{1}}s_{k_{2}} … s_{k_{l}} is reduced. We need to show that f is well defined, that is, if

s_{i_{1}}s_{i_{2}} … s_{i_{l}} = s_{j_{1}}s_{j_{2}} … s_{j_{l}}

are reduced, then

m_{i_{1}}m_{i_{2}} … m_{i_{l}} = m_{j_{1}}m_{j_{2}} … m_{j_{l}}.

Theorem 2 Assume now that W is a the GGR associated with the root system Δ having fundamental system Π ⊂ Δ_{+} and fundamental reflection set S = {s_{j} : s_{j} ∈ Π}. Then W is a Coxeter group.

Proof. If G is any other group with elements g_{1}, g_{2}, …, g_{n} satisfying (g_{i}g_{j})^{m_{i,j}} = 1, we need to show that there is a homomorphism f : W → G with f (s_{i}) = g_{i}. This will show that

W = ⟨s_{1}, s_{2}, …, s_{n} : (s_{i}s_{j})^{m_{i,j}} = 1⟩ . 1

Theorem 1 provides such a map. We need to show that f is a homomorphism. For this, is suffices to show that f (ws_{i}) = f (w) f (s_{i}).

Let w = s_{i_{1}}s_{i_{2}} … s_{i_{k}} be an expression. If s_{i}w is reduced, then

f (s_{i}w) = f (s_{i}s_{i_{1}}s_{i_{2}} … s_{i_{k}}) = m_{i}m_{i_{1}}m_{i_{2}} … m_{i_{k}} = f (s_{i}) f (w)

by the properties of f.

If s_{i}w is not reduced, then let w′ = s_{i}w and take w̃′ to be a reduced expression for w′. Notice that ℓ (w′) = k − 1 since we always have ℓ (s_{i}w) = ℓ (w) ± 1. Then ℓ (s_{i}w̃′) = k since s_{i}w′ = w and we know that ℓ (w) = k. This says that s_{i}w̃′ is reduced. Then by the above argument, we have

f (w) = f (s_{i}w̃′) = f (s_{i}) f \binom{}{w′} = f (s_{i}) f (s_{i}w) .

Since f (s_{i}) is an involution, we have f (s_{i}) f (w) = f (s_{i}w).

Next, we develop a converse to Theorem 2. Now let W be a Coxeter group with generator set S = {s_{1}, s_{2} … s_{n}}.

Definition 3 The Coxeter graph of a Coxeter group W is the graph with vertices S having an edge between s_{i} and s_{j} if s_{i} does not commute with s_{j}. We label this edge m_{i,j} = |s_{i}s_{j}|.

Definition 4 W is indecomposable if its graph is connected.

Observation 1 For two roots α_{i}, α_{j} ∈ Δ, we have ⟨α_{i}, α_{j}⟩ = 0 iff s_{i} commutes with s_{j}. To see this, recall that V = ⟨α_{i}⟩ ⊕ ⟨α_{i}⟩^{⊥} and assume ⟨α_{i}, α_{j}⟩ = 0. Then

s_{i}s_{j}α_{i} = s_{i} (α_{i} − 0) = −α_{i}
s_{j}s_{i}α_{i} = s_{j} (−α_{i}) = −α_{i} − 0

and if β ∈ ⟨α_{i}⟩^{⊥}, then

s_{i}s_{j}β = s_{i} (β − Xα_{j}) = β − Xα_{j} = s_{j} (β)
s_{j}s_{i}β = s_{j} (β)

so that s_{i} and s_{j} commute on all of V. Conversely, if s_{i} and s_{j} commute on all of V, then Therefore, the Coxeter graph divides Δ into mutually orthogonal subroot systems.

Observation 2 The number m_{i,j} labeling the edge between s_{i} and s_{j} is > 2 since s_{i} and s_{j} commute iff m_{i,j} = 2.

Let V be a vector space over ℝ of dimension n = |S| with basis {v_{1}, v_{2}, … v_{n}}. Define the bilinear form B : V × V → ℝ by

B (e_{i}, e_{j}) = − cos (\frac{π}{m_{i,j}} )

for 1 ≤ i, j ≤ n and define the linear transformation T (s_{j}) ∈ GL (V) by

T (s_{j}) (v) = v − 2B (v, e_{j}) e_{j}

for 1 ≤ j ≤ n.

Theorem 3 T : S → End (V) can be extended to a faithful representation T : W → GL (V). T is called the reflection representation