Josh Walton

About Me

Hello! I am a final year PhD student, researching maximal subgroups and representation theory of finite groups of Lie type through the means of the computational algebra at the University of Manchester, U.K. My supervisor is Professor Peter Rowley. My thesis title is: Potential Maximal Subgroups of ${\mathrm{E}}_8(2)$, ${\mathrm{E}}_8(3)$ and The Maximal Tori of Finite Exceptional Groups of Lie Type.

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.

Files

• 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:
$2\mathrm{A}\cong {\mathrm{SL}}_2(3)\times {\mathrm{E}}_7(3{)}_{sc}$ $2\mathrm{B}\cong 2.{\mathrm{\Omega }}_{16}^{+}(3).2$ $4\mathrm{B}\cong 4.({\mathrm{\Omega }}_{14}^{-}(3).2.2)$ $4\mathrm{F}\cong 4.({\mathrm{U}}_8(3).2.2)$ $4\mathrm{G}\cong 4.(({\mathrm{U}}_4(3)\times {\mathrm{\Omega }}_{10}^{-}(3)).4)$ $5\mathrm{A}\cong 10.({\mathrm{\Omega }}_{12}^{-}(3).2)$ $5\mathrm{B}\cong 5.{\mathrm{U}}_5(9)$ $7\mathrm{A}\cong {7.}^2{\mathrm{E}}_6(3)$ $7\mathrm{B}\cong 7.{(}^3{\mathrm{D}}_4(3).{\mathrm{U}}_3(3))$ $8\mathrm{A}\cong {8.}^2{\mathrm{E}}_6(3)$ $8\mathrm{C}\cong 8.{\mathrm{\Omega }}_{12}^{-}(3).2$ $8\mathrm{E}\cong 8.({\mathrm{\Omega }}_{10}^{+}(3)\times \mathrm{Alt}(6)).2$ $8\mathrm{H}\cong 8.({\mathrm{\Omega }}_8^{-}(3)\times {\mathrm{L}}_4(3)).2$ $8\mathrm{K}\cong 8.({\mathrm{\Omega }}_8^{-}(3)\times {\mathrm{U}}_3(3)).2$ $8\mathrm{O}\cong 8.({\mathrm{L}}_4(9).4)$ $8\mathrm{P}\cong 8.({\mathrm{U}}_5(3)\times \mathrm{Alt}(6)).2$ $8\mathrm{Q}\cong 8.({\mathrm{U}}_3(3)\times {\mathrm{L}}_4(3)\times \mathrm{Alt}(6)).2$ $10\mathrm{A}\cong 10.({\mathrm{\Omega }}_{12}^{-}(3).2)$ $10\mathrm{B}\cong 20.(({\mathrm{\Omega }}_8^{+}(3)\times \mathrm{Alt}(6)){.2}^2)$ $10\mathrm{C}\cong (20.{\mathrm{PSO}}_4^{+}(3)).({\mathrm{\Omega }}_8^{-}(3).2)$ $10\mathrm{D}\cong 10.({\mathrm{U}}_4(9).2)$ $10\mathrm{E}\cong 10.({\mathrm{U}}_4(9).2)$ $10\mathrm{F}\cong 10.(({\mathrm{U}}_3(9)\times \mathrm{Alt}(6)).2)$ $11\mathrm{A}\cong 121\times {\mathrm{L}}_5(3)$ $13\mathrm{AB}\cong 13.{\mathrm{E}}_6(3)$ $13\mathrm{CD}\cong 13.{(}^3{\mathrm{D}}_4(3).{\mathrm{L}}_3(3))$ $13\mathrm{E}\cong {13}^2{\times }^3{\mathrm{D}}_4(3)$ $13\mathrm{F}\cong {13}^2\times {\mathrm{L}}_3(3{)}^2$ $14\mathrm{A}\cong 28.({\mathrm{\Omega }}_{10}^{-}(3).4)$ $14\mathrm{B}\cong (7\times {\mathrm{SL}}_2(3)).({\mathrm{U}}_6(3).2)$ $14\mathrm{C}\cong (7:(4.\mathrm{Sym}(4))){.}^3{\mathrm{D}}_4(3)$ $14\mathrm{D}\cong (7\times {\mathrm{SL}}_2(3)).({\mathrm{U}}_3(3)\times {\mathrm{L}}_2(27)).2$ $14\mathrm{E}\cong ((2\times 28).(\mathrm{Alt}(4)\times \mathrm{Sym}(4))).({\mathrm{L}}_2(27).2)$ $20\mathrm{A}\cong 40.({\mathrm{\Omega }}_{10}^{+}(3).2)$ $20\mathrm{B}\cong 20.({\mathrm{\Omega }}_{10}^{-}(3).4)$ $20\mathrm{C}\cong ({\mathrm{SL}}_2(3):40).({\mathrm{\Omega }}_8^{-}(3):2)$ $20\mathrm{D}\cong ({\mathrm{SL}}_2(3):40).({\mathrm{\Omega }}_8^{-}(3):2)$ $20\mathrm{E}\cong 20.(({\mathrm{L}}_4(3)\times {\mathrm{U}}_4(3)).4)$ $20\mathrm{F}\cong (2\times 20).((\mathrm{Alt}(6)\times {\mathrm{U}}_4(3))).(2\times 4)$ $20\mathrm{G}\cong (2\times 20).((\mathrm{Alt}(6)\times {\mathrm{U}}_4(3))).(2\times 4)$ $20\mathrm{H}\cong {\mathrm{Q}}_8.(40\times 3:4).({\mathrm{L}}_4(3).2)$ $20\mathrm{I}\cong {\mathrm{Q}}_8.(2\times 60).2.({\mathrm{U}}_4(3):4)$ $20\mathrm{J}\cong (4\times 80).({\mathrm{U}}_4(3):4)$ $20\mathrm{K}\cong 80\times {\mathrm{U}}_3(9)$ $20\mathrm{L}\cong (10\times 20).({\mathrm{L}}_4(3):2)$ $20\mathrm{M}\cong 2.{\mathrm{Q}}_8^2.6\times 6.60.2.(\mathrm{Alt}(6):2)$ $20\mathrm{N}\cong {\mathrm{Q}}_8.(10\times 3:4).{\mathrm{L}}_2(81)$ $20\mathrm{O}\cong 80.({\mathrm{L}}_2(81):2)$ $20\mathrm{P}\cong (10\times 80).(\mathrm{Alt}(6):2)$ $22\mathrm{A}\cong 242.({\mathrm{L}}_4(3):2)$ $22\mathrm{B}\cong 121\times {\mathrm{GL}}_2(3)\times {\mathrm{L}}_3(3)$ $26\mathrm{AB}\cong 26.({\mathrm{\Omega }}_{10}^{+}(3).2)$ $26\mathrm{CD}\cong (13\times {\mathrm{SL}}_2(3)).({\mathrm{L}}_6(3).2)$ $26\mathrm{EF}\cong (13\times {\mathrm{GL}}_2(3)){.}^3{\mathrm{D}}_4(3)$ $26\mathrm{GH}\cong (13\times {\mathrm{SL}}_2(3)).({\mathrm{L}}_2(27)\times {\mathrm{L}}_3(3):2)$ $26\mathrm{IJ}\cong {\mathrm{Q}}_8.(39\times {\mathrm{Q}}_8).6.({\mathrm{L}}_2(27):2)$ $26\mathrm{K}\cong ({13}^2\times {\mathrm{SL}}_2(3)).({\mathrm{L}}_2(27):2)$ $26\mathrm{LM}\cong {13}^2\times {\mathrm{GL}}_2(3)\times {\mathrm{L}}_3(3)$ $26\mathrm{N}\cong {\mathrm{Q}}_8^2{.3}^2.13.26.2$ $35\mathrm{A}\cong 280.({\mathrm{L}}_4(3).2)$ $41\mathrm{AE}\cong 82.({\mathrm{\Omega }}_8^{-}(3).2)$ $41\mathrm{FJ}\cong 82.({\mathrm{L}}_2(81).2)$ $41\mathrm{KO}\cong 2^2\times {41}^2$
• 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) ${\mathrm{\Omega }}_{10}^{-}(2)$ $\mathrm{Sym}(3)\times {\mathrm{U}}_6(2)$ $(3\times {\mathrm{\Omega }}_8^{+}(2)):3):2$ $3^{1+6}{.2}^{3+6}.(\mathrm{Sym}(3)\times 3)$ $3^2{\mathrm{Q}}_8\times {\mathrm{U}}_3(3):2$

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

• 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

Office: 2.210, Alan Turing Building

University of Manchester