  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.
