## Affine Braids, Markov Traces and the Category $𝒪$

Last update: 22 December 2013

## Markov Traces

A Markov trace on the affine braid group is a trace functional which respects the inclusions ${\stackrel{\sim }{ℬ}}_{1}\subseteq {\stackrel{\sim }{ℬ}}_{2}\subseteq \cdots$ where $ℬ∼k ↪ ℬ∼k+1 b 1 ⋯ k ⟼ b 1 ⋯ k k+1 (5.1)$ More precisely, a Markov trace on the affine braid group with parameters $z,{Q}_{1},{Q}_{2},\dots \in ℂ$ is a sequence of functions ${\text{mt}}_{k}:{\stackrel{\sim }{ℬ}}_{k}⟶ℂ$ such that

 (1) ${\text{mt}}_{1}\left(1\right)=1,$ (2) ${\text{mt}}_{k+1}\left(b\right)={\text{mt}}_{k}\left(b\right),$ for $b\in {\stackrel{\sim }{ℬ}}_{k},$ (3) ${\text{mt}}_{k}\left({b}_{1}{b}_{2}\right)={\text{mt}}_{k}\left({b}_{2}{b}_{1}\right),$ for ${b}_{1},{b}_{2}\in {\stackrel{\sim }{ℬ}}_{k},$ (4) ${\text{mt}}_{k+1}\left(b{T}_{k}\right)=z{\text{mt}}_{k}\left(b\right),$ for $b\in {\stackrel{\sim }{ℬ}}_{k},$ (5) ${\text{mt}}_{k+1}\left(b{\left({\stackrel{\sim }{X}}^{{\epsilon }_{k+1}}\right)}^{r}\right)={Q}_{r}{\text{mt}}_{k}\left(b\right),$ for $b\in {\stackrel{\sim }{ℬ}}_{k},$
where $X∼εk+1= TkTk-1⋯T2 Xε1T2-1⋯ Tk-1-1 Tk-1= 1 2 ⋯ k+1$

If $M$ is a finite dimensional $U={U}_{h}𝔤$ module and $a\in {\text{End}}_{U}\left(M\right)$ the quantum trace of $a$ on $M$ (see [LRa1977, §3] and [CPr1994, Def. 4.2.9]) is the trace of the action of ${e}^{h\rho }a$ on $M,$ $trq(a)=Tr (ehρa,M), anddimq(M) =trq(idM)= Tr(ehρ,M) (5.2)$ is the quantum dimension of $M\text{.}$ The first step of the standard argument for proving Weyl’s dimension formula [BtD1985, VI Lemma 1.19], shows that the quantum dimension of the finite dimensional irreducible ${U}_{h}𝔤\text{-module}$ $L\left(\mu \right)$ is $dimq(L(μ)) =Tr(ehρ,L(μ)) = ∏α>0 eh2⟨μ+ρ,α∨⟩- e-h2⟨μ+ρ,α∨⟩ eh2⟨ρ,α∨⟩- e-h2⟨ρ,α∨⟩ = ∏α>0 [⟨μ+ρ,α∨⟩] [⟨ρ,α∨⟩] , (5.3)$ where $q={e}^{h/2}$ and $\left[d\right]=\left({q}^{d}-{q}^{-d}\right)/\left(q-{q}^{-1}\right)$ for a positive integer $d\text{.}$

Let $\mu ,\nu \in {P}^{+}$ be dominant integral weights. Let $M=L\left(\mu \right)$ and $V=L\left(\nu \right)$ and let ${\Phi }_{k}$ be the representation of ${\stackrel{\sim }{ℬ}}_{k}$ defined in Proposition 3.1. Then the functions $mtk: ℬ∼k ⟶ ℂ b ⟼ trq(Φk(b)) dimq(M)dimq(V)k$ form a Markov trace on the affine braid group with parameters $z= q⟨ν,ν+2ρ⟩ dimq(V)$ and $Qr=∑λ qr(⟨λ,λ+2ρ⟩-⟨μ,μ+2ρ⟩-⟨ν,ν+2ρ⟩) dimq(L(λ))cμνλ dimq(L(μ)) dimq(L(ν)) ,$ where the positive integers ${c}_{\mu \nu }^{\lambda }$ and the sum in the expression for ${Q}_{r}$ are as in the tensor product decomposition $L(μ)⊗L(ν)= ⨁λL(λ)⊕cμνλ.$

 Proof. The fact that ${\text{mt}}_{k}$ as defined in the statement of the Theorem satisfies (1)–(4) in the definition of a Markov trace follows exactly as in [LRa1977, Theorem 3.10c]. The formula for the parameter $z$ is derived in [LRa1977, (3.9) and Thm. 3.10(2)]. It remains to check (5). The proof is a combination of the argument used in [Ore1999, Theorem 5.3] and the argument in the proof of [LRa1977, Theorem 3.10c]. Let ${\epsilon }_{k}:{\text{End}}_{U}\left(M\otimes {V}^{\otimes k}\right)\to {\text{End}}_{U}\left(M\otimes {V}^{\otimes \left(k-1\right)}\right)$ be given by $εk(z)= (idM⊗V⊗(k-1)⊗ě) ∘(z⊗id) (5.4)$ where $ě: V⊗V* ⟶ ℂ x⊗ϕ ⟼ dimq(V)-1ϕ(ehρx).$ If $V$ is simple then $ě$ is the unique ${U}_{h}𝔤\text{-invariant}$ projection onto the invariants in $V\otimes {V}^{*}\text{.}$ Pictorially, $εk ( z 1 ⋯ k ) = z = εk(z) 1 ⋯ k-1 .$ The argument of [LRa1977, Theorem 3.10b] shows that $mtk(b)= mtk-1 (εk-1(b)) ,if b∈ℬ∼k. (5.5)$ Since ${\epsilon }_{1}\left({\left({X}^{{\epsilon }_{1}}\right)}^{r}\right)$ is a ${U}_{h}𝔤\text{-module}$ homomorphism from $M$ to $M$ and, since $M$ is simple, Schur’s lemma implies that $r loops { =ε1((Xε1)r) =ξ·idM,for some ξ∈ℂ.$ Let ${\stackrel{\sim }{R}}_{i}={\text{id}}_{V}^{\otimes \left(i-1\right)}\otimes {Ř}_{VM}\otimes {\text{id}}_{V}^{\left(k+1\right)-i}\text{.}$ Then $(X∼εk+1)r= (R∼k⋯R∼1)-1 (Xε1)r (R∼k⋯R∼1)$ and $mtk+1(b(X∼εk+1)r) = mtk(εk(b(X∼εk+1)r)) = mtk ( εk ( b (R∼k⋯R∼1)-1 (Xε1)r (R∼k⋯R∼1) ) ) = mtk ( b(R∼k⋯R∼1)-1 ε1((Xε1)r) (R∼k⋯R∼1) ) = mtk ( b(R∼k⋯R∼1)-1 ξ·idM R∼k⋯R∼1 ) =ξ·mtk(b).$ This last calculation is more palatable in a pictorial format, $mtk+1 ( b (X∼εk+1)r 1 ⋯ k ) = mtk+1 ( b (X∼εk+1)r 1 ⋯ k ) = mtk ( b (X∼εk+1)r 1 ⋯ k ) = ξ·mtk ( b 1 ⋯ k ) = ξ·mtk ( b 1 ⋯ k ) .$ It remains to calculate the constant $\xi \text{.}$ By (2.14), $(Xε1)r= (Ř02)r= (∑λqc(λ)Pμνλ)r =∑λqrc(λ) Pμνλ,$ where $c\left(\lambda \right)=⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩$ and ${P}_{\mu \nu }^{\lambda }$ is the projection onto the ${L\left(\lambda \right)}^{\oplus {c}_{\mu \nu }^{\lambda }}$ component in the decomposition of $M\otimes V=L\left(\mu \right)\otimes L\left(\nu \right)\text{.}$ Thus $ξ = mt0(ξ·idM)= mt1((Xε1)r)= 1dimq(M)dimq(V) trq(∑λqrc(λ)Pμνλ) = 1dimq(M)dimq(V) trq(∑λqrc(λ)cμνλidL(λ)) = ∑λqrc(λ) cμνλ dimq(L(λ)) dimq(L(μ)) dimq(L(ν)) .$ $\square$

There is another formula [TWe1217386, Lemma (3.51)], for the constant ${Q}_{1}$ in Theorem 5.1, namely, $Q1= ∑w∈W (-1)ℓ(w) q⟨μ+ρ,w(ν+ρ)⟩ ∑w∈W (-1)ℓ(w) q⟨μ+ρ,wρ⟩ , (5.6)$ where $W$ is the Weyl group of $𝔤\text{.}$

Let ${\text{mt}}_{k}$ be as in Theorem 5.1 and let ${\stackrel{\sim }{𝒵}}_{k}={\text{End}}_{U}\left(M\otimes {V}^{\otimes k}\right)\text{.}$ Then ${\text{mt}}_{k}$ is the restriction of the linear functional $mtk: 𝒵∼k ⟶ ℂ a ⟼ trq(a) dimq(M) dimq(V⊗k) (5.7)$ to ${\Phi }_{k}\left({\stackrel{\sim }{ℬ}}_{k}\right)\text{.}$ Since $M\otimes {V}^{\otimes k}$ is a finite dimensional semisimple module ${\stackrel{\sim }{𝒵}}_{k}$ is a finite dimensional semisimple algebra. The weights of the Markov trace mt are the constants ${t}_{\lambda /\mu }$ defined by $mtk=∑λ tλ/μ χ𝒵∼kλ/μ, (5.8)$ where ${\chi }_{{\stackrel{\sim }{𝒵}}_{k}}^{\lambda /\mu }$ are the irreducible characters of ${\stackrel{\sim }{𝒵}}_{k}\text{.}$

Let $M=L\left(\mu \right)$ and $V=L\left(\nu \right)$ be finite dimensional irreducible ${U}_{h}𝔤\text{-modules.}$ The weights of the Markov trace on the affine braid group defined in Theorem 5.1 are $tλ/μ= dimq(L(λ)) dimq(L(μ)) dimq(V)k .$

 Proof. Since $M\otimes {V}^{\otimes k}$ is finite dimensional and semisimple the algebra ${\stackrel{\sim }{𝒵}}_{k}={\text{End}}_{U}\left(M\otimes {V}^{\otimes k}\right)$ is a finite dimensional semisimple algebra. Schur’s lemma can be used to show that, as a $\left({U}_{h}𝔤,{\stackrel{\sim }{𝒵}}_{k}\right)$ bimodule, $M⊗V⊗k≅ ⨁λL(λ)⊗ ℒλ/μ, (5.9)$ where the ${ℒ}^{\lambda /\mu }$ are the irreducible ${\stackrel{\sim }{𝒵}}_{k}$ modules. In the notation of (4.1), ${ℒ}^{\lambda /\mu }={F}_{\lambda }\left(L\left(\mu \right)\right)$ and ${\chi }_{{\stackrel{\sim }{𝒵}}_{k}}^{\lambda /\mu }$ is the character of ${ℒ}^{\lambda /\mu }\text{.}$ Taking the quantum trace on both sides of (5.9) gives $tr1(a)=Tr (e-hρa)= ∑λTr (e-hρ,L(λ)) χ𝒵∼kλ/μ (a)=∑λtr′ (L(λ)) χ𝒵∼kλ/μ (a).$ The result follows by dividing both sides by ${\text{dim}}_{q}\left(L\left(\mu \right)\right){\text{dim}}_{q}{\left(V\right)}^{k}\text{.}$ $\square$

## Notes and references

This is a typed version of the paper Affine Braids, Markov Traces and the Category $𝒪$ by Rosa Orellana and Arun Ram*.

*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.

This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.