# Rigid structure in noncommutative, geometric and combinatorial problems

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics

### Abstract

The proposed research of this proposal is in representation theory, a field of pure mathematics with strong interactions with other sciences including computing science, chemistry and physics. A basic idea of mathematics is to distill the most crucial properties from naturally occurring phenomena, leaving simply the essence of the situation to be studied and understood. We all know this idea already: long ago people did not think of numbers as abstract quantities, but as way of describing specific quantities. One sheep, two sheep, three sheep. It was a remarkable step forward to think of numbers abstractly: they are objects to which we can do things - we can apply algebraic operations to them, such as adding and subtracting, we can compare them, etc - but they are not always objects that we can visualise or specify in the real world. What does the number 1 billion look like? 1 trillion? We know that they are big numbers because we can compare them to other numbers, but how big? Our feeling for such numbers comes essentially from our ability to treat them just like any other number, and in particular ones for which we do have a very good intuition. Abstraction abounds in mathematics. For instance, the study of symmetry is encoded abstractly in the notion of a group. Groups are collections of elements which satisfy certain axioms, axioms which obviously hold for symmetries. However, a group is abstract by definition and need not be presented as symmetries of any particular object, but only as an object satisfying the given list of axioms. The axiomatic approach is a very powerful method which, in this instance, allows one to prove many general theorems, all of which can be applied to any group. Given the general abstract definition of a group, one is lead to think about simple groups, the building blocks from which every group can be built. For a long time group theorists wanted to classify simple groups, and around twenty five they came up with a comprehensive list. Many items on the list had been known since the birth of group theory, but there were a small number of exceptions. These were new groups, absolutely fundamental since they were basic building blocks, but they had not been observed earlier as symmetries of some well-known mathematical object. Where did they come from? Were they symmetries of something? This is where representation theory comes in: it studies how a group (or other abstract mathematical structures) can be the symmetry of some naturally occurring object. It weds mathematical reality and abstraction.Representation theory is thus a powerful tool that is of interest to researchers in many different fields. Within pure mathematics it is important when studying abstract systems, but it is also a very useful for understanding the orginal objects which display the symmetry. It is also used further afield, for instance in chemistry to help to study the symmetry of molecules, in physics when studying the nature of space, or in fluids to help to solve differential equations. In this proposal I intend to build mathematical tools using the rigid structure arising from the representation theory of noncommutative algebras which can then be applied to solve problems in a number of different fields and which will also be of intrinsic interest to representation theorists. In doing this, I will interact with researchers from many different topics, both in the UK and abroad. This activity will have benefits for mathematics in Edinburgh, the UK and beyond.

### ORCID iD

Iain Gordon (Principal Investigator)

### Publications

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Addington N (2015) The Pfaffian-Grassmannian equivalence revisited in Algebraic Geometry

Addington N (2016) Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences in Journal of the London Mathematical Society

Addington N (2019) Mukai flops and P-twists in Journal für die reine und angewandte Mathematik (Crelles Journal)

Addington N (2015) Mukai flops and P-twists

Brochier A (2023) Gaudin algebras, RSK and Calogero-Moser cells in Type A in Proceedings of the London Mathematical Society

Description The original research proposal had five objectives. Four have been to some extent completed by a number of mathematicians, including myself. Most of the outputs I list here are involved with these objectives, although there are a couple that I will mention that are involved in other topics.

Objective 1. Establish a connection between Gaudin Model and Cherednik algebras of type A. This was carried out by Mukhin-Tarasov-Varchenko over a number of papers, and followed up by my PhD student (during part of the fellowship) Gwyn Bellamy. All predicted was essentially correct, and I have on-going work with one of the RAs from the fellowship which pushes this much further and is the subject of the next Objective.

Objective 2. Confirm conceptually the generalised Calogero-Moser space conjectures of [22]. A book by Bonnafe-Rouquier is in press that sets out a generalisation of the conjectures of [22] to a geometric construction of left cells for complex reflection groups. Together with my RA, Adrien Brochier, we are writing up a paper which makes definitive progress on this problem, exactly through the methods described in the research proposal, plus several new ingredients: cactus group, asymptotic version of Drinfeld-Kohno theorem, theory of crystal bases.

Objective 3. Develop tools of hamiltonian reduction, microlocal analysis and noncommutative geometry on the Hilbert scheme". This topic has moved rapidly in the last 6 years, with great progress in understanding my own localisation theorem, placing it in a more conceptual setting that applies to a wide class of symplectic singularities, particularly in work of McGerty-Nevins and Braden-Proudfoot-Webster. I proved what was referred in my proposal as a _key_ goal of the theory of Cherednik algebras by giving a new conceptual proof of the existence and properties of the Procesi bundle (this is publication Macdonald positivity via the Harish-Chandra D-module). My paper Auslander-Gorenstein property for Z-algebras establishes many basic homological properties for the noncommutative algebraic geometry of the Hilbert scheme, and the paper Differential Operators and Cherednik Algebras proves the fundamental theorem that allows comparison of the algebraic and geometric approaches to localisation for Hilbert schemes.

Objective 4. Extend application to other important representation theoretic settings. Together with Losev, I wrote the most important paper on rational Cherednik algebras of type G(r,1,n), namely On category O for cyclotomic rational Cherednik algebras. This paper confirmed 2 conjectures (of Etingof and of Rouquier) and unified several techniques from deformation quantization, algebraic combinatorics, Lie theory in order to prove the definitive results for this topic.

Objective 5. Create a general definition of G-Hilb. There has been little progress on the topics mentioned within this objective.

Beyond these objectives I would also like to mention my paper, written jointly with Stephen Griffeth, on Catalan numbers for complex reflection groups which settled and proved a more general version of a beautiful conjecture of Bessis-Reiner on generalising of the classical Catalan numbers to complex reflection groups. This has been an impetus for the founding of a new topic Rational Catalan combinatorics which was recently the focus of a week-long conference at AIM in Palo Alto.
Exploitation Route The various papers I have written, talks I have given, and people I have discussed mathematics with, will lead to my findings being taken forward in various topics within mathematics: geometric representation theory; algebraic combinatorics; noncommutative algebra; algebraic geometry; deformation quantization; Lie theory.
Sectors Other