Published online by Cambridge University Press: 20 November 2018
Let $F$ be a totally real number field and let
$\text{G}{{\text{L}}_{n}}$ be the general linear group of rank
$n$ over
$F$. Let
$\mathfrak{p}$ be a prime ideal of
$F$ and
${{F}_{\mathfrak{p}}}$ the completion of
$F$ with respect to the valuation induced by
$\mathfrak{p}$. We will consider a finite quotient of the affine building of the group
$\text{G}{{\text{L}}_{n}}$ over the field
${{F}_{\mathfrak{p}}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.