An equitable partition for the distance-regular graph of the bilinear forms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Terwilliger, Paul Williford, Jason |
| author_facet | Terwilliger, Paul Williford, Jason |
| contents | We consider a type of distance-regular graph $Γ=(X, \mathcal R)$ called a bilinear forms graph. We assume that the diameter $D$ of $Γ$ is at least $3$. Fix adjacent vertices $x,y \in X$. In our first main result, we introduce an equitable partition of $X$ that has $6D-2$ subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to $x$ and equidistant to $y$. This equitable partition is called the $(x,y)$-partition of $X$. By definition, the subconstituent algebra $T=T(x)$ is generated by the Bose-Mesner algebra of $Γ$ and the dual Bose-Mesner algebra of $Γ$ with respect to $x$. As we will see, for the $(x,y)$-partition of $X$ the characteristic vectors of the subsets form a basis for a $T$-module $U=U(x,y)$. In our second main result, we decompose $U$ into an orthogonal direct sum of irreducible $T$-modules. This sum has five summands: the primary $T$-module and four irreducible $T$-modules that have endpoint one.
We show that every irreducible $T$-module with endpoint one is isomorphic to exactly one of the nonprimary summands. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12125 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An equitable partition for the distance-regular graph of the bilinear forms Terwilliger, Paul Williford, Jason Combinatorics 05E30 We consider a type of distance-regular graph $Γ=(X, \mathcal R)$ called a bilinear forms graph. We assume that the diameter $D$ of $Γ$ is at least $3$. Fix adjacent vertices $x,y \in X$. In our first main result, we introduce an equitable partition of $X$ that has $6D-2$ subsets and the following feature: for every subset in the equitable partition, the vertices in the subset are equidistant to $x$ and equidistant to $y$. This equitable partition is called the $(x,y)$-partition of $X$. By definition, the subconstituent algebra $T=T(x)$ is generated by the Bose-Mesner algebra of $Γ$ and the dual Bose-Mesner algebra of $Γ$ with respect to $x$. As we will see, for the $(x,y)$-partition of $X$ the characteristic vectors of the subsets form a basis for a $T$-module $U=U(x,y)$. In our second main result, we decompose $U$ into an orthogonal direct sum of irreducible $T$-modules. This sum has five summands: the primary $T$-module and four irreducible $T$-modules that have endpoint one. We show that every irreducible $T$-module with endpoint one is isomorphic to exactly one of the nonprimary summands. |
| title | An equitable partition for the distance-regular graph of the bilinear forms |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2512.12125 |