The 0-Rook Monoid and its Representation Theory

Joël Gay ; Florent Hivert

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

We show that a proper degeneracy at q=0 of the q-deformed rook monoid of Solomon is the algebra of a monoid R_n⁰ namely the 0-rook monoid, in the same vein as Norton's 0-Hecke algebra being the algebra of a monoid H_n⁰ := H_n⁰(A) (in Cartan type A). As expected, R_n⁰ is closely related to the latter: it contains the H_n⁰(A) monoid and is a quotient of H_n⁰(B). It shares many properties with H_n⁰, in particular it is J-trivial. It allows us to describe its representation theory including the description of the simple and projective modules. We further show that R_n⁰ is projective on H_n⁰ and make explicit the restriction and induction along the inclusion map. A more surprising fact is that there are several non classical tower structures on the family of (R_n⁰)_{n in N} and we discuss some work in progress on their representation theory.