Traces for braid groups and Hecke algebras of type B

Knot theories and root systems. Part III

Tammo tom Dieck

1. Traces on groups

Let G be a group and 𝔎 a commutative ring. A (𝔎-valued) trace on G is a function T: G → 𝔎 such that for all g, h ∈ G

(1.1) T (gh) = T (hg).

Equivalently, a trace is a function constant on conjugacy classes. A trace extends to a 𝔎-linear map T: 𝔎G → 𝔎 from the group algebra 𝔎G such that (1.1) holds for any two elements g, h in the group algebra.

Suppose τ: G → G is an automorphism and T a trace. We call T (strongly) τ-invariant if for all g, h ∈ G the relation

(1.2) T (g · τ(h)) = T (g · h)

holds. If we set g = 1, we have the ordinary τ-invariance T (τ(h)) = T (h). If T is τ_{1}- and τ_{2}-invariant, then also τ_{1}^{−1}- and τ_{1}τ_{2}-invariant. If Γ is a group of automorphisms of G, then a trace is called Γ-invariant, if T is τ-invariant for each τ ∈ Γ. It suffices to check Γ-invariance for a generating set of Γ.

Suppose T_{i} is a trace on G_{i} (i = 1, 2). Then (g_{1}, g_{2}) ↦ T_{1}(g_{1})T_{2}(g_{2}) is a trace on G_{1} × G_{2}. We want to generalize this to semi-direct products.

Let α: Γ → Aut G be a group of G-automorphisms. The semi-direct product G ×_{α} Γ is a group structure on the set G × Γ defined by

(g, σ)(h, τ) := (g · σ(h), στ).

In this group structure we have

(1.3) (1, σ)(g, 1)(1, σ)^{−1} = (σ(g), 1).

We will use the following fact several times.

(1.4) Lemma. A pair of group homomorphisms λ: G → H and μ: Γ → H defines via (g, σ) ↦ λ(g)μ(σ) a homomorphism φ: G ×_{α} Γ → H if and only if for all g ∈ G and σ ∈ Γ the relation λ(σ(g)) = μ(σ)λ(g)μ(σ)^{−1} holds. Each homomorphsm φ has this form for a unique pair (λ, μ). □

Is α: Γ → Aut (G) is an antihomorphism, we define the semi-direct product Γ_{α}× G with multiplication (σ, g)(τ, h) = (στ, τ(g)h).

The following is immediately verified from the definitions.

(1.5) Proposition. Let S be a Γ-invariant trace on G and U a trace on Γ. Then

T: G ×_{α} Γ → 𝔎, (g, σ) ↦ S(g)U(σ)

is a trace on G ×_{α} Γ. □

If φ: G → H is a group homomorphism and T a trace on H, then T ∘ φ is a trace on G. Any function T: G → 𝔎 on an abelian group G is a trace. Characters of finite dimensional representations are traces.

T. tom Dieck  2. Braid groups of type B  3

2. Braid groups of type B

The braid group ZB_{n} associated to the Coxeter graph B_{n} is, by definition, the group generated by t, g_{1}, …, g_{n−1} with relations

(2.1) \begin{matrix}(1) & g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j}, & |i − j| = 1 \\ (2) & g_{i}g_{j} & = & g_{j}g_{i}, & |i − j| ≥ 2 \\ (3) & tg_{i} & = & g_{i}t, & i ≥ 2 \\ (4) & tg_{1}tg_{1} & = & g_{1}tg_{1}t. \end{matrix}

For certain applications we need other presentations of this group.

Let Z′B_{n} be the group with generators c, g_{1}, …, g_{n−1} and relations

(2.2) \begin{matrix}(1) & g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j}, & |i − j| = 1 \\ (2) & g_{i}g_{j} & = & g_{j}g_{i}, & |i − j| ≥ 2 \\ (3) & cg_{i} & = & g_{i−1}c, & i ≥ 2, \\ (4) & c^{2}g_{1} & = & g_{n−1}c^{2}. \end{matrix}

We abbreviate g = g_{n−1}g_{n−2} ⋯ g_{1}.

(2.3) Proposition. The assignment φ(g_{i}) = g_{i}, 1 ≤ j ≤ n−1, and φ(t) = g^{−1}c defines an isomorphism φ: ZB_{n} → Z′B_{n}.

Proof. The relations (1) and (2) yield in both groups

(2.4)  g_{i−1}g = gg_{i},  i > 1.

We define in ZB_{n} (resp. Z′B_{n}) an element c (resp. t) by gt = c. From (1), (2) and (2.4) we see that the relations cg_{i} = g_{i−1}c and g_{i}t = tg_{i} are equivalent for i > 1.

We set h = g_{n−1} ⋯ g_{2}, k = g_{n−2} ⋯ g_{1} and infer from (2.4)

(2.5) gh = kg.

We use this to show that c^{2}g_{1} = g_{n−1}c^{2} and tg_{1}tg_{1} = g_{1}tg_{1}t are equivalent, provided (1), (2), and (3) hold. We compute

g_{n−1}^{−1}c^{2}g_{1} = g_{n−1}^{−1}g_{n−1}kthg_{1}tg_{1} = khtg_{1}tg_{1}
c^{2} = gthg_{1}t = ghtg_{1}t = kgtg_{1}t = khg_{1}tg_{1}t

and see the equivalence. □

The braid group ZÃ_{n−1} of the Coxeter graph with n vertices Ã_{n−1} has, by definition, generators g_{1}, …, g_{n} and relations

(2.6) \begin{matrix}g_{i}g_{j}g_{i} & = & g_{j}g_{i}g_{j}, & m(i, j) = 3 \\ g_{i}g_{j} & = & g_{j}g_{i}, & m(i, j) = 2. \end{matrix}

Indices will be considered mod n in this case. We have m(i, j) = 3 if and only if i ≡ j ± 1 mod n. All this holds for n ≥ 3. For n = 2, the group is the free group generated by g_{1} and g_{2}.

The graph Ã_{n−1} has an automorphism which permutes the vertices cyclically. We have an induced automorphism s of ZÃ_{n−1} given by

s(g_{i}) = g_{i−1}, i mod n.

The n-th power of s is the identity.

We use s to form the semi-direct product

(2.7) ZÃ_{n−1} → G_{n} → ZZ;

the generator 1 ∈ ZZ acts through s on ZÃ_{n−1}. The semi-direct product is the group structure on the set ZÃ_{n−1}×ZZ defined by (x, m)·(y, n) = (x·s^{m}(y), m+n). The group G_{n} has the following description by generators and relations. Let G′_{n} denote the group with generators s, g_{1}, …, g_{n} and relations (2.6) together with

(2.8)  sg_{i} = g_{i−1}s,  i mod n.

(2.9) Proposition. The assignment ψ(g_{i}) = (g_{i}, 0) and ψ(s) = (e, 1) yields an isomorphism ψ: G′_{n} → G_{n} (neutral element e).

Proof. One verifies that ψ is compatible with relations (2.6) and (2.8). This is obvious for (2.6). The relation (e, 1)(x, 0)(e, 1)^{−1} = (s(x), 0) is used to show compatibility with (2.8).

An element x ∈ ZÃ_{n−1} has an image x′ ∈ G′_{n}, induced by g_{i} ↦ g_{i}. This assignment has the property (s(x))′ = sx′s^{−1}. We have the Homomorphism G_{n} → G′_{n}, (x, m) ↦ x′s^{m} by (1.4). It is inverse to psi. □

(2.10) Proposition. The assignment α(g_{i}) = g_{i}, 1 ≤ i ≤ n − 1, and α(c) = s defines an isomorphism α: Z′B_{n} → G′_{n}.

Proof. The assignment is compatible with the relations of Z′B_{n}, since

α(c^{2}g_{1}c^{−2}) = s^{2}g_{1}s^{−2} = sg_{n}s^{−1} = g_{n−1}.

An inverse to α is induced by the assignment β(g_{i}) = g_{i}, β(g_{n}) = cg_{1}g^{−1}, and β(s) = c. In order to see that β is well defined, one has to check, in particular, the relations

g_{n−1}g_{n}g_{n−1} = g_{n}g_{n−1}g_{n}, g_{1}g_{n}g_{1} = g_{n}g_{1}g_{n}.

In the first case, this amounts to the equalitiy of

g_{n−1}cg_{1}c^{−1}g_{n−1} = c^{2}g_{1}c^{−1}g_{1}cg_{1}c^{−2}

and

cg_{1}c^{−1}g_{n−1}cg_{1}c^{−1} = cg_{1}cg_{1}c^{−1}g_{1}c^{−1}.

We compute

cg_{1}g_{2}g_{1}c^{−1} = cg_{2}g_{1}g_{2}c^{−1} = cg_{2}c^{−1}cg_{1}c^{−1}cg_{2}c^{−1} = g_{1}cg_{1}c^{−1}g_{1}

and hence

c(g_{1}cg_{1}c^{−1}g_{1})c^{−1} = c^{2}g_{1}g_{2}g_{1}c^{−2}.

On the other hand, g_{1}c^{−1}g_{1}cg_{1} = g_{1}g_{2}g_{1}. This yields the desired equality. The second relation above leads to the same situation. □

If we combine the foregoing, we obtain a semi-direct product

(2.11) ZÃ_{n−1} → ZB_{n} → ZZ.

In terms of the original generators, the inclusion ZÃ_{n−1} ⊂ ZB_{n} is given by

(2.12) g_{n} ↦ gtg_{1}t^{−1}g^{−1}; g_{i} ↦ g_{i}, 1 ≤ i ≤ n − 1.

The homomorphism ZB_{n} → ZZ in (2.14) is given by g_{i} ↦ 0 and t ↦ 1.

Different types of Weyl groups (= Coxeter groups) are related to these braid groups. We have the Coxeter groups W Ã_{n−1} and W B_{n} associated to the graphs Ã_{n−1} and B_{n}. In addition, we will also use a group W^{∞}B_{n}. It is obtained from ZB_{n} by adding the relations g_{j}^{2} = 1, but no relation for t. The reason for introducing this group is a semi-direct product in analogy to (2.14). The arguments which lead to (2.14) also give a semi-direct product

W Ã_{n−1} → W^{∞}B_{n} → ZZ.

We give another interpretation and describe these groups as groups of permutations.

Let t_{n}: ZZ → ZZ, x ↦ x+n be the translation by n. Let P_{n} denote the group of t_{n}-equivariant permutations σ: ZZ → ZZ. Equivariance means σ(i + n) = σ(i) + n. Hence σ induces \overline{σ}: ZZ/n → ZZ/n, and σ ↦ \overline{σ} is a homomorphism π: P_{n} → S_{n} onto the symmetric group S_{n}.

(2.13) Proposition. The kernel of π is isomorphic to ZZ^{n}. The group P_{n} is isomorphic to the semi-direct product ZZ^{n} → P′_{n} → S_{n} in which S_{n} acts on ZZ^{n} by permutations.

Proof. Let σ_{1} ∈ P_{n}. Then there exists a permutation α of {1, …, n} and an ntuple (k_{1}, …, k_{n}) ∈ ZZ^{n} such that σ(i+tn) = α(i)+(k_{i}+t)n. We denote this map by σ_{1} = σ(α; k_{1}, …, k_{n}). Suppose σ_{2} = σ(β; l_{1}, …, l_{n}) is another permutation written in this form. Then

σ_{2} ∘ σ_{1} = σ(βα; l_{α(1)} + k_{1}, …, l_{α(n)} + k_{n}).

If we think of P′_{n} = S_{n} × ZZ^{n} as sets, then the desired isomorphism is given by (α; k_{1}, …, k_{n}) ↦ σ(α; k_{1}, …, k_{n}). □

The semi-direct product P′_{n} has a normal subgroup Q′_{n} which is given as a semi-direct product

(2.14) N → Q′_{n} → S_{n} with N = {(x_{1}, …, x_{n}) | ∑ x_{i} = 0} ⊂ ZZ^{n}. The homomorphism

ε: P′_{n} → ZZ, (α; k_{1}, …, k_{n}) ↦ ∑ k_{i}

is a surjection with kernel Q′_{n}. The canonical sequence

(2.15) Q′_{n} → P′_{n} → ZZ

is itself a semi-direct product; the assignment 1 ↦ (id; 1, 0, …, 0) gives a splitting of ε. Under the isomorphism (2.13) the subgroup Q′_{n} corresponds to the subgroup

Q_{n} = {σ ∈ P_{n} | 1 + 2 + ⋯ + n = σ(1) + ⋯ + σ(n)}.

(2.16) Proposition. The groups W^{∞}B_{n} and P_{n} are isomorphic. The groups W Ã_{n−1} and Q_{n} are isomorphic. The element g_{i} is mapped to the transposition (i, i + 1), i ∈ nZZ. The element t is mapped to σ(i) = i + n for i ≡ 1 mod n and σ(j)j otherwise.

The proof is given after the proof of (2.21). In the proof of (2.17) we use the following:

(2.17) Lemma. The elements

t_{0} = t, t_{1} = g_{1}tg_{1}, …, t_{n−1} = g_{n−1} … g_{2}g_{1}tg_{1}g_{2} … g_{n−1}

of the braid group ZB_{n} pairwise commute.

Proof. We set

g(i, j) = g_{i}g_{i+1} … g_{j}, i ≤ j
g(i, j) = g_{i}g_{i−1} … g_{j}, i ≥ j.

The braid relations imply immediately

g(1, j)g_{j+1}g(j, 1) = g(j + 1, 2)g_{1}g(2, j + 1)

and (2.5)

g(2, j + 1)g(1, j + 1) = g(1, j + 1)g(1, j).

By commutativity of g_{j}-elements, it suffices to show t_{i}t_{i+1} = t_{i+1}t_{i}. We compute

t_{j}t_{j+1} = g(j, 1)tg(1, j)g_{j+1}g(j, 1)tg(1, j + 1)
= g(j, 1)tg(j + 1, 2)g_{1}g(2, j + 1)tg(1, j + 1)
= g(j, 1)g(j + 1, 2)tg_{1}tg(2, j + 1)g(1, j + 1)
= g(j, 1)g(j + 1, 2)[tg_{1}tg_{1}]g(2, j + 1)g(1, j).

A similar computation works for t_{j+1}t_{j}. □

The semi-direct product relation (2.13, (2.17) between W^{∞}B_{n} and W A_{n−1} has a counterpart for the braid groups. The homomorphism

λ: K_{n} → ZA_{n−1}, g_{j} ↦ g_{j}, t ↦ 1

splits by g_{j} ↦ g_{j}. Therefore we have a semi-direct product

(2.18) ZP_{n} → ZB_{n} → ZA_{n−1}.

The elements

y_{0} = t, , y_{1} = g_{1}tg_{1}^{−1}, …, y_{n−1} = g_{n−1} … g_{1}tg_{1}^{−1} … g_{n−1}^{−1}

are contained in the kernel K_{n} of λ.

(2.19) Lemma. The elements y_{j} have the following conjugation properties with respect to ZA_{n−1}:

(1) g_{k}^{−1}y_{j}g_{k} = y_{j}, k > j, k < j − 1

(2) g_{k}^{−1}y_{k}g_{k} = y_{k−1},

(3) g_{k}^{−1}y_{k−1}g_{k} = y_{k−1}y_{k}y_{k−1}^{−1}. Proof. (2) follows directly from the definitions.

(1) If k > j, then g_{k} commutes with every generator in the definition of y_{j}. In the case k < j − 1 one uses the commutation relation between generators and g_{k+1}g_{k}g_{k+1}^{−1} = g_{k}^{−1}g_{k+1}g_{k} (and the inverse) to cancel g_{k}^{−1} and g_{k}.

(3) is proved by induction on k. The verification for k = 0 is easy. We calculate with (1) and (2)

g_{k}^{−1}y_{k}y_{k+1}y_{k}^{−1}g_{k} = y_{k−1}y_{k+1}y_{k−1}^{−1} = g_{k+1}y_{k−1}y_{k}y_{k−1}^{−1}g_{k+1}^{−1}.

On the other hand, by (1) and (2)

g_{k+1}^{−1}g_{k}^{−1}g_{k+1}^{−1}y_{k}g_{k+1}g_{k}g_{k+1} = g_{k}^{−1}g_{k+1}^{−1}g_{k}^{−1}y_{k}g_{k}g_{k+1}g_{k}
= g_{k}^{−1}g_{k+1}^{−1}y_{k−1}g_{k+1}g_{k}
= g_{k}^{−1}y_{k−1}g_{k}.

This yields the induction step. □

(2.20) Proposition. The group K_{n} is the free group generated by y_{0}, …, y_{n−1}.

Proof. By the previous Lemma, the group K_{n}^{0} generated by the y_{0}, …, y_{n−1} is invariant under conjugation by elements of ZA_{n−1}. Since t ∈ K_{n}^{0} and t together with ZA_{n−1} generates ZB_{n}, we must have equality K_{n}^{0} = K_{n}.

Let F_{n} denote the free group generated by y_{0}, …, y_{n−1}. We define homomorphisms γ_{1}, …, γ_{n−1}: F_{n} → F_{n} by immitating (2.20):

(1) γ_{k}(y_{j}) = y_{j}, k > j, k < j − 1

(2) γ_{k}(y_{k}) = y_{k−1},

(3) γ_{k}(y_{k−1}) = y_{k−1}y_{k}y_{k−1}^{−1}. We claim:

(2.21) Lemma. The γ_{j} are automorphisms and satisfy the braid relations

γ_{i}γ_{j}γ_{i} = γ_{j}γ_{i}γ_{j},  |i − j| = 1, and γ_{i}γ_{j} = γ_{j}γ_{i},  |i − j| ≥ 2.

Proof. First we check that the homomorphism δ_{k}: F_{n} → F_{n} (1) δ_{k}(y_{j}) = y_{j}, k > j, k < j − 1

(2) δ_{k}(y_{k−1}) = y_{k},

(3) δ_{k}(y_{k}) = y_{k}^{−1}y_{k−1}y_{k} is inverse to γ_{k}. Hence γ_{k} is an isomorphism. Since γ_{k} fixes y_{j} for j ∉ {k − 1, k}, the second braid relation is obviously satisfied. For the first relation, the reader may check the following values of γ_{1}γ_{2}γ_{1} and γ_{2}γ_{1}γ_{2} on y_{0}, y_{1}, y_{2}:

y_{0} ↦ y_{0}y_{1}y_{2}y_{1}^{−1}y_{0}^{−1}, y_{1} ↦ y_{0}y_{1}y_{1}^{−1}, y_{2} ↦ y_{0}.

We use this Lemma to define a semi-direct product

(2.22) F_{n} → Γ_{n} → ZA_{n−1},

in which g_{j} ∈ ZA_{n−1} acts on F_{n} through δ_{j}. By (2.19) and K_{n}^{0} = K_{n}, we have a canonical epimorphism μ: Γ_{n} → ZB_{n}. We show that μ is an isomorphism. As a set, Γ_{n} = F_{n} × ZA_{n−1}. An inverse to μ has to send g_{j} ↦ (1, g_{j}) and t ↦ (y_{0}, 1). We have to check that this assignment is compatible with the relations of ZB_{n}. This is obvious for the g_{j}. Moreover:

tg_{1}tg_{1} ↦ (y_{0}, 1)(1, g_{1})(y_{0}, 1)(1, g_{1})
= (y_{0}, g_{1})(y_{0}, g_{1})
= (y_{0}δ_{1}(y_{0}), g_{1}^{2})
= (y_{0}y_{1}, g_{1}^{2})

g_{1}tg_{1}t ↦ (1, g_{1})(y_{0}, 1)(1, g_{1})(y_{0}, 1)
= (y_{1}, g_{1})(y_{1}, g_{1})
= (y_{1}δ(y_{1}), g_{1}^{2})
= (y_{0}y_{1}, g_{1}^{2}).

This finishes the proof of Proposition (2.21). □

Proof of (2.17). The elements t_{j} of (2.18) and the elements y_{j} coincide in W^{∞}B_{n}, since g_{j} = g_{j}^{−1} in this group. Lemma (2.20) shows that conjugation y ↦ g_{k}^{−1}yg_{k} acts on the set (y_{0}, …, y_{n−1}) by interchanging y_{k−1} and y_{k}. The proof of (2.21) is now easily adapted to show the isomorphism W^{∞}B_{n} ≅ P′_{n}. This isomorphism restricts to an isomorphism W Ã_{n−1} ≅ Q′_{n}. □

We now apply the previous results to Hecke algebras. We have the Hecke algebras HA_{n−1}, HÃ_{n−1}, and HB_{n} associated to the corresponding Coxeter graphs. We consider algebras over the ground ring 𝔎. The first one is given by generators g_{1}, …, g_{n−1}, the braid relations between them and the quadratic relations g_{j}^{2} = (q − 1)g_{j} + q with a parameter q ∈ 𝔎. The second one has generators g_{1}, …, g_{n}, the braid relations (2.8) and the same quadratic relations. The algebra HB_{n} has generators t, g_{1}, …, g_{n−1}, the braid relations (2.1), the quadratic relations above for the g_{j} and t^{2} = (Q − 1)t + Q with another parameter Q ∈ 𝔎.

If we omit the quadratic relation for Q, then we obtain the definition of H^{∞}B_{n}. This is not a Hecke algebra in the formal sense, i. e. associated to a Coxeter graph. It is a deformation of the group algebra of W^{∞}B_{n}.

We know from Hecke algebra theory that an additive basis of the Hecke algebra is in bijective correspondence with the elements of the Coxeter group. There is a similar relation between W^{∞}B_{n} and H^{∞}B_{n}. In order to derive it, we relate HÃ_{n−1} and H^{∞}B_{n}.

The algebra HÃ_{n−1} has an automorphism τ given by τ(g_{i}) = g_{i−1} (indices mod n). We define the twisted tensor product over the ground ring 𝔎

(2.23) HÃ_{n−1} ⊗ 𝔎[τ, τ^{−1}] =: H_{n}^{∞}

by the multiplication rule (x ⊗ τ^{k}) · (y ⊗ τ^{l}) = (x · τ^{k}(y), τ^{k+l}) for k, l ∈ ZZ and x, y ∈ HÃ_{n−1}.

(2.24) Proposition. The algebra (2.24) is canonically isomorphic to H^{∞}B_{n}.

Proof. We use the isomorphism (2.3) to redefine the algebra H^{∞}B_{n} by generators c, g_{1}, …, g_{n−1} relations (2.2) and the quadratic relations for the g_{j}. The assignment g_{j} ↦ g_{j}⊗1, c ↦ 1⊗τ induces a homomorphism H^{∞}B_{n} → HÃ_{n−1}⊗H_{n}^{∞}. We have a homomorphism HÃ_{n−1} → H^{∞}B_{n}, x ↦ x′ induced by g_{j} ↦ g_{j} with g_{n} = gtg_{1}t^{−1}g^{−1} in H^{∞}B_{n} (see (2.12)). This extends to a homomorphism H_{n}^{∞} → H^{∞}B_{n} by x ⊗ τ^{k} ↦ x′ · c^{k}, since τ(y)′ = cy′c^{−1}. These homomorphisms are inverse to each other. □

(2.25) Corollary. Suppose (b_{j} | j ∈ J) is a 𝔎-basis of HÃ_{n−1}. Then (b′_{j}c^{k} | j ∈ J, k ∈ ZZ) is a 𝔎-basis of H^{∞}B_{n}. □

3. Markov traces on braid groups

We use the semi-direct product (2.19), (2.21)

F_{n} → ZB_{n} → ZA_{n−1}

in order to construct traces on ZB_{n} by (1.5). This requires a ZA_{n−1}-invariant trace on F_{n}.

Let s: ZZ → 𝔎 be any function with s(0) = 1, called parameter function. We define a trace T_{s}: V_{n} → 𝔎 on the free abelian multiplicative group V_{n} with basis y_{0}, …, y_{n−1} by

(3.1) T_{s}(y_{0}^{k(0)} ⋯ y_{n−1}^{k(n−1)}) = ∏_{j=0}^{n−1} s(k(j)).

Let F_{n} → V_{n} be the abelianization. The trace T_{s} on V_{n} lifts to a trace T_{s} on F_{n}.

(3.2) Proposition. The trace T_{s} on F_{n} is ZA_{n−1}-invariant.

Proof. It suffices to check invariance for the generators g_{i}. This is obvious from (2.20). □

If U is any trace on ZA_{n−1} and T_{s} the trace (3.2), we call the induced trace (1.5) on ZB_{n} the s-extension U_{s} of U.

Most important for applications to knot theory are Markov traces on the groups ZA_{n}. We recall: A sequence U = (U^{n}) of traces U^{n} on ZA_{n−1} is called a Markov trace on ZA = (ZA_{n}), provided

(3.3) \begin{matrix}U^{n+1}|ZA_{n−1} & = & U^{n} \\ U^{n+1}(xg_{n}^{±1}) & = & α^{±1}β^{−1}U^{n}(x) \end{matrix}

for x ∈ ZA_{n−1} with units α, β ∈ 𝔎^{∗}.

Markov traces on ZB = (ZB_{n}) have been constructed by Lambropoulou and Przytycki [10]. The next Proposition states that the s-extension of a Markov trace (U^{n}) is a Markov trace in their sense.

(3.4) Proposition. Let (T^{n}) = (U_{s}^{n}) denote the family of s-extensions of a Markov trace (U^{n}) on ZA. Then the following holds:

(1) T^{n+1}|ZB_{n} = T^{n}.

(2) T^{n+1}(xg_{n}^{±1}) = z_{±}T^{n}(x) for x ∈ ZB_{n}, z_{±} = α^{±1}β_{−1}.

(3) T^{n+1}(xy_{n}^{k}) = s(k)T^{n}(x) for x ∈ ZB_{n}. Proof. In terms of the semi-direct product ZB_{n} = F_{n} · ZA_{n−1}, the inclusion ZB_{n} → ZB_{n+1} is given by g_{i} ↦ g_{i} and y_{i} ↦ y_{i} for 0 ≤ i ≤ n − 1. The s-traces T_{s} on F_{n} are compatible with F_{n} → F_{n+1}, y_{i} ↦ y_{i}. This yields (1).

Suppose x = (h, σ) ∈ F_{n} · ZA_{n−1}. Then

xy_{n}^{k} = (h, σ)(y_{n}^{k}, 1) = (h · σ(y_{n}^{k}), σ).

T. tom Dieck 3. Markov traces on braid groups 11

But σ ∈ ZA_{n−1} acts trivially on y_{n}^{k}. Therefore

T (h · σ(y_{n}^{k})) = T (hy_{n}^{k}) = T_{(}h)s(k).

This shows (3); and (2) is equally simple. □

The generalized Hecke algebra H^{∞}B_{n} is the quotient of the group algebra 𝔎ZB_{n} by the ideal generated by the ζ_{i} := g_{i}^{2} − (q − 1)g_{i} − q, provided q ∈ 𝔎^{∗}.

(3.5) Proposition. Suppose the trace U: ZA_{n−1}to𝔎 factors over the quotient maps 𝔎ZA_{n−1} → HA_{n−1}. Then T = U_{s}: 𝔎ZB_{n} → 𝔎 factors over the quotient map 𝔎ZB_{n} → H^{∞}B_{n}.

Proof. We have to show T (xζ_{i}y) = T (yxζ_{i}) = 0 for all y, x ∈ 𝔎ZB_{n}. Write yx = z in the form ∑_{j} λ_{j}u_{j}v_{j} with λ_{j} ∈ 𝔎, u_{j} ∈ F_{n}, and v_{j} ∈ ZA_{n−1}. Then

T (zζ_{i}) = ∑ λ_{j}T (u_{j}v_{j}ζ_{i}) = ∑ λ_{j}T_{s}(u_{j})U(v_{j}ζ_{i}).

By hypothesis, U(v_{j}ζ_{i}) = 0. □

Lambropoulou and Przytycki show that there is a unique trace on the family H^{∞}B_{n} with the properties of the previous Proposition (3.4), normalized by T (1) = 1. Their proof requires the construction of an inductive basis for the family H^{∞}B_{n}. We give a geometric interpretation of their result in section 4.

4. Braids of type B

We use a theorem of Brieskorn [??] to derive some geometric interpretations of the braid group ZB_{n}. The starting point is the reflection representation of the Weyl group W B_{n}. This group is a semi-direct product

(4.1) (ZZ/2)^{n} → W B_{n} → S_{n}.

It acts on complex n-space C^{n} as follows:

(1) S_{n} acts by permuting the coordinates.

(2) (ZZ/2)^{n} act by sign changes (z_{1}, …, z_{n}) ↦ (ε_{1}z_{1}, …, ε_{n}z_{n}), ε_{i} ∈ {±1}. This group contains the reflections in the hyperplanes

z_{i} = ±z_{j}, i ≠ j; and z_{j} = 0.

Let X denote the complement of these hyperplanes. From the theory of finite reflection groups it is known, that W = W B_{n} acts freely on X. Brieskorn [??] shows:

(4.2) Theorem. The braid group ZB_{n} is isomorphic to the fundamental group π_{1}(X/W) of the orbit space X/W. □

If we think of W B_{n} as the Coxeter group with generators t, g_{1}, …, g_{n−1}, then g_{j} acts as the transposition (j, j + 1) and t as z_{1} ↦ −z_{1}.

We use (5.2) to give several interpretations of ZB_{n} by braids.

We remove the hyperplanes z_{j} = 0 from C^{n}. It remains the n-fold product C^{∗} × ⋯ × C^{∗} = C^{∗n}. Removal of the remaining reflection hyperplanes yields the space X of n-tuples (z_{j}) ∈ C^{∗n} with pairwise different squares z_{j}^{2}.

The configuration space C^{n}(C^{∗}) is the space of subsets of C^{∗} with cardinality n. As topological space it is defined as Y/S_{n} where Y ⊂ C^{∗n} is the set of n-tuples (y_{j}) with pairwise different components.

(4.3) Proposition. X/W is homeomorphic to C^{n}(C^{∗}).

Proof. We arrive at X/W in two steps: First we form Y′ = X/(ZZ/2)^{n} and then we divide out by the S_{n}-action. The map (z_{j}) ↦ (z_{j}^{2}) yields an S_{n}-equivariant homeomorphism Y′ → Y. □

By (5.2) and (5.3), ZB_{n} ≅ π_{1}(C^{n}(C^{∗})). The elements of π_{1}(C^{n}(C^{∗})) will be interpreted as braids in the cylinder (cylindrical braids). We take (1, ω, ⋯, ω^{n−1}), ω = exp(2πi/n), as base point in C^{n}(C^{∗}). A loop in C^{n}(C^{∗}) lifts to a path

w: I → Y, t ↦ (w_{1}(t), …, w_{n}(t))

with this initial point. Thus we have (1) w(0) = (1, ω, …, ω^{n−1}).

(2) w(1) = (σ(1), …, σ(ω^{n−1})), with a permutation σ of the set ZZ/n = {1, ω, …, ω^{n−1}}.

(3) For j ≠ k we have w_{j}(t) ≠ w_{k}(t).

T. tom Dieck  4. Braids of type B  13

These data yield a braid z_{w} with n strings in C^{∗} × [0, 1] from ZZ/n × 0 to ZZ/n × 1

z_{w}(t) = {w_{1}(t), …, w_{n}(t)} × t.

Homotopy classes of loops correspond to isotopy classes of such braids. Multiplication of loops corresponds to concatenation of braids, as usual. Thus we have:

(4.4) Theorem. The braid group ZB_{n} is the group of n-string braids in the cylinder C^{∗} × [0, 1]. □

A second interpretation is by symmetric braids in C × [0, 1]. This was already used in [3]. We take the base point (1, 2, …, n) ∈ X. We lift a loop in X/W to a path

w: I → X, t ↦ (w_{1}(t), …, w_{n}(t)).

Then we have:

(1) w(0) = (1, 2, …, n).

(2) w(1) = (±σ(1), …, ±σ(n)) with a permutation σ of {1, …, n}.

(3) For j ≠ k we have w_{j}(t) ≠ ±w_{k}(t).

(4) w_{j}(t) ≠ 0. Let [±n] = {−n, …, −1, 1, …, n}. The data yield a braid with 2n strings in C × [0, 1] from [±n] × 0 to [±n] × 1, namely

t ↦ {−w_{n}(t), …, −w_{1}(t), w_{1}(t), …, w_{n}(t)} × t.

These braids are ZZ/2-equivariant with respect to (z, t) ↦ (−z, t) and are therefore called symmetric. The theorem of Brieskorn thus gives:

(4.5) Theorem. The group ZB_{n} is isomorphic to the group of symmetric braids with 2n strings. □

Symmmetric braids are drawn as ordinary braids but with additional symmetry with respect to the axis 0 × [0, 1]. Here are figures for the generators t and g_{j}.

??

The symmetry is not the reflection in the axis, but corresponds to a spacial rotation about this axis. The relation tg_{1}tg_{1} = g_{1}tg_{1}t appears in this context as a generalized Reidemeister move.

Braids in the cylinder with n strings can be visualized as ordinary braids with n + 1 strings — the axis of the cylinder is the additional string. This method has been used by Lambropoulou [??]. It allows the reduction of B_{n}-type braids to ordinary Artin braids, also with respect to proofs. The theorem of Brieskorn is then not used.

The twofold covering, ramified along the axis, of the cylinder produces a symmetric braid from a cylindrical one — and vice versa.

The cylinder C^{∗} ×[0, 1] has the universal covering C×[0, 1]. Lifting cylindrical braids with n strings produces n-periodic infinite braids in C × [0, 1] from ZZ × 0 to ZZ × 1. They are invariant with respect to the translation (z, t) ↦ (z + n, t). This gives yet another interpretation of ZB_{n} by n-periodic braid pictures.

The relation between ZB_{n} and ZÃ_{n−1} has the following geometric origin or counterpart. The map

C^{∗n} → C^{∗}, (z_{1}, …, z_{n}) ↦ z_{1} · . . . · z_{n}

is S_{n}-equivariant and induces therefore a map from the configuration space

α: C^{n}(C^{∗}) → C^{∗}.

(4.6) Lemma. The map α is a fibre bundle.

Proof. Let

H = {(z_{1}, …, z_{n}) ∈ C^{∗n} | ∏ z_{j} = 1}.

This is an S_{n}-invariant subset. The map

γ: C^{∗} ×_{ZZ/n} H → C^{∗n}, (z, z_{1}, …, z_{n}) ↦ (zz_{1}, …, zz_{n})

is an S_{n}-equivariant homeomorphism. Thus γ is the fibre bundle with fibre H assoziated to the ZZ/n-principal bundle C^{∗} → C^{∗}, z ↦ z^{n}. In C^{∗n} we have to remove the subset

C = {(z_{1}, …, z_{n}) | there exists i ≠ j such that z_{i} = z_{j}}.

Let D = H ∩ C. Then γ induces an S_{n}-equivariant homeomorphism

γ: C^{∗} ×_{ZZ/n} (H \ D) → C^{∗n} \ S.

This yields the fibre bundle description

C^{∗} ×_{ZZ/n} (H \ T)/S_{n} → C^{∗}

for the configuration space. □

We apply the fundamental group to this fibration and obtain the exact sequence

1 → kernel α_{∗} → ZB_{n} → ZZ → 0.

It can be shown that this is the sequence (2.11), i. e. ZÃ_{n−1} is the fundamental group of the fibre of α.

Our next aim is to describe an additive basis of the Hecke algebra H^{∞}B_{n} by geometric means, i. e. by specifying a certain canonical set of basic braids.

A cylindrical braid with n strings is called descending, if for i < j the i-th string is always overcrossing the j-th string. The i-th string is the one starting at ω^{i}, 0 ≤ i ≤ n − 1. Overcrossing means the following: We look radially and orthogonally from infinity onto the axis. The braid is in general position if we only see transverse double points. The first string we meet, coming from infinity, is the overcrossing one.

(4.7) Theorem. The descending braids form a 𝔎-basis of the algebra H^{∞}B_{n}. The descending braids with winding number zero form a 𝔎-basis of the algebra HÃ_{n−1}.

We use (2.11) to reduce the first statement to the second. For the latter Hecke algebra we have the canonical basis related to the elements of reduced form in the Weyl group, and elements of the Weyl group will be shown to correspond to descending braids. We use the description of the Weyl group elements as nperiodic permutations of ZZ. We represent such a permutation by n straight lines c_{1}, …, c_{n} in the strip IR × [0, 1] starting at {1, …, n} × 0 such that c_{i} and c_{j} have at most one crossing, and then repeat with period n. By slightly moving the endpoints of the c_{j} we can assume that the curves are in general position. The resulting crossings are used to write the permutation as a product of reflections. This product is reduced in the sense of Coxeter group theory (see (??)). It is geometrically obvious that the same configuration of crossings can be realized by a descending braid.

(4.8) Proposition. The set

ℭ = {y_{n−1}^{k}g_{n−1}g_{n−2} … g_{j} | k ∈ ZZ, 1 ≤ j ≤ n}

is a system of representatives for the left cosets of the inclusion W^{∞}B_{n−1} ⊂ W^{∞}B_{n}.

Proof. This is an immediate consequence of the semi-direct product description. The powers of y_{n−1} are representatives for cosets of V_{n−1} ⊂ V_{n}, and the products g_{n−1} … g_{j} are representatives for the cosets of S_{n−1} ⊂ S_{n}. □

We use this Proposition to derive the following result of Lambropoulou and Przytycki which was proved by them in a purely algebraic manner. The relation to standard Hecke algebra bases and the interpretation by descending braids seems more transparent, though.

(4.9) Theorem. Let 𝔅 be the canonical basis of H^{∞}B_{n−1}. Then {bc | b ∈ 𝔅, c ∈ ℭ} is a basis of H^{∞}B_{n}.

Proof. Represent a basis element of H^{∞}B_{n} by a descending braid. □

Recall the construction and definition of a Markov trace in section 2. The last Theorem gives immediately the uniqueness of a Markov trace with given parameters.

(4.10) Corollary. There exists a unique Markov trace on H^{∞}B_{n} with given parameters (s(k) | k ∈ ZZ) and z. □

From a Markov trace (U^{n}) on ZA one obtains a link invariant. Let x̂ denote the Alexander closure of the braid x ∈ ZA_{n−1}. Write x as a product of symbols (crossings) g_{1}, g_{1}^{−1}, …, g_{n−1}, g_{n−1}^{−1}, and let w(x) denote the resulting sum of exponents (writhe of x). Then a link invariant P is obtained by setting

P (x̂) := α^{−w(x)}β^{n}U^{n}(x)

for x ∈ ZA_{n−1}. Related are Markov traces T r = (T r_{n}) on Hecke algebras HA = (HA_{n}). These are 𝔎-linear maps T r_{n}: HA_{n−1} → 𝔎 such that

(1) T r_{n+1}|HA_{n−1} = T r_{n},

(2) T R_{n+1}(xx_{n}) = zT r_{n}(x), x ∈ HA_{n−1} with a parameter z ∈ 𝔎. Here we use the names x_{1}, …, x_{n−1} for the standard generators of the Hecke algebra because we want to distinguish them from the g_{j}. The Hecke algebras are defined with a parameter q ∈ 𝔎^{∗} which enters the quadratic relation x_{j}^{2} = (q − 1)x_{j} + q. The relation between the two notions of Markov traces is the following.

(4.11) Proposition. Let q = p^{2} and β(p − p^{−1}) = α − α^{−1} with p^{2} ≠ 1. Let U = (U^{n}) be a Markov trace on ZA with parameters α, β, as defined in section 3. Let ι: ZA_{n} → (HA_{n})^{∗} be the homomorphism g_{j} ↦ p^{−1}x_{j}. Then there exists a unique Markov trace T r on HA such that T r_{n} ∘ ι = U^{n}. It has parameter z = p^{−1}αβ^{−1}. The corresponding link invariant satisfies the skein relation αP (L+) − α^{−1}P (L_{−}) = (p − p^{−1})P (L_{0}). □

Lambropoulou [??] has proved a Markov theorem for links of type B (symmetric links). The statement is exactly as in the classical case, here called of type A. A Markov trace (T^{n}: T B_{n} → 𝔎) therefore yields an invariant of B-links by setting

P (x̂) = α^{−w(x)}β^{n}T^{n}(x)

for x ∈ ZB_{n}. Here w(x) still counts the exponent sum in terms of the generators g_{j}.

T. tom Dieck 5. Representations of Hecke algebras and R-matrices 17

5. Representations of Hecke algebras and R-matrices 6. References

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Tammo tom Dieck Mathematisches Institut Bunsenstraße 3/5 D – 37073 Göttingen tammo@uni-math.gwdg.de