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.
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