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.
The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $U_q(sl_2)$
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abstract
The adjacency matrix of a symplectic dual polar graph restricted to the eigenspaces of an abelian automorphism subgroup is shown to act as the adjacency matrix of a weighted subspace lattice. The connection between the latter and $U_q(sl_2)$ is used to find the irreducible components of the standard module of the Terwilliger algebra of symplectic dual polar graphs. The multiplicities of the isomorphic submodules are given.
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The Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercubes
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.