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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 *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*_{n}^{0}(*A*) (in Cartan type A). As
expected,
*R*_{n}^{0}
is closely related to the latter: it contains the
*H*_{n}^{0}(*A*)
monoid and is a quotient of
*H*_{n}^{0}(*B*). It shares many
properties with
*H*_{n}^{0},
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*_{n}^{0} is projective on
*H*_{n}^{0} 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*_{n}^{0})_{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.

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