The $0$-Rook Monoid and its Representation Theory

We show that a proper degeneracy at $q=0$ of the $q$-deformed rook monoid of Solomon is the algebra of a monoid $R_n^0$ namely the $0$-rook monoid, in the same vein as Norton's $0$-Hecke algebra being the algebra of a monoid $H_n^0 = H^0(A_{n-1})$ (in Cartan type~$A_{n-1}$). As expected, $R_n^0$ is closely related to the latter: it contains the $H^0(A_{n-1})$ monoid and is a quotient of $H^0(B_{n})$. We give a presentation for this monoid as well as a combinatorial realization as functions acting on the classical rook monoid itself. On the way we get a Matsumoto theorem for the rook monoid a result which was conjectured by Solomon. The $0$-rook monoid shares many combinatorial properties with the Hecke monoid: its Green right preorder is an actual order, and moreover a lattice (analogous to the right weak order) which has some nice combinatorial, and geometrical features. In particular the $0$-rook monoid is J-trivial. Following Denton-Hivert-Schilling-Thi\'ery, it allows us to describe its representation theory including the description of the simple and projective modules. We further show that $R_n^0$ is projective on $H_n^0$ and make explicit the restriction and induction functors along the inclusion map. We finally give a (partial) associative tower structures on the family of $(R_n^0)$ and we discuss its representation theory.