Geometric Matrices and the Symmetric Group

TL;DR

This paper develops a matrix representation of geometric algebras using Kronecker products of null vectors, linking algebraic structures with symmetric group representations in a geometric framework.

Contribution

It introduces a novel construction of matrices from null vectors in geometric algebra, enabling a new perspective on symmetric group representations.

Findings

Matrix representations for geometric algebras with p+q <= 2n+1

Unique sum-of-products form for matrices in the null vector basis

Properties of symmetric group irreducible representations in geometric algebra context

Abstract

We construct real and complex matrices in terms of Kronecker products of a Witt basis of 2n null vectors in the geometric algebra over the real and complex numbers. In this basis, every matrix is represented by a unique sum of products of null vectors. The complex matrices provide a direct matrix representation for geometric algebras with signatures p+q <= 2n+1. Properties of irreducible representations of the symmetric group are presented in this geometric setting.