We can fit other objects into circulant loops and lines. Kneemo tells me to think about the Fano star diagram. The star has $9$ vertices as shown, with lines of three vertices. With the unmarked central triangle $(j_1, j_2, j_3)$, there are many triangles in the diagram. Choosing $2$ separated sets of three vertices, we can form the off diagonal $1$-circulant positions of a $3 \times 3$ matrix. The $2$-circulant entries also form lines in the diagram.

Alternatively, we could have filled the matrix $1$-circulants with the three obvious triangles: small $j_i$, large $J_i$ and $I$. This can be done so that the $2$-circulants fit the remaining three long lines.

8 years ago

Observe the similarity between this diagram and the permutohedron Fano diagram, which hinges on the usual seven vertices, plus three extra vertices.

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