On framed rook algebras
We introduce and study the framed rook algebra, a structure that unifies two significant generalizations of the Iwahori-Hecke algebra. The first one, introduced by Solomon, extends the Hecke algebra to the full matrix monoid, yielding the rook monoid algebra. The second one, developed by Yokonuma, replaces the Borel subgroup with the unipotent subgroup, resulting in the Yokonuma-Hecke algebra. Our concrete algebra is constructed from the double cosets of the unipotent subgroup within the full matrix monoid. We show that this double coset decomposition is indexed by the framed symmetric inverse monoid. We also define the Rook Yokonuma-Hecke algebra as an abstract structure using generators and relations. We then prove the main isomorphism theorem, which establishes that this abstract algebra is isomorphic to the framed rook algebra under a specific parameter specialization. To complete our characterization, we provide a faithful representation on a tensor space and establish a linear basis for the Rook Yokonuma-Hecke algebra.