Representations of the rook monoid

Let n be a positive integer and let n= {1, . . . , n}. Let R be the set of all oneto-one mapsσ with domainI (σ ) ⊆ n and rangeJ (σ) ⊆ n. If i ∈ I (σ ) let iσ denote the image of i underσ . There is an associative product (σ, τ ) → στ onR defined by composition of maps: i(στ)= (iσ )τ if i ∈ I (σ ) andiσ ∈ I (τ ). Thus the domainI (στ) consists of alli ∈ I (σ ) such thatiσ ∈ I (τ ). The setR, with this product, is a monoid (semigroup with identity) called the symmetric inverse semigroup. We agree that R contains a map with empty domain and range which behaves as a zero element. Let F be a field. LetMn(F ) denote the algebra of n× n matrices overF . There is a one-to-one map R→Mn(F ) defined by σ → [σ ] = ∑