Elementary methods for splitting representations of rook monoids: A gentle introduction to groupoids

In this paper, we show that the algebra of the colored rook monoid [Formula: see text], i.e. the monoid of [Formula: see text] matrices with at most one non-zero entry (an [Formula: see text]th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a [Formula: see text]-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.