We take a graph theoretic approach to the problem of finding generators for those prime ideals of O q (M m,n (K)) which are invariant under the torus action (K * ) m+n . Launois (2004) [15] has shown that the generators consist of certain quantum minors of the matrix of canonical generators of O q (M m,n (K)) and inLaunois (2004) [14] gives an algorithm to find them. In this paper we modify a classic result of Lindström (1973) [17] and Gessel and Viennot (1985) [6] to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex-disjoint directed paths in an associated directed graph.

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Abstract. We study, from a combinatorial viewpoint, the quantized coordinate ring of m × n matrices over an infinite field K, Oq(Mm,n(K)) (often simply called quantum matrices).The first part of this paper shows that Oq(Mm,n(K)), which is traditionally defined by generators and relations, can be seen as a subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of Oq(Mm,n(K)) as a sum over paths in the graph, each path being assigned an element of the quantum torus. The Oq(Mm,n(K)) relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon's deleting derivations algorithm.The second part of this paper applies the above to the theory of torus-invariant prime ideals of Oq(Mm,n(K)). We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only when char(K) = 0 and with q transcendental over Q. Our strategy is to prove the stronger result that the quantum minors in a given torus-invariant ideal form a Gröbner basis.

The aim of this paper is to study the representation theory of quantum Schubert cells. Let g be a simple complex Lie algebra. To each element w of the Weyl group W of g, De Concini, Kac and Procesi have attached a subalgebra U q OEw of the quantised enveloping algebra U q .g/. Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of U q OEw. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux-Cauchon and Yakimov, that the primitive spectrum of U q OEw admits a stratification indexed by those elements v 2 W with v Ä w in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair v Ä w.

Bonnington and Richter defined the cycle space of an infinite graph to consist of the sets of edges of subgraphs having even degree at every vertex. Diestel and Kühn introduced a different cycle space of infinite graphs based on allowing infinite circuits. A more general point of view was taken by Vella and Richter, thereby unifying these cycle spaces. In particular, different compactifications of locally finite graphs yield different topological spaces that have different cycle spaces.In this work, the Vella-Richter approach is pursued by considering cycle spaces over all fields, not just Z 2 . In order to understand "orthogonality" relations, it is helpful to consider two different cycle spaces and three different bond spaces. We give an analogue of the "edge tripartition theorem" of Rosenstiehl and Read and show that the cycle spaces of different compactifications of a locally finite graph are related.