Powers of Principal Q-Borel ideals

Fix a poset Q on {x(1), ... , x(n)}. A Q-Borel monomial ideal I subset of K[x(1), ... , x(n)] is a monomial ideal whose monomials are closed under the Borel-like moves induced by Q. A monomial ideal I is a principal Q-Borel ideal, denoted I = Q (m ), if there is a monomial m such that all the minimal generators of I can be obtained via Q-Borel moves from m. In this paper we study powers of principal Q-Borel ideals. Among our results, we show that all powers of Q(m) agree with their symbolic powers, and that the ideal Q( m) satisfies the persistence property for associated primes. We also compute the analytic spread of Q(m) in terms of the poset Q.