Rank t H-primes in quantum matrices

Abstract

Let K be a (commutative) field and consider a nonzero element q in K which is not a root of unity. Goodearl and Lenagan have shown that the number of H-primes in the algebra R of m times p quantum matrices which contain all (t+1) times (t+1) quantum minors but not all t times 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!)^2 S(n+1,t+1)^2, 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 by Arakawa-Kaneko and Kaneko.