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Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. Pπ is the permutation P with elements j and j+1 switched. I need to get an explicit expression of A(P) for the recursion relation:

$$\frac{A(P\pi)}{A(P)} = - \exp(-i(k_{p_j}- k_{p_{j+1}}))$$

kj are numbers, and $p_j$ is the j-th element of the permutation.

I hope someone can give me a hint or advice to solve this.


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You might simplify it by considering the function B(P) = ilog(A(P). I assume ii + 1 = 0 here. Also, this may be "too localized" to be appropriate for Math Overflow. Gerhard "Ask Me About System Design" Paseman, 2010.06.23 – Gerhard Paseman Jun 23 '10 at 18:53
Hm ok, it's a problem I came up on physics but thought it might fit more on mathoverflow. This comes up on solving a many-body model with Bethe-ansatz. I might ask this question on "physics overflow" then. – Tobias Jun 23 '10 at 20:44
This isn't as "localized" as you might think Gerhard. But I would want to encourage Tobias to edit his equation to provide a couple examples (unless it's a standard homework problem in a first year grad course, in which case you shouldn't have asked it in the first place). – Alexander Woo Jun 24 '10 at 0:01
Another quick note on clarity: When you say "P$\pi$ is P with j and j+1 switched", it sounds like you're possibly describing P conjugated by $\pi$, rather than P$\pi$. Of course that's not what you wrote, and that wouldn't lead to a sensible recursion, I don't think, but still... – Harry Altman Jun 24 '10 at 5:31

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