Path homology of circulant digraphs

TL;DR

This paper develops a computational framework for analyzing the path homology of circulant digraphs using automorphisms and Fourier methods, revealing structural dependencies and stability phenomena.

Contribution

It introduces a Fourier-based reduction technique for path homology computations of circulant digraphs, highlighting prime versus composite distinctions and structural stability.

Findings

Rank computations reduced to finite-dimensional eigenspaces

Dependence of homology on prime vs. composite n

Stability phenomena for natural connection sets

Abstract

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 $\tau$ and a Fourier decomposition, we reduce many rank computations for the GLMY boundary maps to finite-dimensional $\tau$-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.