Schur–Weyl dualities for the rook monoid: an approach via Schur algebras

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon’s Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach.