Structure of Thin Irreducible Modules of a Q-polynomial Distance-Regular Graph

TL;DR

This paper provides a detailed algebraic description of thin irreducible modules of Q-polynomial distance-regular graphs using Leonard pairs, with applications to q-Racah and classical parameter cases.

Contribution

It offers a comprehensive characterization of thin irreducible modules via intersection numbers, dual intersection numbers, and parameter arrays, extending previous Leonard pair results.

Findings

Explicit description of thin irreducible modules in terms of algebraic parameters

Application of Leonard pair theory to module structure analysis

Results specialized to q-Racah type and classical parameter graphs

Abstract

Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D geq 3 and adjacency matrix A. Fix x in X and let A*=A*(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the subalgebra of Mat_X(C) generated by A and A*. Let W denote a thin irreducible T-module. It is known that the action of A and A* on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, these results will be applied to obtain a detailed description of W. In particular, we give a description of W in terms of its intersection numbers, dual intersection numbers and parameter array. Finally, we apply our results to the case in which Gamma has q-Racah type or classical parameters.