The nucleus of a $Q$-polynomial distance-regular graph
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917359851143168 |
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| author | Terwilliger, Paul |
| author_facet | Terwilliger, Paul |
| contents | Let $Γ$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $Γ$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $Γ$ and the dual adjacency matrix $A^*=A^*(x)$ of $Γ$ with respect to $x$. We introduce a $T$-module $\mathcal N = \mathcal N(x)$ called the nucleus of $Γ$ with respect to $x$. We describe $\mathcal N$ from various points of view. We show that all the irreducible $T$-submodules of $\mathcal N$ are thin. Under the assumption that $Γ$ is a nonbipartite dual polar graph, we give an explicit basis for $\mathcal N$ and the action of $A, A^*$ on this basis. The basis is in bijection with the set of elements for the projective geometry $L_D(q)$, where $GF(q)$ is the finite field used to define $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11282 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The nucleus of a $Q$-polynomial distance-regular graph Terwilliger, Paul Combinatorics Quantum Algebra 05E30 Let $Γ$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $Γ$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $Γ$ and the dual adjacency matrix $A^*=A^*(x)$ of $Γ$ with respect to $x$. We introduce a $T$-module $\mathcal N = \mathcal N(x)$ called the nucleus of $Γ$ with respect to $x$. We describe $\mathcal N$ from various points of view. We show that all the irreducible $T$-submodules of $\mathcal N$ are thin. Under the assumption that $Γ$ is a nonbipartite dual polar graph, we give an explicit basis for $\mathcal N$ and the action of $A, A^*$ on this basis. The basis is in bijection with the set of elements for the projective geometry $L_D(q)$, where $GF(q)$ is the finite field used to define $Γ$. |
| title | The nucleus of a $Q$-polynomial distance-regular graph |
| topic | Combinatorics Quantum Algebra 05E30 |
| url | https://arxiv.org/abs/2408.11282 |