Spin representations and binary numbers

Abstract

We consider a construction of the fundamental spin representations of the simple Lie algebras so(n) in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a Z-graded associative algebra (rather than the usual 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.