M Theory Lesson 210

In the last lesson we saw how the 1-circulant eigenvalues $(312)$ and $(231)$ 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 $\mathbb{R}$, 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 $\textrm{cos} \theta \cdot (X,X,X,X,X,X)$. 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.