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The Markov trace on the type-A Hecke tower

Definition

Let $A = Z [v^{\pm 1}]$ be the Laurent polynomial ring and let $Λ : = A [z] = Z [v^{\pm 1}, z]$ be the polynomial ring over $A$ in one indeterminate (The Laurent polynomial ring as the principal localisation of Z[t] at t, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). For $n \geq 1$ let $H (n) : = Λ \otimes_{A} H_{v} (n)$ be the scalar extension of the generic type-A Hecke algebra of The generic type-A Hecke algebra (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), so that $H (n)$ is the unital $Λ$ -algebra with generators $T_{1}, \dots, T_{n - 1}$ subject to $T_{i}^{2} = (v - 1) T_{i} + v, T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}, T_{i} T_{j} = T_{j} T_{i} (∣ i - j ∣ > 1),$ and $H (1) = Λ$ . The $Λ$ -algebra $H (n)$ is free with basis ${T_{w} : w \in S_{n}}$ , where $T_{w}$ is the product of the generators along a reduced word (The standard basis of the generic type-A Hecke algebra, The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition).

The generators induce a $Λ$ -algebra homomorphism $ι_{n} : H (n) \to H (n + 1)$ , $T_{i} \mapsto T_{i}$ ; it is injective, so we identify $H (n)$ with its image, a $Λ$ -subalgebra of $H (n + 1)$ .

A Markov trace on the type-A Hecke tower is a family of $Λ$ -linear maps ${tr}_{n} : H (n) \to Λ$ , $n \geq 1$ , such that

(M1) ${tr}_{n} (1) = 1$ for one (equivalently, by (M2), every) $n$ ;
(M2) ${tr}_{n + 1} \circ ι_{n} = {tr}_{n}$ for all $n \geq 1$ ;
(M3) ${tr}_{n} (x y) = {tr}_{n} (y x)$ for all $x, y \in H (n)$ ;
(M4) ${tr}_{n + 1} (x T_{n}) = z {tr}_{n} (x)$ for all $n \geq 1$ and $x \in H (n)$ .

Here $z \in Λ$ is a formal parameter, distinct from the Hecke parameter $v$ ; no value of $z$ is fixed or inverted by this definition, and the trace takes values in $Λ$ , not in a field.

Caveats. This is a family over the whole tower, not a single functional; the conditions (M1)--(M4) are a definition, so no trace is asserted to exist here (existence and uniqueness is The Ocneanu Markov trace exists and is unique). By (M3), (M4) is equivalent to ${tr}_{n + 1} (u T_{n} v) = z {tr}_{n} (u v)$ for all $u, v \in H (n)$ : applied to $u T_{n} v$ one gets ${tr}_{n + 1} (u T_{n} v) = {tr}_{n + 1} (T_{n} v u) = z {tr}_{n} (v u) = z {tr}_{n} (u v)$ , and conversely take $v = 1$ .

Facts & Assumptions

Given: The Laurent ring $A = Z [v^{\pm 1}]$ , the polynomial ring $Λ = A [z]$ , the generic Hecke algebras $H_{v} (n)$ over $A$ , and the scalar extensions $H (n) = Λ \otimes_{A} H_{v} (n)$ for $n \geq 1$ . No choice principle is used.

[F1]

$H_{v} (n)$ is the unital $A$ -algebra presented by $T_{1}, \dots, T_{n - 1}$ with $T_{i}^{2} = (v - 1) T_{i} + v$ , the braid relations and the distant commutations; $H_{v} (n) = A$ for $n \leq 1$ (The generic type-A Hecke algebra).

[F2]

${T_{w} : w \in S_{n}}$ is a $Λ$ -basis of $H (n)$ for $n \geq 2$ , and $H (1) = Λ$ ; the reduced word $T_{w}$ is well defined (The standard basis of the generic type-A Hecke algebra, The generic type-A Hecke algebra).

[F3]

Scalar extension along $A \to Λ$ presents $H (n)$ by the same generators and relations over $Λ$ , and $Λ = A [z]$ is the polynomial $A$ -algebra in one indeterminate (Algebras over a commutative ring, central structure maps, and algebra homomorphisms, Restriction of scalars and extension of scalars $S \otimes_{R} M$ along a ring homomorphism $R \to S$ , Universal property of the tensor product for balanced maps into abelian groups, The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution); $Λ$ is commutative with unit and $v$ is a unit of it (The Laurent polynomial ring as the principal localisation of Z[t] at t).

[F4]

A $Λ$ -algebra homomorphism is a ring homomorphism that is $Λ$ -linear (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ , Algebras over a commutative ring, central structure maps, and algebra homomorphisms); the universal property of a presented algebra yields a homomorphism from the presented algebra whenever the prescribed images satisfy the defining relations.

Proof

Step 1.1 establishes the well-formedness of the ambient tower and step 2.1 the injectivity and the stated equivalence of (M1); the four conditions are a definition and require no existence proof.

1.1F1F2F3F4given

The tower is well formed. On the tensor product in [F3], define $(λ \otimes h) (μ \otimes k) = λ μ \otimes h k$ : balancing over the central ring $A$ makes this bilinear product well defined, and associativity and the unit $1 \otimes 1$ follow from those of $H_{v} (n)$ . It has the asserted presentation over $Λ$ . Indeed the generators $1 \otimes T_{i}$ satisfy the relators, giving a map from that presented algebra to the tensor product. Conversely the presented $Λ$ -algebra receives an $A$ -algebra map from $H_{v} (n)$ by [F1], and the balanced map $(λ, h) \mapsto λ h$ extends to the tensor product by [F3]. The two maps are inverse on elementary tensors and generators. By [F2] each $H (n)$ is a free $Λ$ -module with the stated basis, so $H (n)$ is a unital $Λ$ -algebra and $H (1) = Λ$ . Since the defining relators of $H (n)$ are literally among the relators of $H (n + 1)$ under $T_{i} \mapsto T_{i}$ , [F4] applies to the assignment on generators and gives a $Λ$ -algebra homomorphism $ι_{n} : H (n) \to H (n + 1)$ with $ι_{n} (T_{i}) = T_{i}$ .

2.1F2step 1.1algebra∎

Injectivity and the equivalence in (M1). Under $ι_{n}$ the basis element $T_{w}$ , $w \in S_{n}$ , is carried to the element of $H (n + 1)$ given by the same reduced word, which is the standard basis element $T_{w}$ for $w$ regarded in $S_{n + 1}$ (fixing $n + 1$ ); these elements are pairwise distinct members of the $Λ$ -basis of $H (n + 1)$ by [F2], hence are linearly independent and $ι_{n}$ is injective. For the parenthetical in (M1): if ${tr}_{m} (1) = 1$ for some $m$ , then for $k \geq 1$ condition (M2) gives ${tr}_{k} (1) = {tr}_{k + 1} (ι_{k} (1)) = {tr}_{k + 1} (1)$ , so ${tr}_{n} (1) = {tr}_{m} (1) = 1$ for every $n$ ; (M2) applies in both directions because $ι_{k} (1) = 1$ . The equivalence of the two forms of (M4) follows from (M3) as displayed in the caveats.

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Sources

Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 4.3 printed pp. 47-49 (Ocneanu's trace theorem and the Markov trace on the Hecke tower) (standard reference, not scraped)
Theo Johnson-Freyd, MATH 448 Reshetikhin-Turaev invariants, lecture notes 15 January 2016, Sections 1-3 (the Ocneanu trace on the Hecke tower) (standard reference, not scraped)