Counting Involutions on Multicomplex Numbers

TL;DR

This paper establishes a bijection between automorphisms of multicomplex numbers and signed permutations, leading to formulas for counting involutions and their variants, revealing new structural insights into multicomplex algebra.

Contribution

It introduces a novel connection between automorphisms of multicomplex numbers and signed permutations, enabling enumeration of involutions and their variants.

Findings

Derived a formula for the number of involutions on multicomplex spaces.

Generalized involution counting to r-involutions and those preserving elementary imaginary units.

Provided new elementary results on multicomplex numbers, including counts of numbers squaring to ±1.

Abstract

We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a formula for the number of involutions on multicomplex spaces which generalizes a recent result on the bicomplex numbers and contrasts drastically with the quaternion case. We also generalize this formula to $r$-involutions and obtain a formula for the number of involutions preserving elementary imaginary units. The proofs rely on new elementary results pertaining to multicomplex numbers that are surprisingly unknown in the literature, including a count and a representation theorem for numbers squaring to $\pm 1$.