Character Formulas for q -Rook Monoid Algebras

Abstract

The q-rook monoid R n(q) is a semisimple ℂ(q)-algebra that specializes when q → 1 to ℂ[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 $$U_q {\mathfrak{g}}{\mathfrak{l}}(r)$$ 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.