Blocked triangular patterns and their symmetry groups

Abstract

For the natural number n≥1 a 2n×2n matrix partitioned into n×n blocks is called a triangular pattern if the entry pattern of each block is one of the triangles determined by the main diagonal or secondary diagonal. The dihedral group of order 8, D4, realized as a subgroup of the group S2n of 2n×2n blocked permutation matrices, acts via conjugation on the set of triangular patterns Δ. Patterns P and Q in Δ are D4-equivalent if there is a permutation Φ∈D4 such that ΦPΦT=Q. The objective of this paper is to examine the action of D4 on Δ. Of particular interest are the orbits of this group action, and certain other subgroups of S2n associated with D4.