Automorphisms of the Cube $n^d$

TL;DR

This paper characterizes the automorphism group of the hypergraph formed by lines in a d-dimensional cube and proves the colored cube isomorphism problem is GI-complete, advancing understanding of symmetries in combinatorial structures.

Contribution

It provides a complete characterization of the automorphism group of the hypergraph of lines in a d-dimensional cube and establishes GI-completeness for the colored cube isomorphism problem.

Findings

Complete automorphism group characterization of the hypergraph $H_n^d$.

Colored Cube Isomorphism problem is GI-complete.

Advances understanding of symmetries in high-dimensional combinatorial cubes.

Abstract

Consider a hypergraph $H_n^d$ where the vertices are points of the $d$-dimensional combinatorial cube $n^d$ and the edges are all sets of $n$ points such that they are in one line. We study the structure of the group of automorphisms of $H_n^d$, i.e., permutations of points of $n^d$ preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding whether for two colorings of the vertices of $H_n^d$ there exists an automorphism of $H_n^d$ preserving the colors. We show that this problem is ${\sf GI}$-complete.