On Magic Finite Projective Space

TL;DR

This paper generalizes the concept of magic squares to finite projective spaces, classifies functions with zero sums along flats, and explores their connections to number theory and modular representation theory.

Contribution

It provides a complete classification of functions on finite projective spaces with flat-sum zero properties, linking combinatorics, number theory, and representation theory.

Findings

Classified all functions with flat sums equal to zero in finite projective spaces.

Established connections between these functions and elementary number theory.

Linked the problem to the modular representation theory of (n,q).

Abstract

This paper studies a generalization of magic squares to finite projective space $\mathbb{P}^n(q)$. We classify at all functions from $\mathbb{P}^n(q)$ into a finite field where the sum along any $r$-flat is $0$. In doing so we show connections to elementary number theory and the modular representation theory of $\operatorname{GL}(n,q)$.