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The Generalized Terwilliger Algebra of the Hypercube

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arxiv 2301.08366 v2 pith:5H35XQH6 submitted 2023-01-20 math.CO

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keywords gammaalgebranaturalisomorphismmathcalterwilligerthencomplete
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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 $\Gamma$.

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  1. The Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercubes

    math.CO 2025-05 conditional novelty 7.0 of 10

    The degree-N polynomials in four variables, the fixed 3-tensors of the hypercube, and the hypercube Terwilliger algebra are all isomorphic as sl4(C)-modules, with explicit maps.

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