An equitable partition for the distance-regular graph of the bilinear forms

TL;DR

This paper introduces an equitable partition called the $(x,y)$-partition for bilinear forms graphs with diameter at least 3, and decomposes the associated $T$-module into orthogonal irreducible components, revealing new algebraic structure.

Contribution

It provides a novel equitable partition for bilinear forms graphs and decomposes the related $T$-module into irreducible modules, advancing understanding of their algebraic properties.

Findings

The $(x,y)$-partition has $6D-2$ subsets with vertices equidistant to fixed vertices.

The $T$-module $U(x,y)$ decomposes into five orthogonal irreducible modules.

Every nonprimary irreducible $T$-module with endpoint one is uniquely characterized.

Abstract

We consider a type of distance-regular graph $\Gamma=(X, \mathcal R)$ called a bilinear forms graph. We assume that the diameter $D$ of $\Gamma$ 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 $\Gamma$ and the dual Bose-Mesner algebra of $\Gamma$ 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.