Representations of the q-rook monoid

The q-rook monoid is a semisimple algebra over that specializes when q→1 to , where Rn is the monoid of n×n matrices with entries from {0,1} and at most one nonzero entry in each row and column. When q is specialized to a prime power, is isomorphic to the Iwahori algebra , where is the monoid of n×n matrices with entries from a finite field having q-elements and B M is the Borel subgroup of invertible upper triangular matrices. In this paper, we (i) give a new presentation for on generators and relations and determine a set of standard words which form a basis; (ii) explicitly construct a complete set of “seminormal” irreducible representations of ; and (iii) show that is the centralizer of the quantum general linear group acting on the tensor product (W V) n, where V is the fundamental module and W is the trivial module.