Year: 2026
Type: article
Abstract: We consider a type of distance-regular graph $$\Gamma =(X, \mathcal {R})$$ Γ = ( X , R ) called a bilinear forms graph. We assume that the diameter D of $$\Gamma $$ Γ is at least 3. Fix adjacent vertices $$x,y \in X$$ x , y ∈ X . In our first main result, we introduce an equitable partition of X that has $$6D-2$$ 6 D - 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 $$T=T(x)$$ T = T ( x ) is generated by the Bose-Mesner algebra of $$\Gamma $$ Γ and the dual Bose-Mesner algebra of $$\Gamma $$ 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 $$U=U(x,y)$$ 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.
Source: Graphs and Combinatorics
Authors: Paul Terwilliger, Jason Williford
Institutions: University of Wisconsin–Madison, University of Wyoming
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,190
Funder: Simons Foundation
Award: G5964215905
OpenAlex ID: W4417448150