Jones-Wenzl projectors are an explicit combinatorial description of the central projectors of the Temperley-Lieb algebra. They also describe very explicitly the failure of certain representations to exist for $q$ a root of unity and generate the negligible ideal (the ideal of elements $a$ such that $\operatorname{tr}(ab) = 0$ for every $b$.)

Is there a similar combinatorial description of such projectors for the Brauer algebra? Here the Brauer algebra is drawn with the same diagrams as Temperley-Lieb, but the strands are allowed to cross. It is a graphical description of the decomposition of tensor powers ofan irrep of a classical orthogonal or symplectic algebra.

I think the answer is "no" at least for all of the properties I mentioned above, but I'd love to be surprised. I also don't think I need all the properties above: just knowing some elements of the negligible ideal (for whichever value of the loop parameter and whatever number of strands) would be useful.

I think this question is related to mine, except that I looked up the book mentioned and didn't find it helpful.
The paper "On central idempotents in the Brauer algebra" shows how to compute *some* of the idempotents, but non-recursively and without a formula for the trace, so it may be that the question I'm asking doesn't even have a partial answer.