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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.02190 |
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| _version_ | 1866910069510111232 |
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| author | Terwilliger, Paul |
| author_facet | Terwilliger, Paul |
| contents | We consider a 2-homogeneous bipartite distance-regular graph $Γ$ with diameter $D \geq 3$. 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 $q \in \mathbb C$ that is not a root of unity. We investigate $Γ$ using an $S_3$-symmetric approach. In this approach one considers $V^{\otimes 3} = V \otimes V \otimes V$ where $V$ is the standard module of $Γ$. We construct a subspace $Λ$ of $V^{\otimes 3}$ that has dimension $\binom{D+3}{3}$, together with six linear maps from $Λ$ to $Λ$. Using these maps we turn $Λ$ into an irreducible module for the nonstandard quantum group
$U^\prime_q(\mathfrak{so}_6)$ introduced by Gavrilik and Klimyk in 1991. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime_q(\mathfrak{so}_6)$ Terwilliger, Paul Combinatorics Quantum Algebra 05E30 We consider a 2-homogeneous bipartite distance-regular graph $Γ$ with diameter $D \geq 3$. 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 $q \in \mathbb C$ that is not a root of unity. We investigate $Γ$ using an $S_3$-symmetric approach. In this approach one considers $V^{\otimes 3} = V \otimes V \otimes V$ where $V$ is the standard module of $Γ$. We construct a subspace $Λ$ of $V^{\otimes 3}$ that has dimension $\binom{D+3}{3}$, together with six linear maps from $Λ$ to $Λ$. Using these maps we turn $Λ$ into an irreducible module for the nonstandard quantum group $U^\prime_q(\mathfrak{so}_6)$ introduced by Gavrilik and Klimyk in 1991. |
| title | 2-Homogeneous bipartite distance-regular graphs and the quantum group $U^\prime_q(\mathfrak{so}_6)$ |
| topic | Combinatorics Quantum Algebra 05E30 |
| url | https://arxiv.org/abs/2506.02190 |