M Theory Lesson 210

In the last lesson we saw how the 1-circulant eigenvalues $\left(312\right)$ and $\left(231\right)$ correspond to an eigenvector in modulo 7 arithmetic. The remaining $3\times 3$ 1-circulant is the identity operator. Observe that for the operator $K$, the identity is an eigenvalue for any vector of the form $(\pm X,\pm X,\pm X,\pm X,\pm X,\pm X)$. For the cyclic group on 7 elements there are roughly $3\times 2^6+1=193$ such vectors, including zero. For elements of $ℝ$, or $\mathbb{Q}$, vectors of a fixed sign sequence also form an eigenline. In general we might call such a sign sequence an eigenpath for the identity. Other phase choices for the Kasteleyn matrix $K$ clearly alter the eigenspace structure. For example, the operator sends the vector $(X,X,X,X,X,X)$ to $\text{cos}\theta \cdot (X,X,X,X,X,X)$. Democratic matrices, with all entries equal to $X$, may also be considered eigenvectors.

Aside: The difference between 192 and some other integers is the source of a very silly argument between Distler and Lisi.