A Specht Module Analog for the Rook Monoid

Abstract

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

On tensor spaces for rook monoid algebras

“…Therefore Grood called R λ the Specht module for rook monoid algebra F R n . We conclude the main results of [4] as follows Theorem 2.5. With notations as above, {R λ | λ ⊢ r, 0 r n} forms a complete set of pairwise non-isomorphic irreducible F R n -modules.…”

“…We refer the reader to [4] for the definition of π which will not be used in our paper. For each partition λ ⊢ r with 0 r n, let us define…”

Presentations for rook partition monoids and algebras and their singular ideals

“…• S n = {α ∈ RP n : rank(α) = n}, the symmetric group [9]; and • R n = {α ∈ RP n : ker(α) = coker(α) = ∆, supp(α) = dom(α) ∪ codom(α) ′ }, the rook monoid [32,72].…”

“…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.…”

Untitled

“…where f λ is the number of |λ|-standard tableaux of shape λ given by the hook formula (see [21], Theorem 3.10.2). The R n -module M λ is studied in [6,14,19]. In [6], C. Grood determines the analog of Young's natural basis for M λ .…”