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When starting this question I was very hesitant - literature on the subject is vast and I thought most likely the answer is already there somewhere.

Then when the list "Questions that may already have your answer" appeared, the first on the list was Constructing Markov traces simply which is from March 2012 and has no answer, and this encouraged me to go ahead and ask:

There are famous Markov traces on (group algebras of) braid groups, providing knot/link invariants and much more, and constructed via similar traces on finite-dimensional algebras like Birman-Wenzl/Kaufmann, Iwahori-Hecke, Temperley-Lieb algebras etc.

The question is the most naïve one - can these traces be realized as plain ordinary traces of matrices in linear representations of algebras in any of these cases?

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    $\begingroup$ In the case of the Markov trace that gives rise to the Jones polynomial the corresponding representation should be (maybe up to some normalizations or changes of variable) the representation of Temperley-Lieb on $V^{\otimes n}$ where $V$ is the defining representation of the quantum group $U_q(\mathfrak{sl}_2)$. Similar quantum group constructions give relatives of the Jones polynomial. $\endgroup$
    – Qiaochu Yuan
    Commented May 13, 2015 at 7:54
  • $\begingroup$ @QiaochuYuan Thanks a lot, as a diletant I strongly feel this is the answer. But I need some aid in working out details. Could you please either slightly extend your comment and make it an answer, or point to a place in the literature where this might be done? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 13, 2015 at 9:18

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You can construct Markov traces explictly in tensor product representations of braid groups. If $V$ is a finite dimensional vector space and $R\in{\rm End}(V\otimes V)$ an invertible solution on the Yang-Baxter equation, represent the generators of the $n$ strand braid group $B_n$ on $V^{\otimes n}$ by $R_k:={\rm id}_V^{\otimes(k-1)}\otimes R\otimes{\rm id}_V^{\otimes(n-k-1)}$. The idea is now to pick $A\in{\rm End}V$ such that $[A\otimes A,R]=0$ and consider the functional $X\mapsto{\rm Tr}_{V^{\otimes n}}(A^{\otimes n}X)$. This is tracial on the representation generated by the $R_i$'s, and a Markov trace if also ${\rm Tr}_2(R\,(A\otimes A))=A$, where ${\rm Tr}_2={\rm id}_{\rm End V}\otimes {\rm Tr}_V$ denotes the partial trace. The pair $(R,A)$ is then called an enhanced Yang-Baxter operator, a good reference on the subject is the 1988 article "The Yang-Baxter equation and invariants of links" by Turaev in Inventiones. In case $A=1$, the condition amounts to $R$ having trivial partial trace, and the Markov trace is given by the usual matrix trace.

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    $\begingroup$ Excellent, thanks! I like it especially because it resembles some recent intriguing construction from "Partial transpose of random quantum states: Exact formulas and meanders" by Fukuda and Śniady in JMP 54 (2013) -- their construction of eigenvalue distributions for random quantum states involves partial traces of certain impressive Wishart matrices; the results are applied, in particular, to meander combinatorics. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 14, 2018 at 2:03
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    $\begingroup$ You're welcome. Partial traces often play a prominent role in this field because of their connection with conditional expectations. As an example, in a recent article of mine (joint work with U. Penning and S. Wood), we used partial traces to classify all involutive R-matrices, see arxiv.org/abs/1707.00196. $\endgroup$
    – Gandalf Lechner
    Commented Mar 14, 2018 at 13:43

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