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

Abstract We consider a type of distance-regular graph $$\Gamma =(X, \mathcal {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> called a bilinear forms graph. We assume that the diameter D of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is at least 3. Fix adjacent vertices $$x,y \in X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> . In our first main result, we introduce an equitable partition of X that has $$6D-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>6</mml:mn> <mml:mi>D</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 $$T=T(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is generated by the Bose-Mesner algebra of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> and the dual Bose-Mesner algebra of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> 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 $$U=U(x,y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>=</mml:mo> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.