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 maps σ with domain I (σ ) ⊆ n and range J (σ) ⊆ n. If i ∈ I (σ ) let iσ denote the image of i under σ . There is an associative product (σ, τ ) → στ on R defined by composition of maps: i(στ)= (iσ )τ if i ∈ I (σ ) and iσ ∈ I (τ ). Thus the domain I (στ) consists of all i ∈ I (σ ) such that iσ ∈ I (τ ). The set R, 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. Let Mn(F ) denote the algebra of n× n matrices over F . There is a one-to-one map R → Mn(F ) defined by σ → [σ ] = ∑