The Generalized Terwilliger Algebra of the Hypercube

TL;DR

This paper proves that for hypercubes, the generalized Terwilliger algebra is isomorphic to the Terwilliger algebra at any vertex, clarifying the algebraic structure of hypercubes within distance-regular graphs.

Contribution

It establishes that the generalized Terwilliger algebra of a hypercube is isomorphic to the vertex-specific Terwilliger algebra, resolving a question about their relationship in this case.

Findings

The algebra homomorphism is an isomorphism for hypercubes.

The result applies to all vertices of the hypercube.

It extends understanding of algebraic structures in distance-regular graphs.

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 $\natural$ from $\mathcal T$ to the Terwilliger algebra $T(x)$. If $\Gamma$ is complete, then $\natural$ is an isomorphism. If $\Gamma$ is not complete, then $\natural$ 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 $\natural:\mathcal T \to T(x)$ is an isomorphism for all vertices $x$ of $\Gamma$.