Representation theory of q-rook monoid algebras

Abstract

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.

The Annihilator of Tensor Space in the -Rook Monoid Algebra

“…The proof of the following lemma is similar to that of [13,Proposition 3] and hence we omit it here.…”

“…where ι is the inclusion of A q into C(q) (see the cellular structure of a q-rook monoid algebra in [13]). On the other hand, since U = L(0) ⊕ L(ε 1 ) is the direct sum of the trivial and natural module for U q (gl m ), both U q (gl m ) and U ⊗n have A q -forms U A q (gl m ) and U ⊗n A q , such that U A q (gl m ) acts on U ⊗n A q .…”

“…Since we work on the field C(q), the q-rook monoid algebra R n (q) is semisimple [17]. By the representation theory of R n (q) [5,13], there exists an element Φ n ∈ R n (q) for n ≥ 2 such that T i Φ n = Φ n T i = (−q) −1 Φ n and P j Φ n = Φ n P j = 0 for all 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n.…”

“…Then the representation theory of q-rook monoid algebras and their specialisation analogues (with q = 1) was taken up in [1,4,5,16]. Paget in [13] considered the modular representation theory of q-rook monoid algebras and proved that the q-rook monoid algebra R n (q) (where q may be a unit root) is a cellular algebra in the sense of Graham and Lehrer [3] (see [2] for the case of q = 1).…”

Presentations for rook partition monoids and algebras and their singular ideals

“…Halverson and Ram [37] attributed their understanding of the "existence and importance" of the algebras CA k+ 1 2 (n) to Cheryl Grood, who studied them in their own right in [33], where they were called rook partition algebras, and given their own diagrammatic interpretation (see Section 2.1 below for details); Grood also noted that these intermediate algebras were used in earlier work of Martin [57,59]. The reason for the name is due to a connection with the so-called rook monoids (and associated algebras and deformations) studied by Halverson, Solomon and others [12,18,32,34,36,68,72]. As noted by Grood [33], Solomon's discovery [72] of a Schur-Weyl duality for rook monoid algebras (see also [50]) led to the investigation of a number of other "rook diagram algebras", such the rook Brauer algebras [35,60], Motzkin algebras [5] and more.…”