Josh Walton - Academic Page

About Me

Hello! I am a former PhD student of the University of Manchester, researching maximal subgroups and representation theory of finite groups of Lie type through the means of the computational algebra. My supervisor is Professor Peter Rowley. My thesis title is: On the Maximal Subgroups of \(\mathrm{E}_8(3)\) and \(\mathrm{E}_8(2)\), and a copy of my thesis can be found here.

My research interests include, but are not limited to, matrix representations of finite simple groups, Lie algebras and their associated root systems and maximal subgroups of finite exceptional groups of Lie type. More generally, I am interested in the development and implementation of mathematical algorithms to solve open problems in finite group theory and representation theory. I conduct most of my current research using Magma.

Other than this, I am a sudoku nerd and I like to travel the world whenever I have the time.

Files

Below are some files that accompany my research / previous work. These are all compatible with Magma, and are currently configured to open in your browser. To use the code, you can simply copy the contents of the .txt files into your Magma session.

Generators for various maximal and non-maximal subgroups inside the 248-dimensional irreducible adjoint representation of \( \mathrm{E}_8(3) \):

\( \mathrm{Th} \) (maximal) \( \mathrm{M}_{10} \) \( \mathrm{L}_{3}(5) \) (maximal) \( \mathrm{L}_{2}(81) \) \( \mathrm{PGL}_{2}(81) \) \( \mathrm{L}_2(11) \) \( \mathrm{G}_2(3) \) \( \mathrm{U}_3(9) \) \( \mathrm{M}_{11} \) \( ^3\mathrm{D}_{4}(2) \)

Generators for various Sylow subgroups of \( \mathrm{E}_8(3) \): \(2\), \(5\), \(7\), \(11\), \(13\), \(41\).
Generators for centralisers in \( \mathrm{E}_8(3) \) for various conjugacy classes:

\(2\mathrm{A} \cong \mathrm{SL}_2(3) \times \mathrm{E}_7(3)_{sc} \) \( 2\mathrm{B} \cong 2.\Omega_{16}^+(3).2 \) \( 4\mathrm{B} \cong 4.(\Omega^-_{14}(3).2.2) \) \( 4\mathrm{F} \cong 4.(\mathrm{U}_8(3).2.2) \) \( 4\mathrm{G} \cong 4.((\mathrm{U}_4(3) \times \Omega^-_{10}(3)).4) \) \(5\mathrm{A} \cong 10.(\Omega_{12}^-(3).2) \) \( 5\mathrm{B} \cong 5.\mathrm{U}_5(9) \) \(7\mathrm{A} \cong 7.^2\mathrm{E}_6(3) \) \(7\mathrm{B} \cong 7.(^3\mathrm{D}_4(3).\mathrm{U}_3(3)) \) \(8\mathrm{A}\cong 8.^2\mathrm{E}_6(3) \) \(8\mathrm{C}\cong 8.\Omega^-_{12}(3).2 \) \(8\mathrm{E}\cong 8.(\Omega_{10}^+(3)\times \mathrm{Alt}(6)).2 \) \(8\mathrm{H}\cong 8.(\Omega^-_8(3)\times \mathrm{L}_4(3)).2 \) \(8\mathrm{K}\cong 8.(\Omega^-_8(3)\times \mathrm{U}_3(3)).2 \) \(8\mathrm{O}\cong 8.(\mathrm{L}_4(9).4) \) \(8\mathrm{P}\cong 8.(\mathrm{U}_5(3)\times \mathrm{Alt}(6)).2 \) \(8\mathrm{Q}\cong 8.(\mathrm{U}_3(3) \times \mathrm{L}_4(3) \times \mathrm{Alt}(6)).2 \) \(10\mathrm{A} \cong 10.(\Omega_{12}^-(3).2) \) \(10\mathrm{B} \cong 20.((\Omega_{8}^+(3)\times \mathrm{Alt}(6)).2^2) \) \(10\mathrm{C} \cong (20.\mathrm{PSO}^+_4(3)).(\Omega^-_8(3).2) \) \(10\mathrm{D} \cong 10.(\mathrm{U}_4(9).2) \) \(10\mathrm{E} \cong 10.(\mathrm{U}_4(9).2) \) \(10\mathrm{F} \cong 10.((\mathrm{U}_3(9) \times \mathrm{Alt}(6)).2) \) \(11\mathrm{A} \cong 121 \times \mathrm{L}_5(3) \) \(13\mathrm{AB} \cong 13.\mathrm{E}_6(3) \) \( 13\mathrm{CD} \cong 13.(^3\mathrm{D}_4(3).\mathrm{L}_3(3)) \) \(13\mathrm{E} \cong 13^2\times ^3\mathrm{D}_4(3) \) \( 13\mathrm{F} \cong 13^2 \times \mathrm{L}_3(3)^2 \) \(14\mathrm{A} \cong 28.(\Omega_{10}^-(3).4) \) \(14\mathrm{B}\) \(14\mathrm{C}\cong (7:(4.\mathrm{Sym}(4))).^3\mathrm{D}_4(3) \) \(14\mathrm{D}\) \(14\mathrm{E}\) \(20\mathrm{A} \cong 40.(\Omega_{10}^+(3).2) \) \(20\mathrm{B} \cong 20.(\Omega_{10}^-(3).4) \) \(20\mathrm{C} \) \(20\mathrm{D} \cong (\mathrm{SL}_2(3):40).(\Omega^-_8(3):2) \) \(20\mathrm{E} \cong 20.((\mathrm{L}_4(3) \times \mathrm{U}_4(3)).4) \) \(20\mathrm{F} \) \(20\mathrm{G} \cong (2\times 20).((\mathrm{Alt}(6)\times \mathrm{U}_4(3))).(2\times 4) \) \(20\mathrm{H} \cong \mathrm{Q}_8.(40\times 3:4).(\mathrm{L}_4(3).2) \) \(20\mathrm{I} \) \(20\mathrm{J} \cong (4\times 80).(\mathrm{U}_4(3):4)\) \(20\mathrm{K} \cong 80 \times \mathrm{U}_3(9)\) \(20\mathrm{L} \cong (10\times 20).(\mathrm{L}_4(3):2)\) \(20\mathrm{M} \cong 2.\mathrm{Q}_8^2.6\times 6.60.2.(\mathrm{Alt}(6):2) \) \(20\mathrm{N} \cong \mathrm{Q}_8.(10\times 3:4).\mathrm{L}_2(81) \) \(20\mathrm{O} \cong 80.(\mathrm{L}_2(81):2)\) \(20\mathrm{P} \cong (10\times 80).(\mathrm{Alt}(6):2)\) \(22\mathrm{A} \cong 242.(\mathrm{L}_4(3):2)\) \(22\mathrm{B} \cong 121 \times \mathrm{GL}_2(3) \times \mathrm{L}_3(3) \) \(26\mathrm{AB} \cong 26.(\Omega_{10}^+(3).2) \) \(26\mathrm{CD}\cong (13\times \mathrm{SL}_2(3)).(\mathrm{L}_6(3).2) \) \(26\mathrm{EF} \) \(26\mathrm{GH} \) \(26\mathrm{IJ} \cong \mathrm{Q}_8.(39\times \mathrm{Q}_8).6.(\mathrm{L}_2(27):2) \) \(26\mathrm{K} \) \(26\mathrm{LM} \cong 13^2\times \mathrm{GL}_2(3)\times \mathrm{L}_3(3) \) \(26\mathrm{N} \cong \mathrm{Q}_8^2.3^2.13.26.2 \) \(35\mathrm{A}\cong 280.(\mathrm{L}_4(3).2) \) \(41\mathrm{AE} \cong 82.(\Omega_8^-(3).2) \) \(41\mathrm{FJ} \cong 82.(\mathrm{L}_2(81).2) \) \(41\mathrm{KO} \cong 2^2 \times 41^2 \)

Generators for various conjugacy classes of maximal subgroups inside the 78-dimensional representation of \( {}^2\mathrm{E}_6(2) \):

\(\mathrm{F}_4(2)\) (3 classes) \( 2^2.2^9.2^{18}:(\mathrm{L}_3(4) \times \mathrm{Sym}(3)) \) \(\mathrm{Fi}_{22}\) (3 classes) \( \Omega^-_{10}(2) \) \( \mathrm{Sym}(3) \times \mathrm{U}_6(2) \) \( (3 \times \Omega^+_8(2)):3):2 \) \( 3^{1+6}.2^{3+6}.(\mathrm{Sym}(3) \times 3) \) \( 3^2\mathrm{Q}_8 \times \mathrm{U}_3(3):2 \)

A 'read me' file (open in browser) and a setup file (open in browser) containing functions used to produce the results in my publication on maximal tori.
My Python coding project (open in browser) which codes the game Othello and an AI opponent which you can play against. The opponent won a prize in a competition sponsored by the N Brown group.

Publications

Javed, M., Parkin, J., Rowley, P., & Walton, J., The Maximal Tori of Finite Exceptional Groups of Lie Type, Communications in Algebra, 2024. (linked here).

Teaching

MATH10111 - Foundations of Pure Mathematics
MATH11112 - Real Analysis
MATH11221 - Mathematical Problem Solving
MATH19681 - Mathematics 1E1
MATH19861 - Mathematics 0N1
MATH42121 - Galois Theory
MATH11022 - Linear Algebra
MATH20622 - Progamming with Python

Please contact me if you wish to see results from student evaluation surveys in relation to the above courses.

I have also supported several outreach events involving the delivery of math-based workshops to young adults. These include:

Mathematical Modelling Day - leading a interactive workshop on the mathematics of card shuffling in 2022, 2023 and 2024. All hosted at The University of Manchester.
Problem Solving Matters Event - Hosted at The University of Manchester in 2022.
Maths in the Real World Event - Hosted at The University of Manchester in 2022.

Contact Information

Email: joshua.walton@manchester.ac.uk or joshwalton342@gmail.com