Representations of the rook monoid

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Representations of the Rook-Brauer Algebra (2012, methodology): “It was to become a article, “BMWISH Algebras” (Birman–Murakami– Wenzl–Iwahori–Solomon–Hecke Algebras), which was never written.”
Möbius functions and semigroup representation theory II: Character formulas and multiplicities (2006, methodology): “Then, for θ a character of In, we obtain the following multiplicity formula: (χµ, θ)In = ∑ X⊆[r] (−1)r−|X|(χµ, θ1X )Sr (6.1) This should be contrasted with [48, Lemma 3.17], where Solomon has a different formula that is more difficult to apply.”
On the representation theory of partial Brauer algebras (2012, background): “:= C⊕ V given by the matrix idC 0 0 0 and for i = 1,2,...,m let εi denote the endomorphism id⊗ id⊗ ···⊗ id⊗ ε⊗ id ⊗id ⊗···⊗id of V⊗m for which ε is in position i from the left (this is taken from [So]). For i = 1,2,...,m − 1 consider also the endomorphism ζi of V ⊗m sending e j1 ⊗ ej2 ⊗ ··· ⊗ ejm,”
Representations of the q-rook monoid (2004, methodology): “2) is the presen tation ofRn found by Solomon in [20].”
Quivers of monoids with basic algebras (2011, methodology): “On a similar note, applications of semigroup representation theory to Schur-Weyl duality can be found in [46,89].”