Parametrized Braid Groups of Chevalley Groups

Jean-Louis Loday^{1} and Michael R. Stein^{2}

Received: February 17, 2005

Communicated by Ulf Rehmann

Abstract. We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. This first part treats the case of the root system A_{n}; in the second part we prove a similar theorem for the root system D_{n}.

2000 Mathematics Subject Classification: Primary 20F36; Secondary 19Cxx; 20F55. Keywords and Phrases: Braid group, Steinberg group, parametrized braid group, root system.

1 Introduction

Suppose that the strands of a braid are painted and that the paint from a strand spills onto the strand beneath it, modifying the color of the lower strand as in the picture below.

[figure: u; u; v + au; v; a = coefficient of spilling]

Figure 1 This gives rise, for any ring A, to the parametrized braid group Br_{n}(A), which is generated by elements y_{i}^{a}, where i is an integer, 1 ≤ i ≤ n − 1, and a is an element of A, subject to the relations

1Partially supported by Northwestern University in 2002.

2Partially supported by Université Louis Pasteur in 1998-99.

Documenta Mathematica 10 (2005) 391–416

(A1)  y_{i}^{a}y_{i}^{0}y_{i}^{b} = y_{i}^{0}y_{i}^{0}y_{i}^{a+b}
(A1 × A1)  y_{i}^{a}y_{j}^{b} = y_{j}^{b}y_{i}^{a}  if |i − j| ≥ 2,
(A2)  y_{i}^{a}y_{i+1}^{b}y_{i}^{c} = y_{i+1}^{c}y_{i}^{b+ac} y_{i+1}^{a}
for any a, b, c ∈ A

A variation of this group first appeared in [L]. The choice of names for these relations will be explained in section 2 below. The derivation of these relations from the painted braid model can be seen in Figures 2 and 3 below.) Observe that when A = {0} (the zero ring), one obtains the classical Artin braid group Br_{n}, whose presentation is by generators y_{i}, 1 ≤ i ≤ n − 1, and relations

(A1 × A1)  y_{i}y_{j} = y_{j}y_{i}  if |i − j| ≥ 2,
(A2)  y_{i}y_{i+1}y_{i} = y_{i+1}y_{i} y_{i+1}

A question immediately comes to mind: does Figure 1 correctly reflect the elements of the parametrized braid group? Up to equivalence, a picture would be completely determined by a braid and a linear transformation of the set of colors. This linear transformation lies in the subgroup E_{n}(A) of elementary matrices. Hence if the elements of the parametrized braid group correspond exactly to the pictures, then the group should be the semi-direct product of E_{n}(A) by Br_{n}. We will show that this is almost the case: we only need to replace E_{n}(A) by the Steinberg group St_{n}(A) (cf. [St] [Stb]).

Theorem. For any ring A there is an isomorphism

Br_{n}(A) ≅ St_{n}(A) ⋊ Br_{n} ,

where the action of Br_{n} is via the symmetric group S_{n}.

The quotient of Br_{n}(A) by the relation y_{i}^{0}y_{i}^{0} = 1 is the group studied by Kassel and Reutenauer [K-R] (in this quotient group, our relation (A1) becomes y_{i}^{a}y_{i}^{0}y_{i}^{b} = y_{i}^{a+b}, which is exactly the relation used in [K-R] in place of (A1)). They show that this quotient is naturally isomorphic to the semi-direct product St_{n}(A) ⋊ S_{n} of the Steinberg group with the symmetric group. So our theorem is a lifting of theirs.

The proof consists in constructing maps both ways. The key point about their existence is a technical result called the Pure Braid Lemma (cf. 2.2.1). It says that a certain type of parametrized braid commute with the pure braid group.

In the first part of the paper we give a proof of the above theorem which corresponds to the family of Coxeter groups A_{n}. In the second part we prove a similar theorem for the family D_{n}, cf. section 3. The Pure Braid Lemma in the D_{n} case is based on a family of generators of the pure braid group found by Digne and Gomi [D-G]. We expect to prove similar theorems for the other Coxeter groups. A related result has been announced in [Bon].

2 The parametrized braid group

We introduce the parametrized braid group of the family of Coxeter groups A_{n−1}.

Definition 2.0.1. Let A be a ring (not necessarily unital nor commutative). The parametrized braid group Br_{n}(A) is generated by the elements y_{i}^{a}, where i is an integer, 1 ≤ i ≤ n − 1, and a is an element of A, subject to the relations

(A1)  y_{i}^{a}y_{i}^{0}y_{i}^{b} = y_{i}^{0}y_{i}^{0}y_{i}^{a+b}
(A1 × A1)  y_{i}^{a}y_{j}^{b} = y_{j}^{b}y_{i}^{a}  if |i − j| ≥ 2,
(A2)  y_{i}^{a}y_{i+1}^{b}y_{i}^{c} = y_{i+1}^{c}y_{i}^{b+ac} y_{i+1}^{a}

for any a, b, c ∈ A.

The geometric motivation for the defining relations of this group, and its connection with braids, can be seen in the following figures in which u, v, w (the colors) are elements of A, and a, b, c are the coefficients of spilling. Relation (A1) comes from Figure 2. Relation (A1 × A1) arises because the actions of y_{i}^{a} and of y_{j}^{b} on the strands of the braid are disjoint when |i − j| ≥ 2, so that these two elements commute. Relation (A2) derives from Figure 3.

394 Jean-Louis Loday and Michael R. Stein

\begin{aligned}u & v & [figure] & [figure] & [figure] & y_{i}^{b} & u & v + bu & [figure] & [figure] & [figure] & y_{i}^{0} & u & v + bu & [figure] & [figure] & [figure] & y_{i}^{a} & u & v + bu + au & = & u & v & [figure] & [figure] & [figure] & y_{i}^{a+b} & v + (a + b)u & u & [figure] & [figure] & [figure] & y_{i}^{0} & u & v + (a + b)u & [figure] & [figure] & [figure] & y_{i}^{0} & v + (a + b)u & u \\ & Figure 2 \end{aligned}

Documenta Mathematica 10 (2005) 391–416

[figure: u; v; w; y_{i}^{c}; v + cu; u; w; y_{i+1}^{b}; v + cu; w + bu; u; y_{i}^{a}; w + bu; +a(v + cu); v + cu; u; =; u; v; w; y_{i+1}^{a}; u; w + av; v; y_{i}^{b+ac}; w + av; +(b + ac)u; u; v; y_{i+1}^{c}; w + av; +(b + ac)u; v + cu; u]

Figure 3

2.1 Technical lemmas

Let Φ be the root system A_{n} or D_{n} and let A be a ring (not necessarily unital) which is supposed to be commutative in the D_{n} case. Let Δ be a simple subsystem of the root system Φ. Elements of Δ are denoted by α or α_{i}. The image of b ∈ Br(Φ) in the Weyl group W (Φ) is denoted \overline{b}. The parametrized braid group Br(D_{n}, A) is defined in 3.1.1.

Lemma 2.1.1. The following relations in Br(Φ, A) are consequences of relation (A1):

\begin{aligned}(y_{α}^{0}y_{α}^{0})y_{α}^{a} & = & y_{α}^{a}(y_{α}^{0}y_{α}^{0}), \\ y_{α}^{a}(y_{α}^{0})^{−1}y_{α}^{b} & = & y_{α}^{a+b}, \\ y_{α}^{−a} & = & y_{α}^{0}(y_{α}^{a})^{−1}y_{α}^{0}. \end{aligned}

Proof. Replacing b by 0 in (A1) shows that y_{α}^{0}y_{α}^{0} commutes with y_{α}^{a}. From this follows

y_{α}^{a}(y_{α}^{0})^{−1}y_{α}^{b} = (y_{α}^{0}y_{α}^{0})^{−1}y_{α}^{a}(y_{α}^{0}y_{α}^{0})(y_{α}^{0})^{−1}y_{α}^{b}
= (y_{α}^{0}y_{α}^{0})^{−1}y_{α}^{a}y_{α}^{0}y_{α}^{b}
= (y_{α}^{0}y_{α}^{0})^{−1}y_{α}^{0}y_{α}^{0}y_{α}^{a+b}
= y_{α}^{a+b}.

Putting b = −a in the second relation yields the third relation.

Lemma 2.1.2. Assume that the Pure Braid Lemma holds for the root system Φ. Suppose that α ∈ Δ and b ∈ Br(Φ) are such that b̄(α) ∈ Δ (where b̄ ∈ W (Φ) denotes the image of b). Then for any a ∈ A one has

by_{α}^{a}(y_{α}^{0})^{−1}b^{−1} = y_{b̄(α)}^{a}(y_{b̄(α)}^{0})^{−1} in Br(Φ, A).

Proof. First let us show that there exists b′ ∈ Br(Φ) such that b′y_{α}^{a}b′^{−1} = y_{b̄(α)}^{a}. The two roots α and b̄(α) have the same length, hence they are connected, in the Dynkin diagram, by a finite sequence of edges with m = 3 (cf. [Car, Lemma 3.6.3]). Therefore it is sufficient to prove the existence of b′ when α and b̄(α) are adjacent. In that case α and b̄(α) generate a subsystem of type A_{2}; we may assume α = α_{1} and b̄(α) = α_{2} ∈ A_{2}; and we can use the particular case of relation (A2), namely

y_{α_{1}}^{0}y_{α_{2}}^{0}y_{α_{1}}^{a} = y_{α_{2}}^{a}y_{α_{1}}^{0}y_{α_{2}}^{0}

to show that

y_{α_{1}}^{0}y_{α_{2}}^{0}y_{α_{1}}^{a}(y_{α_{2}}^{0})^{−1}(y_{α_{1}}^{0})^{−1} = y_{α_{2}}^{a}.

(Here b′ = y_{α_{1}}^{0}y_{α_{2}}^{0}, α = α_{1} = −ϵ_{1} + ϵ_{2}, b̄′(α) = α_{2} = −ϵ_{2} + ϵ_{3}.) To conclude the proof of the Lemma it is sufficient to show that

by_{α}^{a}(y_{α}^{0})^{−1}b^{−1} = y_{α}^{a}(y_{α}^{0})^{−1}

whenever b(α) = α. According to [H, Theorem, p. 22], b̄ is a product of simple reflections σ_{α_{i}} for α_{i} ∈ Δ which are not connected to α in the Dynkin diagram of Δ. Hence we can write b as the product of an element in the pure braid group and generators y_{α_{i}} which commute with y_{α}^{a} by relation (A1 × A1). Since we have assumed that the Pure Braid Lemma holds for Φ, we can thus conclude that by_{α}^{a}(y_{α}^{0})^{−1}b^{−1} = y_{α}^{a}(y_{α}^{0})^{−1} as desired.

2.2 Braid group and pure braid group

The group Br_{n}(0) = Br_{n} is the classical Artin braid group with generators y_{i} , 1 ≤ i ≤ n − 1, and relations

\begin{aligned}y_{i} y_{j} & = & y_{j} y_{i} , |i − j| ≥ 2, \\ y_{i} y_{i+1} y_{i} & = & y_{i+1} y_{i} y_{i+1} . \end{aligned}

The quotient of Br_{n} by the relations y_{i}y_{i} = 1 , 1 ≤ i ≤ n − 1 is the symmetric group S_{n}; the image of b ∈ Br_{n} in S_{n} is denoted by b̄. The kernel of the surjective homomorphism Br_{n} → S_{n} is the pure braid group, denoted P Br_{n}. It is generated by the elements

a_{j,i} := y_{j}y_{j−1} ⋯ y_{i}y_{i} ⋯ y_{j−1}y_{j},

for n ≥ j ≥ i ≥ 1, ([Bir]; see Figure 4 below).

[figure: i; j; ⋯]

Figure 4: the pure braid a_{j,i}

Lemma 2.2.1 (Pure Braid Lemma for A_{n−1}). Let y_{k}^{a} be a generator of Br_{n}(A) and let ω ∈ P Br_{n} = P Br_{n}(0). Then there exists ω′ ∈ Br_{n}, independent of a, such that

y_{k}^{a} ω = ω′ y_{k}^{a}.

Hence for any integer k and any element a ∈ A, the element y_{k}^{a}(y_{k}^{0})^{−1} ∈ P Br_{n}(A) commutes with every element of the pure braid group P Br_{n}.

Notation. Before beginning the proof of Lemma 2.2.1, we want to simplify our notation. We will abbreviate

\begin{aligned}y_{k}^{0} & = & k \\ (y_{k}^{0})^{−1} & = & k^{−1} \\ y_{k}^{a} & = & k^{a} \end{aligned}

Note that k^{−1} does not mean k^{a} for a = −1. If there exist ω′ and ω′′ ∈ Br_{n} such that k^{a}ω = ω′j^{a}ω′′, we will write

k^{a}ω ∼ j^{a}ω′′.

Observe that ∼ is not an equivalence relation, but it is compatible with multiplication on the right by elements of Br_{n}: if η ∈ Br_{n},

k^{a}ω ∼ j^{a}ω′′ ⇔ k^{a}ω = ω′j^{a}ω′′ ⇔ k^{a}ωη = ω′j^{a}ω′′η ⇔ k^{a}ωη ∼ j^{a}ω′′η

For instance

k^{a} k k ∼ k^{a}  by Lemma 2.1.1,
k^{a} (k − 1) k ∼ (k − 1)^{a}  by relation (A2),
k^{a} (k − 1) ∼ (k − 1)^{a} k by relation (A2).

Proof of Lemma 2.2.1. It is clear that the first assertion implies the second one: since ω′ is independent of a, we can set a = 0 to determine that ω′ = y_{k}^{0} ω (y_{k}^{0})^{−1}. Substituting this value of ω′ in the expression y_{k}^{a} ω = ω′ y_{k}^{a} completes the proof.

To prove the first assertion, we must show, in the notation just introduced, that k^{a}ω ∼ k^{a} for every ω ∈ P Br_{n}, and it suffices to show this when ω is one of the generators a_{j,i} = j (j − 1) ⋯ i i ⋯ (j − 1) j above. Since y_{k}^{a} commutes with y_{i}^{0} for all i ≠ k − 1, k, k + 1, it commutes with a_{j,i} whenever k < i − 1 or whenever k > j + 1. So we are left with the following 3 cases:

(1a)  (j + 1)^{a}a_{j,i} ∼ (j + 1)^{a}
(1b)  j^{a}a_{j,i} ∼ j^{a}
(1c)  k^{a}a_{j,i} ∼ k^{a} i − 1 ≤ k ≤ j − 1

Our proof of case (1a) is by induction on the half-length of ω = a_{j,i}. When i = k − 1, ω = (k − 1) (k − 1), and we have

k^{a} (k − 1) (k − 1) = k^{a} (k − 1) k^{−1} k (k − 1)
∼ (k − 1)^{a} k (k − 1)
∼ k^{a} .

and more generally,

k^{a} (k − 1) (k − 2) ⋯ (k − 2) (k − 1)
∼ (k − 1)^{a} k (k − 2) ⋯ (k − 2)(k − 1)
= (k − 1)^{a} (k − 2) ⋯ (k − 2) k (k − 1)
∼ (k − 1)^{a} k (k − 1)  (by induction),
∼ k^{a} .

The proof of case (1b) is also by induction on the half-length of ω = a_{j,i}. When i = k, ω = k k, and we have

k^{a}k k ∼ k^{a} by Lemma 2.1.1.

Then

k^{a}k (k − 1) ⋯ (k − 1) k

= k^{a}k (k − 1) k k^{−1} (k − 2) ⋯ (k − 2) (k − 1) k
∼ k^{a}(k − 1) k (k − 1) (k − 2) ⋯ (k − 2) k^{−1} (k − 1) k
∼ (k − 1)^{a}(k − 1) (k − 2) ⋯ (k − 2) (k − 1) k (k − 1)^{−1}
∼ (k − 1)^{a} k (k − 1)^{−1}  (by induction)
∼ k^{a}.

For case (1c) it is sufficient to check the cases

(2a) ω = (k + 1) (k + 1)
(2b) ω = (k + 1) k k (k + 1)
(2c) ω = (k + 1) k (k − 1) ⋯ (k − 1) k (k + 1)

which are proved as follows:

k^{a}(k + 1) (k + 1) = k^{a}(k + 1) k^{−1} k (k + 1)
∼ (k + 1)^{a} k (k + 1)
∼ k^{a}  Case (2a)
k^{a}(k + 1)kk(k + 1) ∼ (k + 1)^{a}k (k + 1)
∼ k^{a}  Case (2b)

k^{a}(k + 1) k (k − 1) ⋯ (k − 1) k (k + 1)
∼ (k + 1)^{a}(k − 1) ⋯ (k − 1) k (k + 1)
∼ (k + 1)^{a}k (k + 1)
∼ k^{a}  Case (2c)

Proposition 2.2.2. For any ω ∈ Br_{n}(0) the element ωy_{k}^{a}(y_{k}^{0})^{−1}ω^{−1} depends only on the class ω̄ of ω in S_{n}. Moreover if ω̄(j) = k and ω̄(j + 1) = k + 1, then ωy_{k}^{a}(y_{k}^{0})^{−1} ω^{−1} = y_{j}^{a}(y_{j}^{0})^{−1}.

Proof. The first statement is a consequence of Lemma 2.2.1 since the Weyl group W (A_{n−1}) = S_{n} is the quotient of Br_{n} by P Br_{n}. The second part is a consequence of relation (A1 × A1) and the following computation:

2^{−1} 1^{−1} 2^{a}12 = 1^{a}2^{−1} 1^{−1} 2^{−1} 12
= 1^{a}1^{−1} 2^{−1} 1^{−1} 12
= 1^{a}1^{−1} .
□

2.3 The Steinberg group and an action of the Weyl group

When Φ = A_{n−1}, the Steinberg group of the ring A is well-known and customarily denoted St_{n}(A). In that case it is customary to write x_{ii+1}^{a} = x_{ii+1}(a) for the element x_{α}(a), α = ϵ_{i} − ϵ_{i+1} ∈ Δ, and, more generally, x_{ij}^{a} when α = ϵ_{i} − ϵ_{j} ∈ A_{n−1}.

Definition 2.3.1 ([Stb]). The Steinberg group of the ring A, denoted St_{n}(A), is presented by the generators x_{ij}^{a}, 1 ≤ i, j ≤ n, i ≠ j, a ∈ A subject to the relations

(St0)  x_{ij}^{a}x_{ij}^{b} = x_{ij}^{a+b}
(St1)  x_{ij}^{a}x_{kl}^{b} = x_{kl}^{b}x_{ij}^{a},  i ≠ l, j ≠ k
(St2)  x_{ij}^{a}x_{jk}^{b} = x_{jk}^{b}x_{ik}^{ab}x_{ij}^{a},  i ≠ k.

We should make two observations about this definition. First, it follows from (St0) that x_{ij}^{0} = 1. Second, relation (St2) is given in a perhaps unfamiliar form. We have chosen this form, which is easily seen to be equivalent to (R2), because of its geometric significance (cf. [K-S] for the relationship with the Stasheff polytope), and for the simplification it brings in computation.

The Weyl group W (A_{n−1}) is the symmetric group S_{n}. Its action on the Steinberg group is induced by the formula

(3) σ · x_{ij}^{a} := x_{σ(i)σ(j)}^{a}, σ ∈ S_{n}, a ∈ A

2.4 The main result

Theorem 2.4.1. For any (not necessarily unital) ring A the map

ϕ : Br_{n}(A) → St_{n}(A) ⋊ Br_{n}

from the parametrized braid group to the semi-direct product of the Artin braid group with the Steinberg group induced by ϕ(y_{i}^{a}) = x_{i i+1}^{a} y_{i} is a group isomorphism.

Proof. Step (a). We show that ϕ is a well-defined group homomorphism.

• Relation (A1):

ϕ(y_{i}^{a}y_{i}^{0}y_{i}^{b}) = x_{i i+1}^{a}y_{i} y_{i} x_{i i+1}^{b}y_{i} ,  since x_{i i+1}^{0} = 1,
= y_{i} y_{i} x_{i i+1}^{a}x_{i i+1}^{b}y_{i}  since \overline{y_{i} y_{i}} = 1 ∈ S_{n},
= y_{i} y_{i} x_{i i+1}^{a+b}y_{i}  by (St0),
= ϕ(y_{i}^{0}y_{i}^{0}y_{i}^{a+b}).

• Relation (A1 × A1) follows immediately from (St1).

• Relation (A2) is proved by using the relations of Br_{n} and the 3 relations (St0), (St1), (St2) as follows:

ϕ(y_{i}^{a}y_{i+1}^{b}y_{i}^{c}) = x_{i i+1}^{a}y_{i} x_{i+1 i+2}^{b} \underbrace{y_{i+1} x_{i i+1}^{c}} y_{i}
= x_{i i+1}^{a}y_{i} \underbrace{x_{i+1 i+2}^{b}x_{i i+2}^{c}} y_{i+1} y_{i} ,
= x_{i i+1}^{a} \underbrace{y_{i} x_{i i+2}^{c}} x_{i+1 i+2}^{b} y_{i+1} y_{i} ,
= \underbrace{x_{i i+1}^{a}x_{i+1 i+2}^{c}} \underbrace{y_{i} x_{i+1 i+2}^{b}} y_{i+1} y_{i} ,
= x_{i+1 i+2}^{c}x_{i i+2}^{ac} \underbrace{x_{i i+1}^{a}x_{i i+2}^{b}} \underbrace{y_{i} y_{i+1} y_{i}} ,
= x_{i+1 i+2}^{c}x_{i i+2}^{ac+b} \underbrace{x_{i i+1}^{a}y_{i+1}} y_{i} y_{i+1} ,
= x_{i+1 i+2}^{c} \underbrace{x_{i i+2}^{ac+b} y_{i+1}} \underbrace{x_{i i+2}^{a} y_{i} y_{i+1}} ,
= \underbrace{x_{i+1 i+2}^{c}y_{i+1}} \underbrace{x_{i i+1}^{ac+b}y_{i}} \underbrace{x_{i+1 i+2}^{a} y_{i+1}} ,
= ϕ(y_{i+1}^{c} y_{i}^{b+ac} y_{i+1}^{a}).

Step (b). This is the Pure Braid Lemma 2.2.1 for A_{n}.

Step (c). We construct a homomorphism ψ : St_{n}(A) ⋊ Br_{n} → Br_{n}(A). We first construct ψ : St_{n}(A) → Ker π, where π is the surjection Br_{n}(A) → Br_{n}, by setting

ψ(x_{ij}^{a}) := ω y_{k}^{a}(y_{k}^{0})^{−1} ω^{−1}

where ω is an element of Br_{n} such that ω̄(k) = i and ω̄(k + 1) = j (for instance, ψ(x_{12}^{a}) = y_{1}^{a}(y_{1}^{0})^{−1} and ψ(x_{13}^{a}) = y_{2}^{0}(y_{1}^{a}(y_{1}^{0})^{−1})(y_{2}^{0})^{−1}). Observe that this definition does not depend on the choice of ω̄ (by Lemma 2.1.2), and does not depend on how we choose a lifting ω of ω̄ (by the Pure Braid Lemma 2.2.1). In order to show that ψ is a homomorphism, we must demonstrate that the Steinberg relations are preserved.

• Relation (St0): it suffices to show that ψ(x_{12}^{a}x_{12}^{b}) = ψ(x_{12}^{a+b}),

ψ(x_{12}^{a}x_{12}^{b}) = y_{1}^{a}(y_{1}^{0})^{−1}y_{1}^{b}(y_{1}^{0})^{−1} = y_{1}^{a}(y_{1}^{0})^{−2}y_{1}^{0}y_{1}^{b}(y_{1}^{0})^{−1}
= (y_{1}^{0})^{−2}y_{1}^{a}y_{1}^{0}y_{1}^{b}(y_{1}^{0})^{−1}  by 2.2.1,
= y_{1}^{a+b}(y_{1}^{0})^{−1}  by (A1),
= ψ(x_{12}^{a+b}).

• Relation (St1): it suffices to show that ψ(x_{12}^{a}x_{34}^{b}) = ψ(x_{34}^{b}x_{12}^{a}) and that ψ(x_{12}^{a}x_{13}^{b}) = ψ(x_{13}^{b}x_{12}^{a}). The first case is an immediate consequence of the Pure Braid Lemma 2.2.1 and of relation (A1 × A1). Let us prove the second case, which relies on the Pure Braid Lemma 2.2.1 and relation (A2):

ψ(x_{12}^{a}x_{13}^{b}) = \underbrace{y_{1}^{a}(y_{1}^{0})^{−1}} y_{2}^{0}y_{1}^{b}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}
= (y_{1}^{0})^{−2}y_{1}^{a} \underbrace{y_{1}^{0}y_{2}^{0}y_{1}^{b}}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}
= (y_{1}^{0})^{−2} \underbrace{y_{1}^{a}y_{2}^{b}y_{1}^{0}} \underbrace{y_{2}^{0}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}}
= (y_{1}^{0})^{−2}y_{2}^{0}y_{1}^{b} \underbrace{y_{2}^{a}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}} y_{1}^{0}
= (y_{1}^{0})^{−2}y_{2}^{0}y_{1}^{b}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1} \underbrace{y_{1}^{a}y_{1}^{0}}
= \underbrace{(y_{1}^{0})^{−2}y_{2}^{0}y_{1}^{b}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}(y_{1}^{0})^{2}} \underbrace{y_{1}^{a}(y_{1}^{0})^{−1}}
= ψ(x_{13}^{b})ψ(x_{12}^{a}).

• Relation (St2): it suffices to show that ψ(x_{12}^{a}x_{23}^{b}) = ψ(x_{23}^{b}x_{13}^{ab}x_{12}^{a}).

ψ(x_{12}^{a}x_{23}^{b}) = y_{1}^{a}(y_{1}^{0})^{−1} \underbrace{y_{2}^{b}}(y_{2}^{0})^{−1}
= y_{1}^{a} \underbrace{(y_{1}^{0})^{−1}y_{2}^{b}y_{1}^{0}}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}
= \underbrace{y_{1}^{a}y_{2}^{0}y_{1}^{b}} \underbrace{(y_{2}^{0})^{−1}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}}
= y_{2}^{b}  y_{1}^{ab}  y_{2}^{a}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}(y_{1}^{0})^{−1}
= y_{2}^{b}(y_{2}^{0})^{−1}y_{2}^{0}y_{1}^{ab}(y_{1}^{0})^{−1} \underbrace{y_{1}^{0}y_{2}^{a}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}}(y_{1}^{0})^{−1}
= \underbrace{y_{2}^{b}(y_{2}^{0})^{−1}} \underbrace{y_{2}^{0}y_{1}^{ab}(y_{1}^{0})^{−1}(y_{2}^{0})^{−1}} \underbrace{y_{1}^{a}(y_{1}^{0})^{−1}}
= ψ(x_{23}^{b})ψ(x_{13}^{ab})ψ(x_{12}^{a}),

as a consequence of relation (A2). From 2.2.2 it follows that the action of an element of Br_{n} by conjugation on Ker π depends only on its class in S_{n}. The definition of ψ on St_{n}(A) makes clear that it is an S_{n}-equivariant map. Defining ψ on Br_{n} by ψ(y_{α}) = y_{α}^{0} ∈ Br_{n}(0) yields a group homomorphism

ψ : St_{n}(A) ⋊ Br_{n} → Ker π ⋊ Br_{n} = Br_{n}(A).

The group homomorphisms ϕ and ψ are clearly inverse to each other since they interchange y_{α}^{a} and x_{α}^{a}y_{α}. Hence they are both isomorphisms, as asserted. □

Corollary 2.4.2 (Kassel-Reutenauer [K-R]). The group presented by generators y_{i}^{a}, 1 ≤ i ≤ n − 1, a ∈ A, and relations

(y_{i}^{0})^{2} = 1
y_{i}^{a}(y_{i}^{0})^{−1}y_{i}^{b} = y_{i}^{a+b}
y_{i}^{a}y_{j}^{b} = y_{j}^{b}y_{i}^{a}  if |i − j| ≥ 2
y_{i}^{a}y_{i+1}^{b}y_{i}^{c} = y_{i+1}^{c}y_{i}^{b+ac} y_{i+1}^{a}

a, b, c ∈ A, is isomorphic to the semi-direct product St_{n}(A) ⋊ S_{n}.

Observe that when the first relation in this Corollary is deleted, the second relation has several possible non-equivalent liftings. The one we have chosen, (A1), is what allows us to prove Theorem 2.4.1.

3 The parametrized braid group in the D_{n} case

In this section we discuss the parametrized braid group Br(D_{n}, A) for a commutative ring A and prove that it is isomorphic to the semi-direct product of the Steinberg group St(D_{n}, A) by the braid group Br(D_{n}, 0).

3.1 The braid group and the parametrized braid group for D_{n}

Let Δ = {α_{2}, α_{2′}, α_{3}, …, α_{n}} be a fixed simple subsystem of a root system of type D_{n}, n ≥ 3. We adopt the notation of [D-G] in which the simple roots on the fork of D_{n} are labeled α_{2}, α_{2′}. The system D_{n} contains 2 subsystems of type A_{n−1} generated by the simple subsystems {α_{2}, α_{3}, …, α_{n}} and {α_{2′}, α_{3}, …, α_{n}}, and, for n ≥ 4, a subsystem of type D_{n−1} generated by the simple subsystem {α_{2}, α_{2′}, α_{3}, …, α_{n−1}}.

[figure: 2′; 2; 3; 4; n; •; •; •; •; •; •; •; …]

Dynkin diagram of D_{n}

The Weyl group W (D_{n}) is generated by the simple reflections {σ_{i} = σ_{α_{i}}|α_{i} ∈ Δ}, with defining relations

\begin{aligned}σ_{i}^{2} & = & 1 \\ (σ_{i}σ_{j})^{m(i,j)} & = & 1 \end{aligned}

for i, j ∈ {2, 2′, 3, …, n}, where

m(i, j) = \begin{cases}2 & if α_{i}, α_{j} are not connected in the Dynkin diagram, \\ 3 & if α_{i}, α_{j} are connected in the Dynkin diagram \end{cases}

Since the only values for m(α, β) are 1, 2 and 3, the group Br(D_{n}, A) involves only relations (A1), (A1 × A1) and (A2).

Definition 3.1.1. The parametrized braid group of type D_{n} with parameters in the commutative ring A, denoted Br(D_{n}, A), is generated by the elements y_{α}^{a}, where α ∈ Δ and a ∈ A. The relations are, for a, b ∈ A and α, β ∈ Δ

(A1)  y_{α}^{a}y_{α}^{0}y_{α}^{b} = y_{α}^{0}y_{α}^{0}y_{α}^{a+b}
(A1 × A1)  y_{α}^{a}y_{β}^{b} = y_{β}^{b}y_{α}^{a}  if m(α, β) = 2
(A2)  y_{α}^{a}y_{β}^{b}y_{α}^{c} = y_{β}^{c}y_{α}^{b+ac}y_{β}^{a}  if m(α, β) = 3 and α < β

Note that the simple roots in D_{n} are ordered so that α_{2′} < α_{3}.

3.2 The Steinberg group of D_{n} and the main result

The roots of D_{n} are {±ϵ_{i} ± ϵ_{j}|1 ≤ i ≠ j ≤ n}, [Car],[H]. The Weyl group W (D_{n}) ≅ (ℤ/2)^{n−1} ⋊ S_{n} [Bour, p. 257, (X)] acts on the roots by permuting the indices (action of S_{n}) and changing the signs (action of (ℤ/2)^{n−1}). For the simple subsystem Δ we take α_{i} = −ϵ_{i−1} +ϵ_{i} for i = 2, ⋯, n, and α_{2′} = ϵ_{1} +ϵ_{2}. If u and v are positive integers and α, β two roots, the linear combination uα + vβ is a root if and only if u = 1 = v, α = ±ϵ_{i} ± ϵ_{j}, β ∓ ϵ_{j} ± ϵ_{k} and ±ϵ_{i} ± ϵ_{k} ≠ 0. In this case the definition of the Steinberg group is as follows.

Definition 3.2.1 ([Stb][St]). The Steinberg group of type D_{n} with parameters in the commutative ring A, denoted St(D_{n}, A), is generated by elements x_{α}^{a}, where α ∈ Φ and a ∈ A, subject to the relations (for a, b ∈ A and α, β ∈ Φ)

(St0)  x_{α}^{a}x_{α}^{b} = x_{α}^{a+b}
(St1)  x_{α}^{a}x_{β}^{b} = x_{β}^{b}x_{α}^{a}  if α + β ∉ D_{n} and α + β ≠ 0,
(St2)  x_{α}^{a}x_{β}^{b} = x_{β}^{b}x_{α+β}^{ab}y_{α}^{a}  if α + β ∈ D_{n}.

The Weyl group W (D_{n}) acts on St(D_{n}, A) by σ · x_{α}^{a} = x_{σ(α)}^{a}, and we can construct the semi-direct product St(D_{n}, A) ⋊ Br(D_{n}) with respect to this action.

Theorem 3.2.2. For any commutative ring A the map

ϕ : Br(D_{n}, A) → St(D_{n}, A) ⋊ Br(D_{n})

induced by ϕ(y_{α}^{a}) = x_{α}^{a} y_{α} is a group isomorphism.

Corollary 3.2.3. The group presented by generators y_{i}^{a}, i = 2′, 2, 3, ⋯, n, a ∈ A and relations

(y_{i}^{0})^{2} = 1
y_{i}^{a}(y_{i}^{0})^{−1}y_{i}^{b} = y_{i}^{a+b}
y_{i}^{a}y_{j}^{b} = y_{j}^{b}y_{i}^{a}  if |i − j| ≥ 2, or i = 2, j = 2′,
y_{i}^{a}y_{i+1}^{b}y_{i}^{c} = y_{i+1}^{c}y_{i}^{b+ac} y_{i+1}^{a}  where i + 1 = 3 when i = 2′

for a, b, c ∈ A, is isomorphic to the semi-direct product St(D_{n}, A) ⋊ W (D_{n}).

Proof of Corollary. For each simple root α_{i} ∈ D_{n}, write y_{i}^{a} for y_{α_{i}}^{a}.

Proof of Theorem 3.2.2. Step (a). Since the relations involved in the definitions of Br(D_{n}, A) and St(D_{n}, A) are the same as the relations in the case of A_{n−1}, the map ϕ is well-defined (cf. Theorem 2.4.1). Step (b). The proof of the Pure Braid Lemma in the D_{n} case will be given below in 3.3. Step (c). Let π : Br(D_{n}, A) → Br(D_{n}) be the projection which sends each a ∈ A to 0 (as usual we identify Br(D_{n}, 0) with Br(D_{n})). We define

ψ : St(D_{n}, A) ⋊ Br(D_{n}) → Br(D_{n}, A) ≅ Ker π ⋊ Br(D_{n})

on the first component by ψ(x_{α}^{a}) = y_{α}^{a}(y_{α}^{0})^{−1} ∈ Ker π for α ∈ Δ. For any α ∈ D_{n} there exists σ ∈ W (D_{n}) such that σ(α) ∈ Δ. Let σ̃ ∈ Br(D_{n}) be a lifting of σ, and define ψ(x_{α}^{a}) = σ̃^{−1}ψ(x_{σ(α)}^{a})σ̃ ∈ Ker π. This element is welldefined since it does not depend on the lifting of σ by the Pure Braid Lemma for D_{n} (Lemma 3.3.2), and does not depend on the choice of σ by Lemma 2.1.2. In order to show that ψ is a well-defined group homomorphism, it suffices to show that the Steinberg relations are preserved. But this is the same verification as in the A_{n−1} case, (cf. Theorem 2.4.1.)

The group homomorphisms ϕ and ψ are inverse to each other since they interchange y_{α}^{a} and x_{α}^{a}y_{α}. Hence they are both isomorphisms. □

3.3 The Pure Braid Lemma for D_{n}

3.3.1 Generators for the Pure Braid Group of D_{n}

In principle, the method of Reidemeister-Schreier [M-K-S] is available to deduce a presentation of P Br(D_{n}) from that of Br(D_{n}). The details have been worked out by Digne and Gomi [D-G], although not in the specificity we need here. From their work we can deduce that the group P Br(D_{n}) is generated by the elements y_{α}^{2}, α ∈ Δ, together with a very small set of their conjugates. For example, P Br(D_{4}) is generated by the 12 elements

2^{2}, 2^{′2}, 3^{2},^{3}2^{2},^{3}2^{′2},^{322′}3^{2}, 4^{2},^{4}3^{2},^{43}2^{′2},^{43}2^{2},^{4322′}3^{2},^{4322′3}4^{2}

where a prefixed exponent indicates conjugation: ^{h}g = hgh^{−1}. Here (and throughout) we use the simplified notations k^{a} = y_{α_{k}}^{a} and k = y_{α_{k}}^{0} similar to those of 2.2.

Proposition 3.3.1. For n ≥ 4, P Br(D_{n}) is generated by the elements

• a_{j,i} = j(j − 1) … (i + 1)ii(i + 1) … (j − 1)j, n ≥ j ≥ i ≥ 2, and

• b_{j,i} = j(j − 1) … 322′3 … (i − 1)ii(i − 1) … 32′23(j − 1)j, n ≥ j ≥ i ≥ 3, where i + 1 = 3 when i = 2′.

Note. Since the notation can be confusing, let us be clear about the definition of these generators in certain special cases:

• When i = j, a_{j,i} = i^{2}.

• When i = 3, b_{j,3} = j(j − 1) … 322′332′2 … (j − 1)j.

Proof of (3.3.1). We work in the case where W = W (D_{n}) in the notation of [D-G]. In the proof of [D-G, Corollary 2.7], we see that P_{W} = U_{n} ⋊ P_{W_{I_{n−1}}} ; taking I_{n} = {s_{1}, s_{2}, s_{2′}, …, s_{n−1}}, n ≥ 4, as on [D-G, p. 10], we see that their P_{W} is equal to (our) P Br(D_{n}) and their P_{W_{I_{n−1}}} is equal to (our) P Br(D_{n−1}). It follows that a set of generators for P Br(D_{n}) can be obtained as the union of a set of generators for P Br(D_{n−1}) with a set of generators for U_{n}. This sets the stage for an inductive argument, since D_{3} = A_{3} (with {α_{2}, α_{3}, α_{2′}} ⊂ D_{3} identified with {α_{1}, α_{2}, α_{3}} ⊂ A_{3}). Because W (D_{n}) is a finite Weyl group, it follows from [D-G, Proposition 3.6], that U_{n} is generated (not just normally generated) by the elements a_{b,s}, and a list of these generators in our case is given on [D-G, p. 10]. The calculations necessary to prove the Pure Braid Lemma for D_{n} are simpler if we replace the Digne-Gomi generators by the equivalent set in which conjugation is replaced by reflection; that is, we replace a generator ^{h}g = hgh^{−1} by hgh′, where if h = y_{i_{1}} … y_{i_{k}}, h′ = y_{i_{k}} … y_{i_{1}}. (We already used this trick in the case of A_{n−1}.) For D_{4}, this procedure yields as generators of P Br(D_{4}) the set

\begin{matrix}2^{2}, 2^{′2}, 3^{2}, 32^{2}3, 32^{′2}3, 322′3^{2}2′23, 4^{2}, 43^{2}4, 432^{2}34, 432^{′2}34, \\ 4322′3^{2}2′234, 4322′34^{2}32′234, \end{matrix}

and, more generally, P Br(D_{n}), n ≥ 4 is generated by the elements stated in the Proposition.

Lemma 3.3.2 (Pure Braid Lemma for D_{n}). Let α_{k} ∈ D_{n}, let y_{k}^{a} be a generator of Br(D_{n}, A), and let ω ∈ P Br(D_{n}). Then there exists ω′ ∈ Br(D_{n}), independent of a, such that

y_{k}^{a} ω = ω′ y_{k}^{a}.

Hence for any integer k and any element a ∈ A, the element y_{k}^{a}(y_{k}^{0})^{−1} ∈ Br(D_{n}, A) commutes with every element of the pure braid group P Br(D_{n}).

Proof. Let us show that the first assertion implies the second one. Let ω ∈ P Br(D_{n}) ⊂ P Br(D_{n}, A). By the first assertion of the Lemma we have

y_{k}^{a}ω = ω′y_{k}^{a}

for some ω′ ∈ Br(D_{n}, 0), independent of a. Setting a = 0 tells us that ω′ = y_{k}^{0}ω(y_{k}^{0})^{−1}. Thus

y_{k}^{a}(y_{k}^{0})^{−1}ω = ωy_{k}^{a}(y_{k}^{0})^{−1}

as desired.

Before beginning the proof of the first assertion, we recall some notation introduced in 2.2. We abbreviate y_{α_{k}}^{a} by k^{a} and y_{α_{k}}^{0} by k. Whenever there exist ω′ and ω′′ ∈ Br(D_{n}) such that k^{a}ω = ω′j^{a}ω′′, we will write k^{a}ω ∼ j^{a}ω′′. This is not an equivalence relation, but it is compatible with multiplication on the right by elements of Br(D_{n}): if η ∈ Br(D_{n}),

k^{a}ω ∼ j^{a}ω′′ ⇔ k^{a}ω = ω′j^{a}ω′′ ⇔ k^{a}ωη = ω′j^{a}ω′′η ⇔ k^{a}ωη ∼ j^{a}ω′′η

From defining relations (A1), (A1 × A1), and (A2) of 2, we can deduce the following:

(4a)  k^{a}kk ∼ k^{a}
(4b)  k^{a} (k − 1) k ∼ (k − 1)^{a}
(4c)  k^{a} (k − 1) k^{−1} ∼ (k − 1)^{a}
(4d)  k^{a}(k + 1)k ∼ (k + 1)^{a}
(4e)  (k + 1)^{a}k(k + 1) ∼ k^{a}
(4f) (k + 1)^{a}k(k + 1)^{−1} ∼ k^{a}
(4g)  k^{a}(k + 1)k^{−1} ∼ (k + 1)^{a}
(4h)  k^{a}(k + 1)^{−1}k^{−1} ∼ (k + 1)^{a}
(4i) (k + 1)^{a}k^{−1}(k + 1)^{−1} ∼ k^{a}
(4j)  k^{a}k(k − 1) ∼ (k − 1)^{a}(k − 1)k^{−1}

Proof, continued. In the notation just introduced, we must show, for every ω ∈ P Br(D_{n}), that k^{a}ω ∼ k^{a}, and it suffices to show this when ω is one of the generators a_{j,i} or b_{j,i} of 3.3.1. That is, we must show

(5a) k^{a}a_{j,i} ∼ k^{a} n ≥ j ≥ i ≥ 2, 1 ≤ k ≤ n
(5b) k^{a}b_{j,i} ∼ k^{a} n ≥ j ≥ i ≥ 3, 1 ≤ k ≤ n

The proofs of (5a) for i ≥ 3 are exactly the same as the corresponding proofs for A_{n−1} (see section 2); the additional case i = 2′ presents no new issues. Thus we shall concentrate on proving (5b); the proof proceeds by induction on n. The case n = 3 is the case of the root system D_{3} = A_{3}, which is part of the Pure Braid Lemma 2.2.1 for A_{n−1}. Hence we may assume n ≥ 4, and that (5b) holds whenever j, k ≤ n − 1. That is, we must prove (5b) in these cases:

k = n, j ≤ n − 1; k ≤ n − 1, j = n; k = n, j = n

which further subdivide into the cases

(6)  k = n, j ≤ n − 2
(7)  k = n, j = n − 1
(8)  k ≤ n − 2, j = n
(9)  k = n − 1, j = n
(10)  k = n, j = n

• Case (6) k = n and j ≤ n − 2. Since i ≤ j ≤ n − 2, it follows from relation (A1 × A1) that n^{a} commutes with every generator which occurs in the expression for b_{j,i}; hence

n^{a}b_{j,i} = n^{a}j(j − 1) … 322′3 … (i − 1)ii(i − 1) … 32′23(j − 1)j
= b_{j,i}n^{a}
∼ n^{a}

as desired.

• Case (7) k = n and j = n − 1.

If i ≤ n − 2, then

n^{a}b_{n−1,i}

= n^{a}(n − 1)(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 2)(n − 1)
= \underbrace{n^{a}(n − 1)n^{−1}} n(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 2)(n − 1)
∼ (n − 1)^{a}n(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 2)(n − 1)
= \underbrace{(n − 1)^{a}(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 2)} n(n − 1)
∼ (n − 1)^{a}n(n − 1)
∼ n^{a}

as desired. The case k = n, i = j = n−1, is considerably more complicated. We first prove some preliminary lemmas.

Lemma 3.3.3.

n^{a}(n − 1)(n − 2) … 322′ ∼ 2^{a}32′4 … (n − 2)(n − 1)n

Proof.

n^{a}(n − 1)(n − 2) … 322′ = n^{a}(n − 1)n^{−1}n(n − 2) … 322′
∼ (n − 1)^{a}n(n − 2) … 322′
= (n − 1)^{a}(n − 2) … 322′n
⋮
∼ 3^{a}22′4 … (n − 2)(n − 1)n
= 3^{a}23^{−1}32′4 … (n − 2)(n − 1)n
∼ 2^{a}32′4 … (n − 2)(n − 1)n

□

Lemma 3.3.4.

32′4354 … (n − 3)(n − 1)(n − 2)nn = 345 … (n − 1)nn2′34 … (n − 2)

Proof.

32′4354 … (n − 3)(n − 1)(n − 2)nn
= 32′4354 … (n − 3)(n − 1)nn(n − 2)
= 32′4354 … (n − 1)nn(n − 3)(n − 2)
⋮
= 345 … (n − 1)nn2′34 … (n − 2)

Parametrized Braid Groups 409

□

We now complete the case k = n, i = j = n − 1.

n^{a}b_{n−1,n−1}

= n^{a}(n − 1)(n − 2) … 322′3 … (n − 2)(n − 1)(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

∼ 2^{a}32′4 … (n − 2)(n − 1)n3 … (n − 2)(n − 1)(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1) (by Lemma 3.3.3)

= 2^{a}32′4 … (n − 2)(n − 1)3 … (n − 2) \underbrace{n(n − 1)n^{−1}} n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a}32′4 … (n − 2) \underbrace{(n − 1)3 …} (n − 2)(n − 1)^{−1}n(n − 1)n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a}32′4 … (n − 2)3 … \underbrace{(n − 1)(n − 2)(n − 1)^{−1}} n(n − 1)n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a}32′4 … (n − 2)3 … \underbrace{(n − 2)^{−1}(n − 1)(n − 2)} n(n − 1)n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1) ⋮

= 2^{a}32′ \underbrace{434^{−1}} 54 … (n − 1)(n − 2)n(n − 1)n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a} \underbrace{32′3^{−1}} 4354 … (n − 1)(n − 2)n(n − 1)n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a} \underbrace{2^{′−1}32′} 4354 … (n − 1)(n − 2)n(n − 1)n(n − 1)(n − 2) …

… 32′23 … (n − 2)(n − 1)

∼ 2^{a}32′4354 … (n − 1)(n − 2)n \underbrace{(n − 1)n(n − 1)} (n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a}32′4354 … (n − 1)(n − 2)n \underbrace{n(n − 1)n} (n − 2) …

… 32′23 … (n − 2)(n − 1)

= 2^{a}345 … (n − 1)nn2′34 … (n − 2)(n − 1)n(n − 2) …

… 32′23 … (n − 2)(n − 1) (by Lemma 3.3.4)

We now manipulate part of this expression so that we can apply induction.

2^{a}345 … (n − 1)nn2′

= 2^{a}345 … (n − 1)nn \underbrace{(n − 1) … 5433^{−1}4^{−1}5^{−1} … (n − 1)^{−1}} 2′
∼ 2^{a}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}2′

Documenta Mathematica 10 (2005) 391–416

410 Jean-Louis Loday and Michael R. Stein

(by the case of A_{n−1} = {α_{2}, α_{3}, …, α_{n}}). Hence

n^{a}b_{n−1,n−1}

= 2^{a}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}2′34 … (n − 2)(n − 1)n(n − 2)(n − 3) …
… 32′23 … (n − 2)(n − 1)
= 2^{a}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}2′34 … (n − 2)(n − 1)(n − 2)n(n − 3) …
… 32′23 … (n − 2)(n − 1)
= 2^{a}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}2′34 … (n − 1)(n − 2)(n − 1)n(n − 3) …
… 32′23 … (n − 2)(n − 1)
= 2^{a}3^{−1}4^{−1}5^{−1} … (n − 2)^{−1}2′34 … (n − 2)(n − 1)n(n − 3) …
… 32′23 … (n − 2)(n − 1)
⋮
= 2^{a}3^{−1}2′34 … (n − 2)(n − 1)n2′23 … (n − 2)(n − 1)
= 2^{a}3^{−1}2′32′4 … (n − 2)(n − 1)n23 … (n − 2)(n − 1)
= 2^{a}3^{−1}32′34 … (n − 2)(n − 1)n23 … (n − 2)(n − 1)
= 2^{a}2′34 … (n − 2)(n − 1)n23 … (n − 2)(n − 1)
∼ 2^{a}34 … (n − 2)(n − 1)n23 … (n − 2)(n − 1)
= 2^{a}324 … (n − 2)(n − 1)n3 … (n − 2)(n − 1)
∼ 3^{a}4 … (n − 2)(n − 1)n3 … (n − 2)(n − 1)
⋮
∼ (n − 1)^{a}n(n − 1)
∼ n^{a}

as desired. □

• Case (8) k ≤ n − 2 and j = n.

k^{a}b_{n,i}

= k^{a}n(n − 1) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)n
∼ \underbrace{k^{a}(k + 1) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (k + 1)} (k + 2) … (n − 1)n
∼ k^{a}(k + 2) … n (by induction)
∼ k^{a}

as desired.

• Case (9) k = n − 1 and j = n.

(n − 1)^{a}b_{n,i} = (n − 1)^{a}n(n − 1) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)n = \underbrace{(n − 1)^{a}n(n − 1)} … 322′3 … (i − 1)ii(i − 1) … 32′ 223 … (n − 1)n

(†) ∼ n^{a}(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)n

Parametrized Braid Groups 411

Suppose first that i ≤ n − 2. Then

(†) = n^{a}(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)n
= (n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 2)n^{a}(n − 1)n
∼ n^{a}(n − 1)n
∼ (n − 1)^{a}

as desired. If i = n − 1

(†) = n^{a}(n − 2) … 322′3 … (n − 2)(n − 1)(n − 1)(n − 2) …
… 32′23 … (n − 1)n
∼ \underbrace{n^{a}(n − 1)(n − 1)} (n − 2) … 32′23 … (n − 1)n
∼ n^{a}(n − 2) … 32′23 … (n − 1)n (by the case A_{2} = {α_{n−1}, α_{n}})
∼ n^{a}(n − 1)n (by relation (A1 × A1))
∼ (n − 1)^{a}

as desired. If i = n

(†) = n^{a}(n − 2) … 322′3 … (n − 1)nn(n − 1) … 32′23 … (n − 1)n
∼ n^{a}(n − 1)nn(n − 1)(n − 2) … 32′23 … (n − 1)n
∼ (n − 1)^{a}n(n − 1)(n − 2) … 32′23 … (n − 1)n
∼ n^{a}(n − 2) … 32′23 … (n − 1)n
∼ n^{a}(n − 1)n
∼ (n − 1)^{a}

as desired.

• Case (10) k = n and j = n.

n^{a}b_{n,i} = n^{a}n(n − 1)(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)n

(†) ∼ (n − 1)^{a}(n − 1)n^{−1}(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)n

If i ≤ n − 2, then

(†) = (n − 1)^{a}(n − 1)(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … \underbrace{n^{−1}(n − 1)n}

= \underbrace{(n − 1)^{a}(n − 1)(n − 2) … 322′3 … (i − 1)ii(i − 1) … 32′23 … (n − 1)} n(n − 1)^{−1}

∼ (n − 1)^{a}n(n − 1)^{−1} (by induction) ∼ n^{a}

412 Jean-Louis Loday and Michael R. Stein as desired. If i = n − 1, then

(†) = (n − 1)^{a}(n − 1)(n − 2) …

… 322′3 … (n − 2)n^{−1}(n − 1)(n − 1)(n − 2) … 32′23 … (n − 1)n ⋮

∼ 3^{a}322′4^{−1}35^{−1}4 …

… (n − 1)^{−1}(n − 2)n^{−1}(n − 1)(n − 1)(n − 2) … 32′23 … (n − 1)n

= 3^{a}322′4^{−1}35^{−1}4 … (n − 1)^{−1}(n − 2) \underbrace{n^{−1}(n − 1)n} n^{−1}(n − 1)(n − 2) …

… 32′23 … (n − 1)n

= 3^{a}322′4^{−1}35^{−1}4 …

… \underbrace{(n − 1)^{−1}(n − 2)(n − 1)} n \underbrace{(n − 1)^{−1}n^{−1}(n − 1)} (n − 2) … … 32′23 … (n − 1)n

= 3^{a}322′4^{−1}35^{−1}4 …

… (n − 3) \underbrace{(n − 2)(n − 1)(n − 2)^{−1}} n \underbrace{n(n − 1)^{−1}} n^{−1}(n − 2) … … 32′23 … (n − 1)n

= 3^{a}322′4^{−1}35^{−1}4 …

… (n − 3)(n − 2)(n − 1)nn(n − 2)^{−1}(n − 1)^{−1}n^{−1}(n − 2) … … 32′23 … (n − 1)n ⋮

= 3^{a}322′ \underbrace{4^{−1}34} … (n − 2)(n − 1)nn4^{−1}5^{−1} …

… (n − 1)^{−1}n^{−1}(n − 2) … 32′23 … (n − 1)n

∼ 2^{a}2 \underbrace{3^{−1}2′3} 43^{−1}5 … (n − 1)nn4^{−1}5^{−1} … (n − 1)^{−1}n^{−1}(n − 2) …

… 32′23 … (n − 1)n

= 2^{a}22′32^{′−1}4 … (n − 1)nn3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}n^{−1}(n − 2) …

… 32′23 … (n − 1)n

= 2^{a}22′34 … (n − 1)nn2^{′−1}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}n^{−1}(n − 2) …

… 32′23 … (n − 1)n

∼ 2^{a}234 … (n − 1)nn2^{′−1}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}n^{−1}(n − 2) …

… 32′23 … (n − 1)n

= 2^{a}234 …

… (n − 1)nn \underbrace{(n − 1) … 4322^{−1}3^{−1}4^{−1} … (n − 1)^{−1}} 2^{′−1}3^{−1}4^{−1}5^{−1} … … (n − 1)^{−1}n^{−1}(n − 2) … … 32′23 … (n − 1)n

Parametrized Braid Groups 413

∼ 2^{a}2^{−1}3^{−1}4^{−1} …

… (n − 1)^{−1}2^{′−1}3^{−1}4^{−1}5^{−1} … (n − 1)^{−1}n^{−1}(n − 2) … 32′23 … (n − 1)n

(by the case of A_{n−1} = {α_{2}, α_{3}, …, α_{n}})
∼ 2^{a}2^{−1}3^{−1}4^{−1} …
… (n − 1)^{−1}2^{′−1}3^{−1}4^{−1}5^{−1} …
… (n − 3)^{−1} \underbrace{(n − 2)^{−1}(n − 1)^{−1}(n − 2)} n^{−1}(n − 3) …
… 32′23 … (n − 1)n
= 2^{a}2^{−1}3^{−1}4^{−1} … (n − 1)^{−1}2^{′−1}3^{−1}4^{−1}5^{−1} … (n − 3)^{−1}
\underbrace{(n − 1)(n − 2)^{−1}(n − 1)^{−1}} n^{−1}(n − 3) …
… 32′23 … (n − 1)n
= 2^{a}2^{−1}3^{−1}4^{−1} …
… (n − 1)^{−1}(n − 1)2^{′−1}3^{−1}4^{−1}5^{−1} …
… (n − 3)^{−1}(n − 2)^{−1}(n − 1)^{−1}n^{−1}(n − 3) …
… 32′23 … (n − 1)n
= 2^{a}2^{−1}3^{−1}4^{−1} … (n − 2)2^{′−1}3^{−1}4^{−1}5^{−1} …
… (n − 3)^{−1}(n − 2)^{−1}(n − 1)^{−1}n^{−1}(n − 3) …
… 32′23 … (n − 1)n
⋮
= 2^{a}2^{−1}3^{−1}2^{′−1}3^{−1}4^{−1} … n^{−1}2′23 … (n − 1)n
= 2^{a}2^{−1}3^{−1} \underbrace{2^{′−1}3^{−1}2′} 4^{−1} … n^{−1}23 … (n − 1)n
= 2^{a}2^{−1}3^{−1}32^{′−1}3^{−1}4^{−1} … n^{−1}23 … (n − 1)n
∼ 2^{a}2^{−1}3^{−1}4^{−1} … n^{−1}23 … (n − 1)n
= 2^{a} \underbrace{2^{−1}3^{−1}2} 4^{−1} … n^{−1}3 … (n − 1)n
= \underbrace{2^{a}32^{−1}} 3^{−1}4^{−1} … n^{−1}3 … (n − 1)n
∼ 3^{a}3^{−1}4^{−1} … n^{−1}3 … (n − 1)n (by (4g))
⋮
∼ (n − 1)^{a}(n − 1)^{−1} \underbrace{n^{−1}(n − 1)n} (by (4g))
= (n − 1)^{a}(n − 1)^{−1}(n − 1)n(n − 1)^{−1}
= (n − 1)^{a}n(n − 1)^{−1}
∼ n^{a}

as desired.

Documenta Mathematica 10 (2005) 391–416

414 Jean-Louis Loday and Michael R. Stein If i = n, then

(†) = (n − 1)^{a}(n − 1)n^{−1}(n − 2) … 322′3 … (n − 1)nn(n − 1) …

… 32′23 … (n − 1)n

= (n − 1)^{a}(n − 1)(n − 2) … 322′3 … \underbrace{n^{−1}(n − 1)n} n(n − 1) …

… 32′23 … (n − 1)n

= (n − 1)^{a}(n − 1)(n − 2) … 322′3 … (n − 2)(n − 1)n \underbrace{(n − 1)^{−1}n(n − 1)} …

… 32′23 … (n − 1)n

= (n − 1)^{a}(n − 1)(n − 2) … 322′3 … (n − 2)(n − 1)nn(n − 1)n^{−1}(n − 2) …

… 32′23 … (n − 1)n

= (n − 1)^{a}(n − 1)(n − 2) … 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2) …

… 32′23 … n^{−1}(n − 1)n

= (n − 1)^{a}(n − 1)(n − 2) … 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2) …

… 32′23 … (n − 1)n(n − 1)^{−1}

∼ (n − 2)^{a}(n − 2)(n − 1)^{−1}(n − 3) …

… 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2) … … 32′23 … (n − 1)n(n − 1)^{−1} (by (4g))

= (n − 2)^{a}(n − 2)(n − 3) …

… 322′3 … (n − 1)^{−1}(n − 2)(n − 1)nn(n − 1)(n − 2) … … 32′23 … (n − 1)n(n − 1)^{−1}

= (n − 2)^{a}(n − 2)(n − 3) …

… 322′3 … (n − 2)(n − 1)(n − 2)^{−1}nn(n − 1)(n − 2) … … 32′23 … (n − 1)n(n − 1)^{−1}

= (n − 2)^{a}(n − 2)(n − 3) …

… 322′3 … (n − 2)(n − 1)nn(n − 2)^{−1}(n − 1)(n − 2) … … 32′23 … (n − 1)n(n − 1)^{−1}

= (n − 2)^{a}(n − 2)(n − 3) …

… 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2)(n − 1)^{−1} … … 32′23 … (n − 1)n(n − 1)^{−1}

= (n − 2)^{a}(n − 2)(n − 3) … 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2) …

… 32′23 … (n − 3)(n − 1)^{−1}(n − 2)(n − 1)n(n − 1)^{−1}

= (n − 2)^{a}(n − 2)(n − 3) … 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2) …

… 32′23 … (n − 3)(n − 2)(n − 1)(n − 2)^{−1}n(n − 1)^{−1}

= (n − 2)^{a}(n − 2)(n − 3) … 322′3 … (n − 2)(n − 1)nn(n − 1)(n − 2) …

… 32′23 … (n − 3)(n − 2)(n − 1)n(n − 2)^{−1}(n − 1)^{−1} ⋮

∼ 2^{a}22′3 … (n − 1)nn(n − 1) …
… 32′23 … (n − 1)n2^{−1}3^{−1} … (n − 1)^{−1} (by (4g))
∼ \underbrace{2^{a}23 … (n − 1)nn(n − 1) … 32} 2′3 … (n − 1)n2^{−1}3^{−1} … (n − 1)^{−1}
∼ 2^{a}2′3 … (n − 1)n2^{−1}3^{−1} … (n − 1)^{−1}
(by the case of A_{n−1} = {α_{2}, α_{3}, …, α_{n}})
∼ 2^{a}3 … (n − 1)n2^{−1}3^{−1} … (n − 1)^{−1}
= 2^{a}32^{−1} … (n − 1)n3^{−1} … (n − 1)^{−1}
= 3^{a}4 … (n − 1)n3^{−1} … (n − 1)^{−1}

⋮
= (n − 1)^{a}n(n − 1)^{−1}
= n^{a}

as desired. □

References

[Bir] Birman, J. S., Braids, links, and mapping class groups. Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1974.

[Bon] Bonis, R., Groupes d’Artin et formes réelles des relations de Yang-Baxter. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 8, 663–666.

[Bour] Bourbaki, N., Groupes et algèbres de Lie, Chapitres IV, V et VI. Actualités Scientifiques et Industrielles, No. 1337 Hermann, Paris 1968.

[Car] Carter, Roger W., Simple groups of Lie type. John Wiley & Sons, Inc., New York, 1989.

[D-G] Digne, F., Gomi, Y., Presentation of pure braid groups. J. Knot Theory Ramifications 10 (2001), no. 4, 609–623.

[H] Humphreys, J. E., Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge, 1990.

[K-S] Kapranov, M.M., Saito, M., Hidden Stasheff polytopes in algebraic Ktheory and in the space of Morse functions. in Higher homotopy structures in topology and mathematical physics (Poughkeepsie, NY, 1996), 191–225, Contemp. Math., 227, Amer. Math. Soc., Providence, RI, 1999.

[K-R] Kassel, Ch. et Reutenauer, Ch., Une variante à la Coxeter du groupe de Steinberg. K-Theory 14 (1998), no. 4, 305–318.

[L] Loday, J.-L., Homotopical syzygies, in “Une dégustation topologique: Homotopy theory in the Swiss Alps”, Contemporary Mathematics no 265 (AMS) (2000), 99–127.

[M-K-S] Magnus, W., Karrass, A., Solitar, D., Combinatorial group theory: Presentations of groups in terms of generators and relations. Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney 1966.

[St] Stein, M. R., Generators, relations and coverings of Chevalley groups over commutative rings. Amer. J. Math. 93 (1971), 965–1004.

[Stb] Steinberg, R., Générateurs, relations et revêtements de groupes algébriques. 1962 Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) pp. 113–127 Librairie Universitaire, Louvain, Gauthier-Villars, Paris.

Jean-Louis Loday Institut de Recherche Mathématique Avancée, CNRS et Université Louis Pasteur 7 rue R. Descartes, 67084 Strasbourg Cedex, France loday@math.u-strasbg.fr Michael R. Stein Department of Mathematics Northwestern University 2033 Sheridan Road Evanston IL 60208–2730 USA mike@math.northwestern.edu