Mini-Workshop: Non-Negativity is a Quantum Phenomenon

In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical grassmannian (Brown, Goodearl and Yakimov); and, torus invariant prime ideals in the quantum grassmannian (Launois, Lenagan and Rigal). The aim of this meeting was to explore the reasons for this coincidence in matrices and the grassmannian in particular, and to explore similar ideas in more general settings Mathematics Subject Classification (2000): 05xx, 14xx, 16xx, 17xx, 20xx. Introduction by the Organisers There were 15 participants in the mini-workshop, with representatives of the three main areas included. Before describing the conduct of the meeting, the next few paragraphs outline the background that led to the proposal to have such a mini-workshop and recent results in the three main areas involved in the programme. Many quantum algebras have a natural action by a torus and a key ingredient in the study of the structure of these algebras is an understanding of the torus invariant objects. For example, the Stratification Theory of Goodearl and Letzter shows that, in the generic case, a complete understanding of the prime spectrum of quantum matrices would start by classifying the (finitely many) torus invariant prime ideals. In a paper in Journal of Algebra in 2003, Cauchon succeeded in 768 Oberwolfach Report 14 counting the number of torus invariant prime ideals in quantum matrices. His method involved a bijection between certain diagrams, now known as Cauchon diagrams, and the torus invariant primes. Considerable progress in the understanding of quantum matrices has been made since that time by using Cauchon diagrams. The semiclassical limit of quantum matrices is the classical coordinate ring of the variety of matrices endowed with a Poisson bracket that encodes the nature of the quantum deformation which leads to quantum matrices. As a result, the variety of matrices is endowed with a Poisson structure. A natural torus action leads to a stratification of the variety via torus orbits of symplectic leaves. In a paper in Advances in Mathematics in 2006, Brown, Goodearl and Yakimov showed that there were finitely many such torus orbits of symplectic leaves. Each torus orbit is defined by certain rank conditions on submatrices. The classification is given in terms of certain pairs of permutations from the relevant symmetric group with restrictions arising from the Bruhat order. The non-negative part of the real grassmannian consists of those points in the grassmannian which can be represented by Plücker coordinates that are all nonnegative. One can specify a cell decomposition of the non-negative grassmannian by specifying exactly which coordinates are to be zero/non-zero. In a paper posted on the arxiv in 2006, Postnikov classified the non-empty cells by means of a bijection with certain diagrams, known as Le-diagrams. By using this bijection, Williams, in a paper in Advances in Mathematics in 2005, succeeded in obtaining a generating function for the number of cells according to their dimensions, and as a consequence, was able to count the number of non-empty non-negative cells. Many interesting combinatorial structures were used in Postnikov’s paper, for example, planar networks, matroids, decorated permutations and Bruhat intervals. Specialisation of the Williams’ generating function leads to interesting combinatorial information concerning Eulerian numbers, Narayana numbers and binomial coefficients. The interesting observation from the point of view of this mini-workshop is that in each of the above three sets of results the combinatorial objects that arise turn out to be the same, although the methods employed to obtain the results are very different! The definitions of Cauchon diagrams and Le-diagrams are the same, and the restricted permutations arising in the Brown-Goodearl-Yakimov study can be seen to lead to Cauchon/Le diagrams via the notion of pipe dreams. Postnikov’s work is largely combinatorial, Brown-Goodearl-Yakimov employ algebraic geometry, while Cauchon’s work is mainly non-commutive algebra. Once one is aware of these connections, one can employ notions arising in any of the three areas to suggest means of progress in the others. An example of this is seen in a paper of the two organisers and Rigal which appeared recently in Selecta Mathematica. Here, inspired by the results of Postnikov and Williams on total non-negativity, the authors extended the work of Cauchon in quantum matrices to the quantum grassmannian, established that there are only finitely many torus invariant prime ideals in the quantum grassmannian and counted them by using Mini-Workshop: Non-Negativity is a Quantum Phenomenon 769 Williams’ result. Useful structural information about the quantum grassmannian was then obtained via these results. The intention of this mini-workshop was to bring together experts in each of the three areas to try to understand the underlying reasons for the similarities arising in these investigations. There were indications that this would involve notions of cluster algebras and canonical bases, and so experts from these areas were involved. Finally, combinatorial questions were seen to be important and so there were also combinatorial experts involved. More specifically, in the longer term, the hope is that one might develop a single setting in which results can be proved and then routinely applied to the separate cases. Alternatively, it may be possible to generate machines that will transfer results from one case to the others. In the case of the link between quantum algebras and their semiclassical limits this is also part of a more general programme that would study many different quantum algebras. On the first day of the meeting, there were four overview talks. The first, a general introduction to the area was given by one of the organisers, Tom Lenagan. This was followed by three lectures; one devoted to introducing each of the three main areas to be explored during the meeting. Ken Goodearl gave a survey talk about Poisson algebras and symplectic leaves. Lauren Williams introduced the topic of total non-negativity in matrices, the grassmannian and more general classes of varieties. Gérard Cauchon talked about quantum algebras and the deleting derivations algorithm. On the second day of the meeting, there were three talks, each one introducing a method that might have a bearing on the connection between the three main areas. Robert Marsh introduced the canonical basis, Bernard Leclerc talked about cluster algebras and Francesco Brenti discussed Bruhat intervals. During the rest of the meeting, each of the other participants gave a talk and there were extensive discussions by various subsets of the participants. Several new collaborations arose from these discussions, including, gratifyingly, some across the boundaries of the three different groups represented. The final lecture, by the other organiser, Stéphane Launois, announced recent work of the two organisers and Ken Goodearl, giving an application of the restoration algorithm (the inverse process to the deleting derivations algorithm), first developed for use in quantum matrices, to the explicit determination of the totally non-negative cells in totally non-negative matrices and the defining minors of torus orbits of symplectic leaves in the classical variety of matrices endowed with the standard Poisson bracket arising from the semiclassical limit of quantum matrices. Mini-Workshop: Non-Negativity is a Quantum Phenomenon 771 Mini-Workshop: Non-Negativity is a Quantum Phenomenon