The 0-rook Monoid and its Representation Theory

Joël Gay (1) , Florent Hivert (1, 2)

1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
2 LRI - Laboratoire de Recherche en Informatique

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

Keywords: combinatorics; representation theory of monoids; symmetric group; Iwahori-Hecke algebras; rook monoid

Domains: Combinatorics [math.CO]

Joël Gay, Florent Hivert. The 0-rook Monoid and its Representation Theory. FPSAC'17 - 29th international conference on Formal Power Series and Algebraic Combinatorics, Jul 2017, London, United Kingdom. ⟨hal-01691266⟩