Séminaire Lotharingien de Combinatoire, 78B.18 (2017), 12 pp.

Joël Gay and Florent Hivert

The 0-Rook Monoid and its Representation Theory

Abstract. We show that a proper degeneracy at q=0 of the q-deformed rook monoid of Solomon is the algebra of a monoid Rn0 namely the 0-rook monoid, in the same vein as Norton's 0-Hecke algebra being the algebra of a monoid Hn0 := Hn0(A) (in Cartan type A). As expected, Rn0 is closely related to the latter: it contains the Hn0(A) monoid and is a quotient of Hn0(B). It shares many properties with Hn0, 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 Rn0 is projective on Hn0 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 (Rn0)n in N and we discuss some work in progress on their representation theory.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

The following versions are available: