Character Formulas for q-Rook Monoid Algebras

The q-rook monoid R_n(q) is a semisimple (q)-algebra that specializes when q 1 to [R_n], where R_n is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. We use a Schur-Weyl duality between R_n(q) and the quantum general linear group to compute a Frobenius formula, in the ring of symmetric functions, for the irreducible characters of R_n(q). We then derive a recursive Murnaghan-Nakayama rule for these characters, and we use Robinson-Schensted-Knuth insertion to derive a Roichman rule for these characters. We also define a class of standard elements on which it is sufficient to compute characters. The results for R_n(q) specialize when q = 1 to analogous results for R_n.