Classical Kloosterman sums: representation theory, magic squares, and Ramanujan multigraphs

TL;DR

This paper explores the representation theory of a finite group related to Kloosterman sums, introduces special matrices with combinatorial and arithmetic properties, and connects them to Ramanujan-like multigraphs.

Contribution

It establishes a novel link between Kloosterman sums, matrix theory, and Ramanujan multigraphs, revealing new algebraic and combinatorial structures.

Findings

Kloosterman sums appear as eigenvalues of specific matrices

Matrices exhibit 'magical' combinatorial properties

These matrices encode arithmetic properties of Kloosterman sums

Abstract

We study the representation theory of a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of "magical" combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.