1 J an 2 00 5 Combinatoric of H - primes in quantum matrices

Let n be a positive integer greater than or equal to 2, and q ∈ C transcendental over Q. In this paper, we give an algorithmic construction of an ordered bijection between the set of H-primes of Oq (Mn(C)) and the sub-poset S of the (reverse) Bruhat order of the symmetric group S2n consisting of those permutations that move any integer by no more than n positions. Further, we describe the permutations that correspond via this bijection to rank t H-primes in Oq (Mn(C)), that is, to those H-invariant prime ideals of Oq (Mn(C)) which contain all (t+1)× (t+1) quantum minors but not all t× t quantum minors. More precisely, we establish the following result. Imagine that there is a barrier between positions n and n + 1. Then a 2n-permutation σ ∈ S corresponds to a rank t H-invariant prime ideal of Oq (Mn(C)) if and only if the number of integers that are moved by σ from the right to the left of this barrier is exactly n − t. The existence of such a bijection (with such properties) was conjectured by Goodearl and Lenagan. 2000 Mathematics subject classification: 16W35, 20G42, 06A07.