The nucleus of the Johnson graph $J(N,D)$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917879812718592 |
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| author | Nomura, Kazumasa Terwilliger, Paul |
| author_facet | Nomura, Kazumasa Terwilliger, Paul |
| contents | In this paper, we describe the nucleus of the Johnson graph $Γ= J(N,D)$
with $N > 2D$. Let $X$ denote the vertex set of $Γ$. Let $A \in \text{Mat}_X({\mathbb C})$ denote the adjacency matrix of $Γ$. Let $\{E_i\}_{i=0}^D$ denote the $Q$-polynomial ordering of the primitive idempotents of $A$. Fix $x \in X$, and consider the corresponding dual adjacency matrix $A^*$ and dual primitive idempotents $\{E^*_i\}_{i=0}^D$. The subalgebra $T$ of $\text{Mat}_X({\mathbb C})$ generated by $A$, $A^*$ is called the subconstituent algebra of $Γ$ with respect to $x$. Let $V={\mathbb C}^X$ denote the standard module of $Γ$. For $0 \leq i \leq D$ define \[ {\mathcal N}_i = (E^*_0 V + E^*_1 V + \cdots + E^*_i V) \cap
(E_0 V + E_1 V + \cdots + E_{D-i} V). \] It is known that the sum ${\mathcal N} = \sum_{i=0}^D {\mathcal N}_i$ is direct, and $\mathcal N$ is a $T$-module. The $T$-module $\mathcal N$ is called the nucleus of $Γ$ with respect to $x$. For $0 \leq i \leq D$ we construct a basis for ${\mathcal N}_i$ and a basis for $E^*_i {\mathcal N}$. From this we obtain two bases of $\mathcal N$. We give a combinatorial interpretation of these two bases. We give the transition matrices between these two bases. We also give the action of $A$, $A^*$ on these bases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The nucleus of the Johnson graph $J(N,D)$ Nomura, Kazumasa Terwilliger, Paul Rings and Algebras 05E30, 15B10 In this paper, we describe the nucleus of the Johnson graph $Γ= J(N,D)$ with $N > 2D$. Let $X$ denote the vertex set of $Γ$. Let $A \in \text{Mat}_X({\mathbb C})$ denote the adjacency matrix of $Γ$. Let $\{E_i\}_{i=0}^D$ denote the $Q$-polynomial ordering of the primitive idempotents of $A$. Fix $x \in X$, and consider the corresponding dual adjacency matrix $A^*$ and dual primitive idempotents $\{E^*_i\}_{i=0}^D$. The subalgebra $T$ of $\text{Mat}_X({\mathbb C})$ generated by $A$, $A^*$ is called the subconstituent algebra of $Γ$ with respect to $x$. Let $V={\mathbb C}^X$ denote the standard module of $Γ$. For $0 \leq i \leq D$ define \[ {\mathcal N}_i = (E^*_0 V + E^*_1 V + \cdots + E^*_i V) \cap (E_0 V + E_1 V + \cdots + E_{D-i} V). \] It is known that the sum ${\mathcal N} = \sum_{i=0}^D {\mathcal N}_i$ is direct, and $\mathcal N$ is a $T$-module. The $T$-module $\mathcal N$ is called the nucleus of $Γ$ with respect to $x$. For $0 \leq i \leq D$ we construct a basis for ${\mathcal N}_i$ and a basis for $E^*_i {\mathcal N}$. From this we obtain two bases of $\mathcal N$. We give a combinatorial interpretation of these two bases. We give the transition matrices between these two bases. We also give the action of $A$, $A^*$ on these bases. |
| title | The nucleus of the Johnson graph $J(N,D)$ |
| topic | Rings and Algebras 05E30, 15B10 |
| url | https://arxiv.org/abs/2412.19389 |