The Erdős-Ko-Rado basis for a Leonard system

Abstract

We introduce and discuss an Erdős-Ko-Rado basis for the underlying vector space of a Leonard system $\Phi =(A;A^{\ast };\{E_i{\}}_{i=0}^d;\{E_i^{\ast }{\}}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and $A^{\ast }$. We describe the transition matrices to/from other known bases, as well as the matrices representing $A$ and $A^{\ast }$ with respect to the new basis. We also discuss how these results can be viewed as a generalization of the linear programming method used previously in the proofs of the "Erdős-Ko-Rado theorems" for several classical families of $Q$-polynomial distance-regular graphs, including the original 1961 theorem of Erdős, Ko, and Rado.

19 pages

References in corpus (4)

• Commutative association schemes
• Some algebra related to $P$-and $Q$-polynomial association schemes
• The Erdős-Ko-Rado theorem for twisted Grassmann graphs
• A bilinear form relating two Leonard systems

Cited by in corpus (1)

• Distance-regular graphs