On tensor spaces for rook monoid algebras

Let m, n ∈ N, and V be an m-dimensional vector space over a field F of characteristic 0. Let U = F ⊕V and R_n be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U^⊗n in FR_n, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of U^⊗n in FR_n.