Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\mathrm{\Gamma }$ gives rise to a finite group ${\mathbb{Z}}^{n-1}\mathrm{/}{\mathbb{Z}}^{n-1}M$ known as the sandpile (or critical) group $S(\mathrm{\Gamma })$ of $\mathrm{\Gamma }$. We determine $S(\mathrm{\Gamma })$ of the generalized de Bruijn graphs $\mathrm{\Gamma }=\mathrm{D}\mathrm{B}(n,d)$ with vertices $0,\dots ,n-1$ and arcs $(i,di+k)$ for $0\le i\le n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in {\mathrm{G}\mathrm{L}}_n(p)$ we show that $S(\mathrm{D}\mathrm{B}(n,p))$ is isomorphic to the quotient by $⟨X⟩$ of the centralizer of $X$ in ${\mathrm{P}\mathrm{G}\mathrm{L}}_n(p)$. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field ${\mathbb{F}}_{p^n}$ from spanning trees in $\mathrm{D}\mathrm{B}(n,p)$.