Commuting Difference Operators and the Combinatorial Gale Transform

We develop the spectral theory of $n$-periodic strictly triangular difference operators $L=T^{-k-1}+\sum_{j=1}^k a_i^j T^{-j}$ and the spectral theory of the “superperiodic” operators for which all solutions of the equation $(L+1)\psi=0$ are (anti)periodic. We show that, for a superperiodic operator $L$ of order $k+1$, there exists a unique superperiodic operator $\mathcal{L}$ of order $n-k-1$ which commutes with $L$ and show that the duality $L\leftrightarrow \mathcal{L}$ coincides, up to a certain involution, with the combinatorial Gale transform recently introduced in [S. Morier-Genoud, V. Ovsienko, R. E. Schwartz, S. Tabachnikov, Linear difference equations, frieze patterns and combinatorial Gale transform, Forum Math. Sigma, 2 (2014), e22].