Research

Combinatorial Representation Theory: Discovering the interfaces of Algebra with Geometry and Topology

1. Vision of the Programme Grant (PG)

An effective strategy for studying a mathematical object is to represent it in another, better understood object. Our vision is to develop a broad and holistic theory of representation, engaging and unifying diverse perspectives, from algebra, geometry and topology to physics, life sciences and engineering, enabling a cross-discipline transfer of ideas. Our aim is thus to develop a unifying representation theory, which hence acts as a conduit for cross-fertilization between these disciplines. We will achieve this by analysing and generalising techniques that are emerging now in these subjects, combining the recent progress of our team and others across these diverse fields. We will in particular provide algebraic tools in the service of disciplines ranging from the pure, such as modular representation theory, to the applied, such as theoretical physics underpinning quantum computing. This particular unlikely pairing is unified through our PG.

Timeliness: Current developments in algebraic representation theory and related areas has given rise to strong opportunities for the transformation of other fields. In particular, geometric and combinatorial phenomena have emerged that lead to powerful new techniques and applications. Our team has been at the forefront of this progress. For example, surface models provide a vital perspective in cluster theory (Baur, Marsh); discrete Lie geometry has been instrumental in deriving a McKay correspondence for reflection groups (Faber); problems in modelling of topological phases have been solved and new topological invariants have been found by lifting Lie-theoretic techniques to two-dimensional category theory (Faria Martins); generalised Lie geometry has been used to compute decomposition matrices (Martin, Parker). There is a natural unification of all four strands: the latter three rely explicitly on the local-global geometric underpinning of Lie theory, which is also one of the original motivations for cluster theory. Our individual advances are touching the limit of what we can achieve without a unified programme. To take this further and bring these phenomena into a common framework requires a combined approach. This is what our PG will achieve, revealing the deep ‘landscape’ connections.

Recent progress in these areas make this the right moment for a PG in combinatorial representation theory. We are in a position to embrace the perspectives of both pure and application-driven mathematics, and with the potential, in the long term, for serving the needs of physical sciences, life sciences and engineering. This unification of perspectives requires a programme-level research structure and algebra is the right core platform for such an ambitious venture.

Key words (in no particular order): cluster algebras, cluster categories, frieze patterns, Grassmannians, Cohen–Macaulay modules, singularities in algebraic geometry, dimer algebras, surface combinatorics, Postnikov diagrams, diagram algebras, Brauer algebras, Hecke algebras, Lie theory, Lie-theoretic combinatorics, Coxeter presentations, Weyl groups, mutation, Kac-Moody Lie algebras, braid groups, cluster presentations, graphs, Cartan matrices, root systems, root basis, monoids, companion bases, extended topological quantum field theory, higher category theory, higher Knizhnik-Zamolodchikov connections, higher gauge Theory, mathematical models for topological phases of matter, topological quantum computing.

Partner universities: NTNU, Trondheim, Norway; Texas A&M University, USA; Uppsala Universitet, Sweden; Universidad de Talca, Chile; Universität zu Köln, Germany; Université du Québec à Montréal, Canada.

2. The Work Packages (WPs) and their interrelations

We propose five work packages (circular nodes in the figure, linked by edges), each led by 3-4 team members.

WP4: Explain key relationships between algebraic representation theory and geometry (Faria Martins, Martin, Parker): We aim to explain and use the geometry arising from physics and topology in the representation theory of combinatorial diagram algebras and categories.

WP5: Developing applications of combinatorial representation theory to topology and physics (all team members): We aim to construct invariants for low dimensional topology by using combinatorial representation theory frameworks including mutation of friezes and cluster algebras as well as higher dimensional category theory. These invariants are of core value in geometric topology, and in applications, including condensed matter physics and quantum computing.

2.1 Inter-relations between work packages

A transverse level of structure in our PG is that of four different themes, T1-T4 below, where points of view and tools common to different WP’s are synergistically grouped. The themes have blurred boundaries and potential for interaction as indicated by the dashed lines and interactions will be facilitated, enabling flexibility in our work (e.g. T4 has the potential to provide new approaches to WPs 2 and 3 when we consider properties which are stable in the limit, as lattices become finer). Regular theme and cross-theme working groups led by the leads will facilitate the flow of ideas and applications across the different WPs and ensure focus on unifying aspects of the PG.

T1 Diagrammatic algebras: Algebras associated to diagrammatic structures arise in many ways, e.g oriented graphs (WP1), sets of permutation diagrams (WP4) and, more generally, tangles (WP5). Common objectives concern which features of such diagrammatic algebras depend only on the topology of the spaces in which the diagrams are embedded, and how the algebras associated to small portions of a diagram can be put together in order to reconstruct the algebra attached to the entire diagram.

T2 Mutation of algebraic and geometric structures: Diagrammatic and combinatorial representations of objects are often not uniquely defined. However, there are combinatorial moves (mutations) which connect any two of them. In T2, we will explore which algebraic properties of objects are stable under mutation; their algebraic geometric representation (WP2), their Lie-theoretic meaning (WPs 1,3), and their topological and physical meaning (WPs 4,5). This should provide algebraic objects that emulate the combinatorial moves of low dimensional topology, contributing to the construction of topological invariants.

T3 Algebraic Geometry of Grassmannians and moduli spaces: The Grassmannian of subspaces of a fixed vector space is a much-studied classical geometric object. Dimer algebras of surfaces (WP1) have been used to form categorical models of this structure, while the corresponding cluster algebras have been used to define new families of integer friezes (WP2) and presentations of reflection groups (WP3). We expect to provide a universal framework encompassing surface algebras, friezes and presentations of objects arising in Lie theory, as well as insights into representation theory.

T4 Computation for Physics: This theme concerns the interface of algebra with applications, including theoretical physics, and the flow of ideas in both directions. Lattice models of space are valuable for modelling topological phases of condensed matter and statistical physics, and the construction of topological quantum field theories including extended topological quantum field theories (WP5). Subcategories of the partition category (WP4) appear in this setting as discretised versions of local operator algebras. With this, we expect to extract field-theoretic physics and topological invariants from lattice models.