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# A Hypergraph with Commuting Partial Laplacians

Published:2001-12-01
Printed: Dec 2001
• Cristina M. Ballantine
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## Abstract

Let $F$ be a totally real number field and let $\GL_{n}$ 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 $\GL_{n}$ 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.
 Keywords: Hecke operators, buildings
 MSC Classifications: 11F25 - Hecke-Petersson operators, differential operators (one variable) 20F32 - unknown classification 20F32

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