The nucleus of the Grassmann graph $J_q(N,D)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916957163356160 |
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| author | Lee, Jae-Ho Park, Jongyook Seong, Ian |
| author_facet | Lee, Jae-Ho Park, Jongyook Seong, Ian |
| contents | Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with vertex set $X$ that consists of the $D$-dimensional subspaces of $\mathcal{V}$. Two vertices are adjacent whenever their intersection has dimension $D-1$. Fix a vertex $x$ in $X$. The Terwilliger algebra $T=T(x)$ of $J_q(N,D)$ with respect to $x$ is the subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^* = A^*(x)$. It is known that an irreducible $T$-module $W$ has certain parameters called the endpoint $r$, the dual endpoint $t$, and the diameter $d$. The displacement of $W$ is defined to be the integer $r+t-D+d$. Let $\mathcal{N}=\mathcal{N}(x)$ denote the span of all irreducible $T$-modules with displacement 0. We call $\mathcal{N}$ the nucleus of $J_q(N,D)$ with respect to $x$. In this paper, we study the structure of $\mathcal{N}$. Specifically, we present a formula for the dimension of $\mathcal{N}$, construct two explicit bases for $\mathcal{N}$, and describe the action of $A$ and $A^*$ on these bases. To obtain these results, we use the projective geometry $P_q(N)$, consisting of all subspaces of $\mathcal{V}$, as a key tool. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15395 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The nucleus of the Grassmann graph $J_q(N,D)$ Lee, Jae-Ho Park, Jongyook Seong, Ian Combinatorics 05E30, 05C50, 06C05 Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $N$ and $D$ denote integers with $N>D \ge 1$. Let $\mathcal{V}$ denote an $N$-dimensional vector space over $\mathbb{F}_q$. The Grassmann graph $J_q(N,D)$ is the graph with vertex set $X$ that consists of the $D$-dimensional subspaces of $\mathcal{V}$. Two vertices are adjacent whenever their intersection has dimension $D-1$. Fix a vertex $x$ in $X$. The Terwilliger algebra $T=T(x)$ of $J_q(N,D)$ with respect to $x$ is the subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^* = A^*(x)$. It is known that an irreducible $T$-module $W$ has certain parameters called the endpoint $r$, the dual endpoint $t$, and the diameter $d$. The displacement of $W$ is defined to be the integer $r+t-D+d$. Let $\mathcal{N}=\mathcal{N}(x)$ denote the span of all irreducible $T$-modules with displacement 0. We call $\mathcal{N}$ the nucleus of $J_q(N,D)$ with respect to $x$. In this paper, we study the structure of $\mathcal{N}$. Specifically, we present a formula for the dimension of $\mathcal{N}$, construct two explicit bases for $\mathcal{N}$, and describe the action of $A$ and $A^*$ on these bases. To obtain these results, we use the projective geometry $P_q(N)$, consisting of all subspaces of $\mathcal{V}$, as a key tool. |
| title | The nucleus of the Grassmann graph $J_q(N,D)$ |
| topic | Combinatorics 05E30, 05C50, 06C05 |
| url | https://arxiv.org/abs/2509.15395 |