Semi-simple partition algebras as centralizers of representations for rook monoids

Abstract: Let $\mathcal{P}_k(\delta)$, where $k$ is a positive integer and $\delta$ some complex parameter, be the classical partition algebra over the complex numbers. In the case when $\delta=n$, it is well-known that the algebra $\mathcal{P}_k(\delta)$ is the centralizer of the symmetric group $S_n$ acting on the $k$-fold tensor space of the natural representation of $S_n$, for $n\geq 2k$. The algebra $\mathcal{P}_k(\delta)$ is semi-simple for generic values of $\delta$. In this paper, we show that semi-simple partition algebras appear as the centralizer algebras for certain representations of the rook monoids given by an iterative restriction-induction of the trivial representation. Along the way, we also give a decomposition of this iterative representation of the rook monoid into various tensor spaces and show that the corresponding dimensions are given by generalized Bell numbers.