Modules and Structures of Planar Upper Triangular Rook Monoids

In this paper, we discuss modules and structures of the planar upper triangular rook monoid B_n. We first show that the order of B_n is a Catalan number, then we investigate the properties of a module V over B_n generated by a set of elements v_S indexed by the power set of {1, ..., n}. We find that every nonzero submodule of V is cyclic and completely decomposable; we give a necessary and sufficient condition for a submodule of V to be indecomposable. We show that every irreducible submodule of V is 1-dimensional. Furthermore, we give a formula for calculating the dimension of every submodule of V. In particular, we provide a recursive formula for calculating the dimension of the cyclic module generated by v_S, and show that some dimensions are Catalan numbers, giving rise to new combinatorial identities.