M Theory Lesson 208
Now consider the $6 \times 6$ Kasteleyn matrix given by 
This matrix is the unique such matrix with eigenvalues $(1) + (2) = (231)$ and $(1) - (2) = (312)$, the elements of $S_{3}$. It satisfies the relation $K^{2} = K + [0,(312) - (231)]$, using the same notation as the last post. This comes close to being idempotent, but the real idempotents are of course Carl's particle operators. A graph for this $K$ 
looks like the tiling by hexagons and triangles, which is a rectification of the hexagonal tiling of the plane. Observe that the six edges of the top left (1) form a hexagon within this graph, as do the other circulant components. The graph can be factored into two Hamiltonian circuits of length 12.
This matrix is the unique such matrix with eigenvalues $(1) + (2) = (231)$ and $(1) - (2) = (312)$, the elements of $S_{3}$. It satisfies the relation $K^{2} = K + [0,(312) - (231)]$, using the same notation as the last post. This comes close to being idempotent, but the real idempotents are of course Carl's particle operators. A graph for this $K$ 
looks like the tiling by hexagons and triangles, which is a rectification of the hexagonal tiling of the plane. Observe that the six edges of the top left (1) form a hexagon within this graph, as do the other circulant components. The graph can be factored into two Hamiltonian circuits of length 12.
posted by Kea | 6:05 PM
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