RANK<i>t</i>ℋ-PRIMES IN QUANTUM MATRICES

Abstract

ABSTRACT Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (2002 Goodearl , K. R. , Lenagan , T. H. ( 2002 ). Prime ideals invariant under winding automorphisms in quantum matrices . Internat. J. Math 13 : 497 – 532 . [CROSSREF] [Crossref] , [Google Scholar]) have shown that the number of ℋ-primes in R = O q (ℳ n (𝕂)) that 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!) 2 S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the 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 Kaneko (1997 Kaneko , M. ( 1997 ). Poly-Bernoulli numbers . J. Théorie Nombres Bordeaux 9 : 221 – 228 .[Crossref] , [Google Scholar]) and Arakawa and Kaneko (1999 Arakawa , T. , Kaneko , M. ( 1999 ). On poly-Bernoulli numbers . Comment Math. Univ. St. Paul 48 ( 2 ): 159 – 167 . [Google Scholar]).