What question did this study set out to answer?

The research aims to analyze a 2-homogeneous bipartite distance-regular graph with specific diameter properties and the associated quantum group.

March 2, 2026Open Access

2-Homogeneous bipartite distance-regular graphs and the quantum group U^ q (so₆)

Key Points

The research aims to analyze a 2-homogeneous bipartite distance-regular graph with specific diameter properties and the associated quantum group.
Considered a 2-homogeneous bipartite distance-regular graph with diameter D ≥ 3.
Used Q-polynomial ordering for the primitive idempotents of the graph.
Applied an S3-symmetric approach to examine the graph’s structure.
Constructed a subspace of the tensor product V⊗3 with dimension derived from the diameter.
Developed six linear maps from the constructed subspace back to itself.
Constructed a subspace with dimension (D+3 choose 3).
Established six linear maps allowing the subspace to function as an irreducible module.
Demonstrated connections between the graph attributes and the nonstandard quantum group Uq(so6).

Abstract

Abstract We consider a 2-homogeneous bipartite distance-regular graph Γ with diameter D 3 D ≥ 3. We assume that Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of Γ. This Q -polynomial ordering is described using a nonzero parameter q C q ∈ C that is not a root of unity. We investigate Γ using an S₃ S 3 -symmetric approach. In this approach one considers V^ 3 = V V V V ⊗ 3 = V ⊗ V ⊗ V where V is the standard module of Γ. We construct a subspace Λ of V^ 3 V ⊗ 3 that has dimension (array{cD+3\\ 3array}) D + 3 3, together with six linear maps from Λ to Λ. Using these maps we turn Λ into an irreducible module for the nonstandard quantum group U^ q (so₆) U

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Paul Terwilliger

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Journal of Algebraic Combinatorics

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2-Homogeneous bipartite distance-regular graphs and the quantum group U^ q (so₆)

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Abstract

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