## People

Dave Benson, Christine Bessenrodt, Simon Blackburn, John Britnell, Peter Cameron, Anton Cox, David Craven, Charles Eaton, Karin Erdmann, Matt Fayers, Eugenio Giannelli, Nick Gill, Simon Goodwin, Jan Grabowski, Anne Henke, Sinead Lyle, Paul Martin, Andrew Mathas, John Murray, Nikolay Nikolov, Rowena Paget, Alison Parker, Sarah Rees, Jeremy Rickard, Sibylle Schroll, Peter Symonds, Matt Towers.

## Recent talks

- An introduction to plethysm.
- From Euclid to Turing: proofs, truths and codes.
- Access to Bristol, November 2023.
- Sutton Trust Summer School, August 2023.

- Stability of plethysms of symmetric functions.
- An introduction to plethysms of symmetric functions and representations of general linear groups (scanned notes).
- Bristol Algebra Seminar, November 2022.

- Prime, partitions and power series: Mathematical truths from Euclid to Ramanujan, October 2022.
- Theoretical Sciences Visiting Program TSVP Talk, OIST. The video is on Youtube.

- Introduction to plethysm (all on board, notes scanned).
- Introduction to modular plethysms (slides with some important parts on board). Research summary and extended abstract.
- Character theory and categorification, Oberwolfach, August 2022.
- Structure of group algebras over local rings, Ambleside August 2022 (cut down version)

- Modular plethysms of symmetric functions (scanned notes).
- Hannover, Colloquium and conference in memory of Christine Bessenrodt, July 2022.

- Untangling the ZX calculus.
- Bristol, December 2021.

- What are quantum states really?. (Rough scanned notes, based in part on this blog post).
- Bristol, December 2021.

- From representations of general linear groups to plethysms of symmetric functions (and back).
- Bristol Algebra Seminar, November 2021.

- Plethysms, polynomial representations of linear groups and Hermite reciprocity over an arbitrary field.
- OIST Representation Theory Seminar, April 2021.

- Why we should all talk about equality and diversity.
- Video for all EPMS students.
- Live: Thursday 24 September 2020, 2pm.

- The counter-intuitive behaviour of high-dimensional spaces.
- Modular plethysms for SL
_{2}(*F*). Extended abstract.- Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory, Oberwolfach, February 2020.

- An introduction to plethysm. Extended abstract.
- Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory, Oberwolfach, February 2020.

- Plethysms and polynomial representations of SL
_{2}(**C**). (All done on board, notes scanned.)- Kent Algebra Seminar, May 2019.
- Oberseminar Algebra, Stuttgart, March 2019.
- Aberdeen Algebra Seminar, February 2019.
- NUS Singapore, informal seminar, January 2019.

- Plethysms of symmetric functions. (All done on board, notes scanned.)
- Hannover Representation Theory Days 2018, Hannover, September 2018

- The mod 2 homology of the simplex and representations of symmmetric groups. (All done on board.)
- The Liar Game: Truths and proofs from Euclid to Turing.
- Ng Kong Beng public lecture, NUS Singapore, December 2017

- The mod 2 homology of the simplex. (All done on board.)
- Representation Theory of Symmetric Groups and Related Algebras, NUS Singapore, December 2017

- Plethysms: permutations, weights and Schur functions.
- Cambridge University, Algebra and Representation Theory Seminar, March 2017
- Bristol University, Algebra and Geometry Seminar, Februrary 2017
- City University, London Algebra Colloquium, December 2016

- Plethysms and decomposition matrices of symmetric groups. (Mostly done on board.)
- Kronecker Coefficient Conferences 2016, City University, London, September 2016

- Minimal and maximal constituents of plethysms of Schur functions.
- Algebraic combinatorics and group actions, Herstmonceux, July 2016

- Generalized Foulkes characters and maximal and minimal constitutents of plethysms. (All done on board.)
- Representation theory of symmetric groups and related topics, Kaiserslautern, February 2016

- Bell numbers, partition moves and eigenvalues of the random-to-top shuffle. (All done on board.)
- York Algebra Seminar, November 2015
- 69th BLOC meeting, University of Leicester, June 2015

- Fast and fun results:
functional programming for mathematicians. Mathematica notebook with solutions to all the exercises.
- Generic skills training, Royal Holloway, December 2018
- Generic skills training, Royal Holloway, November 2017
- Generic skills training, Royal Holloway, March 2016
- Generic skills training, Royal Holloway, October 2014

- Foulkes characters: deflations, twists and algorithms. (All done on board.)
- Representations of symmetric groups, Hecke algebras and KLR algebras, University of Birmingham, July 2014

- The Liar Game.
Also given in half hour version, and given a small refresh in update in 2017 and 2019.
- Exploring Maths 2019, Royal Holloway, June 2019
- Exploring Maths 2018, Royal Holloway, June 2018
- Exploring Maths 2017, Royal Holloway, June 2017
- Exploring Maths 2016, Royal Holloway, June 2016
- Ark Academy, February 2016
- Greenford High School, September 2015
- Exploring Maths 2015, Royal Holloway, July 2015
- Hatch End High School, May 2015
- Applicant Visit Day, Royal Holloway, December 2014, January 2015 (twice)
- Headstart (Exploring Maths & Physics) 2014, Royal Holloway, June 2014
- Exploring Maths 2013 and 2014, Royal Holloway, June 2013 and 2014

- Foulkes modules and decomposition numbers for symmetric groups.
(Mostly done on board.)
- Leeds Algebra Seminar, March 2014
- Modular representation theory of finite
and
*p*-adic groups, National University of Singapore, April 2013

- Derangements in transitive permutation groups. (All done on board.)
- Manchester Algebra Seminar, March 2014

- Orbit coherence in permutation groups. (All done on board)
- Imperial College, London Algebra Colloquium, June 2013
- Oxford Algebra Seminar, December 2012

- Critical evaluation of mathematical writing through peer-marking and
formative assessment.
- Annual Teaching and Learning Symposium, Royal Holloway, April 2013

- Character deflations, wreath products and Foulkes' Conjecture.
(Sections 2 and 3 done on board.)
- Birmingham Algebra Seminar, January 2013
- Birkbeck Pure and Applicable Mathematics Seminar, April 2012
- Bristol Algebra Seminar, December 2011

- The probability that two elements of a finite group are
conjugate. (Almost all done on board.)
- York Algebra Seminar, May 2012

- Vertices of Specht modules.
- Oberwolfach Miniworkshop, May 2011

- Commuting conjugacy
classes in groups: an overview. Given in various different versions at:
- Queen's University, Belfast, April 2011
- Southampton Pure Mathematics Seminar, April 2011
- City University Mathematics Research Seminar, December 2010
- QMUL London Algebra Colloquium, October 2010
- RHUL Pure Mathematics Seminar, October 2010

- A tour of Foulkes' Conjecture.
- The symmetric group: representations and combinatorics, RHUL March 2011

- Knights, spies, games and social networks.
- Bristol Algorithms Day, February 2010.

## Foulkes' Conjecture and plethysms

Let*m*and

*n*be natural numbers. The

*Foulkes character*for

*m*and

*n*is the permutation character of the symmetric group of degree

*mn*given by its action on the collection of all set partitions of a set of size

*mn*into

*n*sets each of size

*m*. Foulkes' Conjecture states that if

*m*<

*n*then the character for the action on

*m*-subsets contains the character for the action on

*n*-subsets.

### Data on Foulkes multiplicities

Proposition 5.1 in this paper with Rowena Paget and
Anton Evseev gives a new recurrence for the character multiplicities in Foulkes' Conjecture. Using this
recurrence we extended
earlier work
by Jurgen Müller and Max Neunhöffer to prove Foulkes' Conjecture whenever the parameters *m*
and *n* satisfy *m* + *n* < 20 and, later in 2016, to *m* + *n* ≤ 20.

- FoulkesMultiplicities.hs,
Plotter.hs and Main.hs.
Haskell source code for computing constituents of Foulkes characters using Proposition 5.1 and drawing graphs
such as the one below. Compile using
`ghc --make -O2 Main.hs`

.Run with

`./Main`

to see help text or (for instance)`./Main 5 7 Single [10,10,10,5]`

to compute the multiplicity of (10,10,10,5) in the Foulkes character for*m*= 5 and*n*= 7. - Constituents of Foulkes characters
for
*m*+*n*≤ 15. - Proof of Foulkes' Conjecture for
*m*+*n*≤ 20.- Multiplicities for
*m*≤ 7 (14.3Mb) - Multiplicities for
*m*= 8 (33.5Mb) - Multiplicities for
*m*= 9 (53.4Mb)

- Multiplicities for

The graph below shows the differences between the base 2 logs of the multiplicities of irreducible
characters labelled by partitions of 56 in the Foulkes characters for *m* = 7, *n* = 8
(larger multiplicity) and *m* = 8, *n* = 7 (smaller multiplicity).

The 31275 partitions of 56 with at most 7 parts are ordered lexicographically: blue dots show multiplicities where the multiplicity in the smaller character is 0. If both multiplicities are zero the dot is drawn below the axis. Vertical lines show changes in the largest part of the partition.

### Minimal constituents of Foulkes characters

The links below give some supporting material for this paper with Rowena Paget.

- SetFamilies.hs. Haskell source code for computing minimal constituents of Foulkes characters using Theorem 1.2 of the paper.
- Examples.hs. Verifies two of the counterexamples presented in the paper.
- Minimals.txt. Minimal constituents of Foulkes modules for
*m*and*n*such that*m + n < 20*

### Minimal and maximal constituents of generalized Foulkes characters

This paper with Rowena Paget describes all minimal and
maximal constituents of the generalized Foulkes characters corresponding to the plethysms
*s*_{ν} ○ s_{μ} where ν and μ are arbitrary partitions.

- TableauFamilies.lhs. Literate Haskell code generating the conjugate-semistandard tableau family tuples that parametrize these constituents. It may also be used to check Example 8.3 and perform Algorithm 9.5.
- Source code (compiled by LaTeX).

### Littlewood's third method

If there is a unique symmetric function *f* such
that 〈*f*, *s*_{(3n)}〉 = 1 and 〈*f*,
*s*_{(1)} *s*_{λ}〉 =
〈*s*_{(n-1)} ○ *s*_{(3)})
*s*_{(2)}, *s*_{λ}〉
for all partitions λ of 3*n*-1 then, as Littlewood observed on page 349 of
this paper,
this *f* is the plethysm *s*_{(n)} ○ *s*_{(3)}.
The Haskell source code linked to below shows there is a unique *f* for *n* ≤ 8 and so Littlewood's
method is effective in these cases.
(Evaluate `claimInPaper` at the Haskell prompt.)

### Plethysms of symmetric functions and representations of SL_{2}(**C**)

Let *E* be the natural representation of SL_{2}(**C**). In this paper with Rowena Paget we study when the representations ∇^{λ }Sym^{r}*E* and ∇^{μ }Sym^{s}*E* are isomorphic. Equivalently, when are *s*_{λ}(1,*q*,…*q*^{l}) and *s*_{μ}(1,*q*,…*q*^{m}) equal up to a power of *q*?

- HookContent.hs. Haskell source code for searching for isomorphisms and checking the classification in Section 10.
- Main.hs. Use to compile HookContent.
- ExceptionalEquivalences.nb. Mathematica notebook that can be used to verify the exceptional isomorphisms in Section 11.

## Involutive random walks

The involutive random walk on the set {0, 1, … *n* - 1}, started at the state *x* chooses *y* ≤ *x* according to a prescribed probability distribution, and then steps to *n* - 1 - *x*. These walks have a remarkably rich theory explored in this paper.

- InvolutiveWalk.nb. Mathematica notebook for computing transition matrices and verifying some of the claims in the paper.
- A related matrix transform with interesting spectral behaviour.

## Simplex homology

Magma code for checking the base cases in Theorem 1.1 and conjectures in Section 7 of my paper The multistep homology of the simplex and representations of symmetric groups.

## Burnside's method

A group *K* is said to be a *B-group* if whenever *K* is a regular subgroup of a permutation
group *G* then *G* is either 2-transitive or imprimitive. Burnside proved that *C _{p}*
is a B-group whenever

*p*is prime. He later published two papers attempting to extend this result to other abelian groups. Both have fatal but interesting flaws. A proof of Conjecture 6.5 in my linked paper will rescue Burnside's proof for cyclic groups of even order.

- RamanujanMatrix.hs has been used to verify Conjecture 6.5 for permutation groups of degree at most 600.

## Coherent subgroups of permutation groups

Let *G* be a permutation group acting on a set Ω. We say that *G* is *join-coherent*
is the set of orbit partitions of elements of *G* acting on Ω is closed under the lattice
operation of taking joins, and *meet-coherent* if it is closed under taking meets.
These definitions were introduced in this paper with
John R. Britnell.

- CoherentSubgroups.mgm can be used to verify the computational assertions in the paper.
- SmallDegreeJoinCoherentGroups.mgm shows that the main theorems in the paper classify all join-coherent groups of degree at most 11.

## Conjugacy probabilities in finite groups

Let *G* be a finite group and let *g* and *h* be chosen independently and uniformly
at random from *G*. Let *κ(G)* be the probability that *g* and *h* are
conjugate, and let *ρ(G)* be the probability that *g* and *h* have conjugates that commute.
The links below give some supporting material for this this paper with
Simon R. Blackburn and John R. Britnell where we study these probabilities for symmetric groups and other finite groups.

- KappaRho.hs and Main.hs.
Haskell source code for computing
*κ*and*ρ*for symmetric groups. Verifies all the computational assertions in the paper. See Main.hs for instructions. - KappaRhoValues.txt. Values of
*κ*for symmetric groups of degree at most 100 (see OEIS A087132), and*ρ*for symmetric groups of degree at most 40 (see OEIS A192983).

## Numerical data on derangements

Numerical data on derangements in symmetric groups acting on *k*-sets.
In the linked table, the entry in position (*n*,*k*) shows the number of derangements in the
symmetric group of degree *n*, its natural action on *k*-sets. The data are also available in
plain text. Computed as part of joint work with
John R. Britnell. This note describes the algorithm used and
its implementation. Haskell source code:

## Generalized Bell and Stirling numbers

This Magma code may be used to verify the claims in Section 7 of this paper with John R. Britnell.

## Knights and spies

The Knights and Spies Problem has its own webpage.

## Expository notes

- Some examples on duality (2002). Shows that if we work with Specht modules defined over the integers, then there is an injective homomorphism from any given Specht module into its dual. Also attempts to explains why there are frequently two different definitions of duality.
- Notes on Bernoulli numbers and Euler's summation formula (2006, revised November 2014).
- Notes on double cosets of Young subgroups of symmetric groups (2007, revised March 2014).
- In Trinity 2008 the Oxford Representation Theory
Advanced Class read Gene Murphy's paper
*The idempotents of the symmetric group and Nakayama's Conjecture*. - Introductory notes on Schur functors and weight spaces (2007, revised January 2014).
- Exercise 13 in Diaconis' book
*Group representations in probability and statistics*(2011). - Notes on impartial games with entailment (2013).
- Notes on the Weyl Character Formula (June 2014).
- A proof that the Solomon Descent Algebra is closed under multiplication (January 2015).
- Diaconis and Fulman have given a beautiful bijective proof of a theorem relating carries in base
*b*-addition to descents in*b*-riffle shuffles. My notes give a detailed proof of the strongest version of their result (June 2017). - A construction of the Weyl module Δ
^{λ}(*V*) as an explicit submodule of an exterior power (February 2020). - My reviews for MathSciNet. My reviews for Zentralblatt.

Some notes on more basic mathematics are on my teaching page.