Path homology of circulant digraphs

We organize and extend a set of computations and structural observations about the Grigoryan–Lin–Muranov–Yau (GLMY) path complex of circulant digraphs $\vec{C}_n^S$ and circulant graphs $C_n^S$. Using the shift automorphism $τ$ and a Fourier decomposition, we reduce many rank computations for the GLMY boundary maps to finite-dimensional $τ$-eigenspaces. This provides a reusable “symbol-matrix” recipe that highlights (i) the dependence on prime versus composite $n$ and (ii) stability phenomena for certain natural choices of connection sets $S$. Several fully worked examples are included, together with a discussion of how the additive structure of $S$ governs low-dimensional chains and Betti numbers.

💡 Research Summary

The paper develops a systematic method for computing the Grigoryan‑Lin‑Muranov‑Yau (GLMY) path homology of circulant digraphs (\vec C_n^S) and their undirected counterparts (C_n^S). After recalling the definition of the GLMY path complex, the authors focus on the special symmetry of circulant graphs: the cyclic group (\mathbb Z_n) acts by translation, giving rise to an automorphism (\tau) that commutes with the boundary operator (\partial). By working over a field that contains all (n)‑th roots of unity (e.g., (\mathbb C)), they diagonalize (\tau) via the discrete Fourier transform. This yields a decomposition \