2 9 O ct 2 00 3 Rank t H - primes in quantum matrices . Stéphane Launois

Let K be a (commutative) field and consider a nonzero element q in K which is not a root of unity. In [5], Goodearl and Lenagan have shown that the number of H-primes in R = Oq (Mn(K)) which contain all (t+1)×(t+1) quantum minors but not all t× t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!)S(n+ 1, t+ 1), where S(n+ 1, t+ 1) denotes the Stirling number of second kind associated to n+1 and t+1. This result was conjectured by Goodearl, Lenagan and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established in [10] and [1]. 2000 Mathematics subject classification: 16W35 (20G42 11B68 11B73).