Thanks. In the "modern" paper they also must assume $Z$ is a multiple of the identity when doing the Jack expansion. So this seems to be a real limitation of this method, which is a bit curious. I would love to get a comment about this from the symmetric function experts.

@wdlang Gutzwiller trace formula is only for chaotic systems, yes. Integrable systems have a different theory.

If you want to go beyond the formal expression, you must have some information about the sequences $\vec{i}$ and $\vec{j}$. Are all indices equal? Are they all different? etc.

@AlexanderGruber hard disagree on the idea of deleting the comments. Simple matter of free speech. If a female user comes here and finds the comments, she will notice that everybody else disagrees, frowns upon them and that the user is in hot water for them. The notion that we must act to "protect" female users, lest them see something that might "upset" them, is patronizing and might be sexist in itself, I think.

@darijgrinberg but that paper doesn't mention computing a polynomial with argument $x^{-1}$. How does it relate?

Which additional factor? I see $(N-2M-1)/2$ in both cases

So the exponent is the same as for the orthogonal group