Duncan Laurie

Research interests

I am currently a PhD student at the Mathematical Institute, University of Oxford, supervised by Professor Kevin McGerty and working in the Algebra research group.

My research is in geometric representation theory, more specifically quantum groups, quantum affine algebras, crystal bases, quantum toroidal algebras, quiver varieties, and related areas.


Email: firstname [dot] lastname [at] maths [dot] ox [dot] ac [dot] uk

There is also a contact form on my Mathematical Institute profile.

Publications and preprints

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

Duncan Laurie (7 Nov 2023)


Abstract: We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types E_6^(1), E_7^(1) and E_8^(1). In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.

Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group

Duncan Laurie (13 Apr 2023)


Abstract: We construct an action of the extended double affine braid group on the quantum toroidal algebra Uq(gtor) in all untwisted types. In the simply laced cases, using this action and certain involutions of we obtain automorphisms and anti-automorphisms of Uq(gtor) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and k0^a0...kn^an up to inverse. This generalises existing results in type A due to Miki, and can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups which Cherednik used to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures.

Other writings

Thesis for the 'Transfer of Status' milestone of my PhD, December 2021

Mini-project for the Oxford Master's course 'C3.4 Algebraic Geometry', December 2020

Master's dissertation, April 2019

Duncan Laurie - C3.4 Algebraic Geometry Miniproject.pdf
Duncan Laurie - Master's Dissertation.pdf




Reading groups


Past conferences





Fig. 1. An illustration of the quantum toroidal algebra and its generators, highlighting the horizontal and vertical quantum affine subalgebras Uh and Uv respectively

Fig. 2. Young wall patterns for types E_6^(1), E_7^(1) and E_8^(1) as constructed in my paper arXiv:2311.03905