Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids

Gérard Henry Edmond Duchamp, Joseph Ben Geloun, Christophe Tollu

We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n\times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each c…