In a paper by Kenyon, Propp and Wilson, the number of spanning trees in a certain graph in the hexagonal lattice is:
$ \prod_{a,b,c} (3 - a-b-c)^{1/6}$
where $a^{3n}=1, (a/b)^n=1,abc=1$ and $a,b,c$ are distinct.
In between the lines, the paper says this product is an integer.
I tried to program it Mathematica
Product[ (3 - e[k/(3 n)] - e[(k + 3 l)/(3 n)] - e[(-2 k - 3 l)/(3 n)])^(1/6) , {k, 1, 3 n}, {l, 1, n - 1}];
The result is a real number but not an integer.
In mathematical terms I wrote:
$ \prod_{k=1}^{3n} \prod_{l=1}^{n-1} (3-\omega^k- \omega^{k+3l}- \omega^{-2k-3l}) $ where $\omega = e^{2\pi i /3n} = \cos \frac{2\pi}{3n} + i \sin \frac{2\pi}{3n}$ but this number has significant figures pas the decimal point.
How do I (correctly) parameterize the product in $a,b,c$ ?
Response to comments:
- it looks like we need $k \neq l, 2l \mod n$ and $l \neq 0 \mod n$.
- the Galois action seems to be $(a,b,c) \mapsto (\omega a , \omega b, \omega^{-2} c)$. Shouldn't the sum be Galois invariant even if $a,b,c$ are not always distinct?
- people who'd had not trouble with it, can you share your code?