Combinatoric of H-primes in quantum matrices

Abstract

Let n be a positive integer greater than or equal to 2, and q a complex number, transcendental over Q. In this paper, we give an algorithmic construction of an ordered bijection between the set of H-primes of n \times n quantum matrices and the sub-poset S of the (reverse) Bruhat order of the symmetric group S_{2n} 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, that is, to those H-invariant prime ideals which contain all (t+1) \times (t+1) quantum minors but not all t \times 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 \sigma which belongs to the sub-poset S corresponds to a rank t H-invariant prime ideal of the algebra of n \times n quantum matrices if and only if the number of integers that are moved by \sigma 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.