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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2404.09346 |
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| _version_ | 1866929358929657856 |
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| author | Nomura, Kazumasa Terwilliger, Paul |
| author_facet | Nomura, Kazumasa Terwilliger, Paul |
| contents | Let $Γ$ denote a $Q$-polynomial distance-regular graph, with vertex set $X$ and diameter $D\geq 3$.
The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$, where ${\hat x}$ denotes column $x$ of the identity matrix $I \in {\rm Mat}_X(\mathbb C)$. Let $E$ denote a $Q$-polynomial primitive idempotent of $Γ$. The eigenspace $EV$ is spanned by the vectors
$\lbrace E {\hat x} \vert x \in X\rbrace$.
It was previously known that these vectors satisfy a condition called the balanced set condition. In this paper, we introduce a variation on the balanced set condition called the Norton-balanced condition. The Norton-balanced condition involves the Norton algebra product on $EV$. We define $Γ$ to be Norton-balanced whenever $Γ$ has a $Q$-polynomial primitive idempotent $E$ such that the set $\lbrace E {\hat x} \vert x \in X\rbrace$ is Norton-balanced.
We show that $Γ$ is Norton-balanced in the following cases:
(i)
$Γ$ is bipartite; (ii) $Γ$ is almost bipartite; (iii) $Γ$ is dual-bipartite; (iv) $Γ$ is almost dual-bipartite; (v) $Γ$ is tight; (vi)
$Γ$ is a Hamming graph; (vii) $Γ$ is a Johnson graph; (viii) $Γ$ is the Grassmann graph $J_q(2D,D)$;
(ix) $Γ$ is a halved bipartite dual-polar graph; (x) $Γ$ is a halved Hemmeter graph; (xi) $Γ$ is a halved hypercube; (xii) $Γ$ is a folded-half hypercube;
(xiii) $Γ$ has $q$-Racah type and affords a spin model.
Some theoretical results about the Norton-balanced condition are obtained, and some open problems are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Norton-balanced condition for $Q$-polynomial distance-regular graphs Nomura, Kazumasa Terwilliger, Paul Combinatorics 05E30 Let $Γ$ denote a $Q$-polynomial distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$, where ${\hat x}$ denotes column $x$ of the identity matrix $I \in {\rm Mat}_X(\mathbb C)$. Let $E$ denote a $Q$-polynomial primitive idempotent of $Γ$. The eigenspace $EV$ is spanned by the vectors $\lbrace E {\hat x} \vert x \in X\rbrace$. It was previously known that these vectors satisfy a condition called the balanced set condition. In this paper, we introduce a variation on the balanced set condition called the Norton-balanced condition. The Norton-balanced condition involves the Norton algebra product on $EV$. We define $Γ$ to be Norton-balanced whenever $Γ$ has a $Q$-polynomial primitive idempotent $E$ such that the set $\lbrace E {\hat x} \vert x \in X\rbrace$ is Norton-balanced. We show that $Γ$ is Norton-balanced in the following cases: (i) $Γ$ is bipartite; (ii) $Γ$ is almost bipartite; (iii) $Γ$ is dual-bipartite; (iv) $Γ$ is almost dual-bipartite; (v) $Γ$ is tight; (vi) $Γ$ is a Hamming graph; (vii) $Γ$ is a Johnson graph; (viii) $Γ$ is the Grassmann graph $J_q(2D,D)$; (ix) $Γ$ is a halved bipartite dual-polar graph; (x) $Γ$ is a halved Hemmeter graph; (xi) $Γ$ is a halved hypercube; (xii) $Γ$ is a folded-half hypercube; (xiii) $Γ$ has $q$-Racah type and affords a spin model. Some theoretical results about the Norton-balanced condition are obtained, and some open problems are given. |
| title | The Norton-balanced condition for $Q$-polynomial distance-regular graphs |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2404.09346 |