Spin Representations and Binary Numbers

Abstract

We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a $\mathbb{Z}$-graded associative algebra (rather than the usual $\mathbb{N}$-filtered Clifford algebra). Our description gives a quick way to write down the spin matrices, and gives a way to encode some extra structure, such as the real structure which is invariant under the compact real form, for some $n$. Additionally we can encode the spin representations combinatorially as (coloured) graphs.