The Terwilliger Algebra of a Distance-Regular Graph of Negative Type
Stefko Miklavic

TL;DR
This paper investigates the structure of the Terwilliger algebra associated with a specific class of distance-regular graphs of negative type, revealing the existence and properties of certain irreducible modules.
Contribution
It characterizes the irreducible modules of the Terwilliger algebra for graphs with classical parameters and negative type, providing explicit bases and actions.
Findings
Exactly two irreducible T-modules with endpoint 1 exist up to isomorphism.
The dimensions of these modules are D and 2D-2.
Explicit bases and actions of A on these bases are provided.
Abstract
Let denote a distance-regular graph with diameter . Assume has classical parameters with . Let denote the vertex set of and let denote the adjacency matrix of . Fix and let denote the corresponding dual adjacency matrix. Let denote the subalgebra of generated by . We call the {\em Terwilliger algebra} of with respect to . We show that up to isomorphism there exist exactly two irreducible -modules with endpoint 1; their dimensions are and . For these -modules we display a basis consisting of eigenvectors for , and for each basis we give the action of
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Taxonomy
TopicsFinite Group Theory Research · Rings, Modules, and Algebras · Geometric and Algebraic Topology
