Structure and Representation Theory for Double Group of Four-Dimensional Cubic Group

TL;DR

This paper explores the structure and representation theory of four-dimensional cubic groups, including their double groups and spinor representations, using Clifford theory to classify all representations.

Contribution

It provides a detailed analysis of the four-dimensional cubic group and its double group, introducing spinor representations and classifying all inequivalent representations.

Findings

Derived all single-valued and spinor representations of $O_4$

Classified conjugate classes of hypercubic groups in any dimension

Applied Clifford theory to decompose induced representations

Abstract

Hypercubic groups in any dimension are defined and their conjugate classifications and representation theories are derived. Double group and spinor representation are introduced. A detailed calculation is carried out on the structures of four-dimensional cubic group $O_4$ and its double group, as well as all inequivalent single-valued representations and spinor representations of $O_4$. All representations are derived adopting Clifford theory of decomposition of induced representations. Based on these results, single-valued and spinor representations of the orientation-preserved subgroup of $O_4$ are calculated.