Commuting difference operators and the combinatorial Gale transform

We study 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 $L$ there exists a unique superperiodic operator ${\cal L}$ of order $(n-k-1)$ which commutes with $L$ and show that the duality $L\leftrightarrow {\cal L}$ coincides up to a certain involution with the combinatorial Gale transform recently introduced in [21].