Combinatorics of H-primes in quantum matrices

For q∈C transcendental over Q, we give an algorithmic construction of an order-isomorphism 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. 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 an order-isomorphism was conjectured by Goodearl and Lenagan.