On tensor spaces for rook monoid algebras (Q287978)

This paper studies Solomon's version of the Schur-Weyl duality for the symmetric inverse semigroup \(\mathrm{IS}_n\) (which is called by the alternative name \textit{rook monoid} in the paper) in which \(\mathrm{IS}_n\) acts on the \(n\)-th tensor power of the direct sum of the natural and the trivial representation of \(\mathrm{GL}_m\). The main result of the paper gives an explicit description of the kernel of this action in the semigroup algebra of \(\mathrm{IS}_n\) in terms of a two-sided ideal generated by an explicitly constructed element of the group algebra.