The combinatorial structure of the prime spectrum of quantum matrices

Herein we study the prime ideals in the algebra of quantum matrices. The main content of this work is the application of combinatorial methods to the analysis of a special subclass of prime ideals, namely, those invariant under the action of an algebraic torus $\C{H}$. We call such ideals $\C{H}$-primes. By the $\C{H}$-stratification theory of Goodearl and Letzter, the set of prime ideals can be partitioned (or ``stratified'') in a manner such the parts (or ``strata'') are in bijective correspondence with the $\C{H}$-primes. Moreover, each stratum satisfies nice topological properties. Thus, to understand the set of prime ideals, a first step is to understand the $\C{H}$-primes. The first problem we approach is the question of finding a generating set for a given $\C{H}$-prime. Launois proved that for almost all quantum matrix algebras, the generating sets consist of certain ``quantum minors'' derived from the generators of the quantum matrix algebra. Launois also provided an algorithm to find such minors, however it is algebraic in nature and somewhat unwieldy. We prove that Launois' algorithm can be considered combinatorial. Specifically, we show that the problem of determining which quantum minors appear in a given $\C{H}$-prime is equivalent to finding sets of non-intersecting paths in a certain graph associated to the $\C{H}$-prime. Each stratum has a notion of ``dimension'' attached to it. In particular, the $\C{H}$-primes in strata of dimension zero are members of an important subclass of the prime ideals, namely, the primitive ideals. We give an easy answer to the problem of determining the dimension. The main result is that the dimension is equal to the number of odd cycles in a certain permutation that is easily found given an $\C{H}$-prime. We are then able to give enumeration formulae for $d$-dimensional strata.