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The Hecke generators satisfy the Artin relations and are units

Statement

Let $H (n)$ be the Hecke tower over $Λ = Z [v^{\pm 1}, z]$ of The Markov trace on the type-A Hecke tower. Then: (1) the elements $T_{1}, \dots, T_{n - 1}$ are units of $H (n)$ with $T_{i}^{- 1} = v^{- 1} T_{i} + (v^{- 1} - 1)$ ; (2) the assignment $σ_{i} \mapsto T_{i}$ descends to a group homomorphism $π_{n} : B_{n} ⟶ H (n)^{\times}, π_{n} (σ_{i}) = T_{i},$ where $H (n)^{\times}$ is the group of units of $H (n)$ ; (3) for every Artin word $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}$ one has $π_{n} (β) = T_{i_{1}}^{ε_{1}} \dots T_{i_{k}}^{ε_{k}}$ , and $π_{n + 1} (ι_{n}^{B r} (β)) = ι_{n} (π_{n} (β))$ for all $β \in B_{n}$ under the standard inclusion of braid groups $B_{n} \to B_{n + 1}$ .

Facts & Assumptions

Given: The Hecke tower over $Λ = Z [v^{\pm 1}, z]$ and an integer $n \geq 1$ . No choice principle is used.

[F1]

$H (n)$ is the $Λ$ -algebra with generators $T_{1}, \dots, T_{n - 1}$ and the relations $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}$ for $1 \leq i \leq n - 2$ and $T_{i} T_{j} = T_{j} T_{i}$ for $∣ i - j ∣ > 1$ (The generic type-A Hecke algebra, The Markov trace on the type-A Hecke tower).

[F2]

Each generator $T_{i}$ is a unit of $H (n)$ with $T_{i}^{- 1} = v^{- 1} T_{i} + (v^{- 1} - 1)$ (The Markov trace of an inverse Hecke generator).

[F3]

$B_{n} = ⟨ σ_{1}, \dots, σ_{n - 1} ∣ Artin relators ⟩$ with the braid and far-commutation relations, $B_{0}, B_{1}$ trivial (The braid group by Artin presentation).

[F4]

Von Dyck: a function from the generators of a presented group to a group $G$ that sends every defining relator to the identity extends to a unique group homomorphism (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group); a group homomorphism satisfies $φ (x_{1} \dots x_{k}) = φ (x_{1}) \dots φ (x_{k})$ and $φ (x^{- 1}) = φ (x)^{- 1}$ (Monoid homomorphism and group homomorphism).

Proof

1.1F2F3F4

The assignment kills the relators. Define $u (σ_{i}) : = T_{i} \in H (n)^{\times}$ , using [F2] to regard each $T_{i}$ as an element of the unit group. At the braid relator $σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}$ both sides are sent to the equal elements $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}$ of [F1]; at a far-commutation relator both sides are sent to $T_{i} T_{j} = T_{j} T_{i}$ by [F1]. Hence every defining relator of [F3] is sent to the identity and [F4] applies, giving a unique group homomorphism $π_{n} : B_{n} \to H (n)^{\times}$ with $π_{n} (σ_{i}) = T_{i}$ . For $n = 1$ the domain $B_{1}$ is trivial and its unique homomorphism into $H (1)^{\times}$ sends the identity to the unit of $H (1)$ .

2.1F3F4step 1.1∎

Words and compatibility. For an Artin word $β = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{k}}^{ε_{k}}$ , [F4] gives $π_{n} (β) = π_{n} (σ_{i_{1}})^{ε_{1}} \dots π_{n} (σ_{i_{k}})^{ε_{k}} = T_{i_{1}}^{ε_{1}} \dots T_{i_{k}}^{ε_{k}}$ , since negative exponents are the inverses from step 1.1; this also shows that the value does not depend on the chosen word, being the value of the homomorphism $π_{n}$ at the element $β$ . The standard inclusion $B_{n} \to B_{n + 1}$ sends each Artin generator $σ_{i}$ with $i \leq n - 1$ to the generator with the same name (The braid group by Artin presentation), and $ι_{n} : H (n) \to H (n + 1)$ sends $T_{i}$ to $T_{i}$ (The Markov trace on the type-A Hecke tower); hence $π_{n + 1} (ι_{n}^{B r} (β))$ and $ι_{n} (π_{n} (β))$ are both the product of the $T_{i}^{ε_{i}}$ computed in $H (n + 1)$ , and they agree.

Depends on

Used by

An unnormalized Hecke trace is not Markov invariant Counterexample
The HOMFLYPT polynomial from the Hecke Markov trace Definition
The Hecke trace skein calculation for a three-crossing braid Example
The Jones specialization of a two-strand closure Example
The Hecke trace construction is an oriented link invariant Theorem
The HOMFLYPT skein relation Theorem

Dependency tree · two levels

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Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.3 printed pp. 47-48 (the map f: B_n to H_n(t) defined on generators and the Hecke relations) (standard reference, not scraped)