Association Schemes and Directed Graphs Determined by Orbitals of General Linear Groups Over Finite Chain Rings

Abstract

Let R be a finite commutative chain ring, 1 ≤ k ≤ n − 1, the set of right invertible k × n matrices, and GL _n (R) the general linear group of degree n over R, respectively. It is clear that Q ↦ QU (, U ∈ GL _n (R)) is a transitive action on and induces the diagonal action of GL _n (R) on defined by (Q ₁, Q ₂)U = (Q ₁ U, Q ₂ U) for and U ∈ GL _n (R), which are then subdivided into orbits under the action of GL _n (R). First, we investigate the three questions: (i) How should the orbits be described? (ii) How many orbits are there? (iii) What are the lengths of the orbits? Then we compute parameters of the association scheme on and give precisely the structures of directed graphs determined by the orbits of diagonal action of GL _n (R) on .

Key Words:

• Association scheme
• Directed graph
• Finite commutative chain ring
• Right invertible matrix
• Orbital

2000 Mathematics Subject Classification:

• Primary 05E15
• Secondary 05E30, 05C20, 15A33

ACKNOWLEDGMENT

This research is supported in part by the NNSFC (No. 10971160).

Notes

Communicated by J. T. Yu.