Structure and representation theory of double group of four-dimensional cubic group

TL;DR

This paper generalizes the cubic group to four dimensions, analyzing its structure, conjugacy classes, and representations, including double and spinor representations, using Clifford theory.

Contribution

It introduces the double group and spinor representations for four-dimensional cubic groups, extending previous three-dimensional theories.

Findings

Derived all inequivalent single-valued representations of O_4

Classified conjugacy classes of the four-dimensional cubic group

Calculated the structure of the double group and spinor representations

Abstract

We generalize the concept of cubic group into any dimension and derive their conjugate classifications and representation theorys. Double group and spinor representation are defined. 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.