The Generalized Terwilliger Algebra of the Hypercube

Abstract

In the year 2000, Eric Egge introduced the generalized Terwilliger algebra $\mathcal{T}$ of a distance-regular graph $\Gamma$. For any vertex $x$ of $\Gamma ,$ there is a surjective algebra homomorphism $♮$ from $\mathcal{T}$ to the Terwilliger algebra $T(x)$. If $\Gamma$ is complete, then $♮$ is an isomorphism. If $\Gamma$ is not complete, then $♮$ may or may not be an isomorphism, and in general the details are unknown. We show that if $\Gamma$ is a hypercube, then the algebra homomorphism $♮:\mathcal{T}\to T(x)$ is an isomorphism for all vertices $x$ of $\Gamma$.

24 pages