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.
15 years ago
Thursday, February 24, 2011
Theory Update 67
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.
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.
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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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