Mark Wildon's Website: Research

My main area of research is the modular representation theory of the symmetric group. My other interests include partition combinatorics, reductive algebraic groups, free Lie algebras, symmetric functions and combinatorial games.

MathSciNet reviews and Zentralblatt reviews of my papers. Most of my papers are also on the arXiv.

Two theorems on the vertices of Specht modules 2003 Arch. Math. (Basel) 81 (2003) 505–511. Classifies all Specht modules with cyclic vertex, finds the vertices of Specht modules labelled by hook partitions, and proves that vertices of Specht modules are preserved by Scopes functors.
Counting partitions on the abacus July 2006 Ramanujan Journal 17 (2008) 355–367. Uses the abacus notation for partitions to give combinatorial proofs of upper and lower bounds on the partitions function. Features the partition diagram above.
Character values and decomposition matrices of symmetric groups October 2006 J. Alg. 319 (2008) 3382–3397. Gives a necessary and sufficient condition for two rows of a decomposition matrix of a symmetric group to be equal.
Labelling the character tables of symmetric and alternating groups December 2006 Q. J. Math. 59 (2008) 123–135. Suppose you are given a character table of a symmetric group with the partitions labelling its rows and columns removed. This paper shows that in all but two cases, there is a unique way to reconstruct the labels.
A combinatorial method for calculating the moments of Lévy area
Coauthor: Daniel Levin
January 2007 Trans. Amer. Math. Soc. 360 (2008) 6695–6709. Gives a combinatorial method for finding the moments, and hence the characteristic function, of the signed area of a Brownian motion.
On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension
Coauthor: John R. Britnell
August 2007 Bull. Lond. Math. Soc. 40 (2008) 897–906. Let G be a finite group and H a normal subgroup such that G/H is cyclic. We define for each G-conjugacy class a centralizing subgroup, which controls how the class splits up when the conjugacy action is restricted to H. We show that the G-conjugacy classes with a given centralizing subgroup are as uniformly distributed between the cosets of H as is possible.
Multiplicity-free representations of symmetric groups March 2009 J. Pure Appl. Alg. 213 (2009) 1464–1477. Completes Saxl's classification of the multiplicity-free permutation characters of symmetric groups and looks at Specht filtrations for the associated permutation modules. A related family of monomial characters is also classified.
Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem to group theory
Coauthor: John R. Britnell
December 2008 J. Group Theory 12 (2009) 795–802. Say that two conjugacy classes of a group commute if they contain representatives that commute. Let G be a finite group containining a normal subgroup of prime index. Each conjugacy class of G lies entirely with some coset of this subgroup. We use Hall's Marriage Theorem to partition these classes into subsets all of whose members commute.
Knights, spies, games and ballot sequences

A version suitable for printing on black and white printers is here.
January 2009 (revised April 2010) Disc. Math. 310 (2010) 2974–2983. In a room there are 100 numbered people. A person may either be a knight or a spy. Knights always tell the truth, but spies may lie or tell the truth as they see fit. There are strictly more knights than spies present. Asking only questions of the form ‘Person i, what is the identity of person j?’, how many questions are needed to guarantee to find everyone's true identity? My webpage on the problem has some extra material.
Vertices of Specht modules and blocks of the symmetric group June 2009 (revised January 2010) J. Alg. 323 (2010) 2243–2256. Gives a large subgroup contained in the vertex of any indecomposable non-projective Specht module. This result is used to give a new way to find the defect groups of blocks of the symmetric groups, and to describe their Brauer correspondents.
On types and classes of commuting matrices over finite fields
Coauthor: John R. Britnell
December 2009 J. Lond. Math. Soc. 83 (2011) 470–492. Say that two similarity classes of matrices over a finite field commute if they contain elements that commute. We reduce the problem of determining all commuting classes to the case of nilpotent classes, and prove a number of new results about this case.
Set families and Foulkes modules
Coauthor: Rowena Paget
July 2010 (revised March 2011) J. Alg. Combinat. 34 (2011) 525–544. We construct a new family of homomorphisms from Specht modules into Foulkes modules, and use these homomorphisms to give a combinatorial description of all the minimal partitions (in the dominance order) that label irreducible characters appearing in Foulkes modules.
The probability that a pair of elements of a finite group are conjugate
Coauthors: Simon R. Blackburn and John R. Britnell
August 2011 J. Lond. Math. Soc. 86 (2012) 755–778. Let κ(G) be the probability that two elements of a finite group G, chosen independently and uniformly at random, are conjugate. We prove two ‘gap’ results classifying all groups for which κ(G) is unusually small or large. We also prove bounds and asymptotic results on κ(G) when G is a symmetric group, and study the related probability that two permutations have conjugates that commute. This Haskell code verifies the computational lemmas in the paper.
Character deflations and a generalization of the Murnaghan–Nakayama rule
Coauthors: Anton Evseev and Rowena Paget
April 2012 (extended November 2013) J. Group Theory 17 (2014) 1034–1070. We define a deflation map that sends characters of the symmetric group Smn to characters of Sn via characters of the wreath product of Sm and Sn. We prove a combinatorial rule for the values taken by these deflated characters that simultaneously generalizes the Murnaghan–Nakayama rule and Young's rule.
Finding a princess in a palace: a pursuit–evasion problem
Coauthor: John R. Britnell
April 2012 Elec. J. Combinat. 20 (2013) #25. We solve a pursuit–evasion problem in which a prince must find a princess constrained to move on each day from one vertex of a finite graph to another.
Orbit coherence in permutation groups
Coauthor: John R. Britnell
May 2012 J. Group Theory 17 (2014) 73–109. We prove a number of classification and structural theorems on permutation groups whose set of orbit partitions is closed under the lattice operations of either meet or join. In particular, we show that the orbit partitions of any centralizer in a finite symmetric group form a lattice. This Magma code may be used to verify the computational claims in the paper.
On types of matrices and centralizers of matrices and permutations
Coauthor: John R. Britnell
September 2013 J. Group Theory 17 (2014) 875–887. The type of a matrix over a finite field was defined by J. A. Green, who also showed that matrices of the same type have conjugate centralizers. We extend this definition and result to arbitrary fields and prove that the converse also holds. We also prove the analogous results for symmetric and alternating groups.
Foulkes modules and decomposition numbers of the symmetric group
Coauthor: Eugenio Giannelli
September 2013 (revised Februrary 2014) J. Pure Appl. Alg. 219 (2015) 255–276. Gives a combinatorial description of certain columns of the decomposition matrices of symmetric groups in odd characteristic. The result applies to blocks of arbitrarily high weights and in many cases of interest determines at least one column completely. The result is obtained using the p-local structure of various twists of a Foulkes permutation module. Summary of notation.
The majority game with an arbitrary majority
Coauthor: John R. Britnell
March 2014 Discrete Appl. Math. 208 (2016) 1–6. Determines the minimum number of binary comparisons needed to identify a ball of a majority colour (white or black) in a sequence of blindfold drawings from a bag of 2s + e balls, known to contain s + e balls of the majority colour.
A combinatorial proof of a plethystic Murnagham–Nakayama rule July 2014 (revised April 2015) SIAM J. Discrete Math. 30 (2016) 1526–1533. Gives a short combinatorial proof using James' abacus of a rule for multiplying a Schur function by the plethysm of a power sum and complete symmetric function.
Minimal and maximal constituents of twisted Foulkes characters
Coauthor: Rowena Paget
September 2014 J. Lond. Math. Soc. 93 (2016) 301–318. Determines the minimal and maximal constituents of the characters of the symmetric groups Smn corresponding to the plethysms of Schur functions sν ○ s(m), where ν is a partition, using two combinatorial rules involving families of subsets and multisubsets of the natural numbers.
Sylow subgroups of symmetric and alternating groups and the vertex of S(kp-p,1p) in characteristic p
Coauthors: Eugenio Giannelli, Kay Jin Lim
November 2014 J. Alg. 455 (2016) 358–385. Shows that a p-subgroup of a symmetric group or alternating group is a Sylow p-subgroup if and only if it contains a conjugate of every elementary abelian p-subgroup. We use this result, the Brauer correspondence, and arguments using generic Jordan type, to determine the vertices of certain Specht modules labelled by partitions of the form (kp-p,1p).
Searching for knights and spies: a majority/minority game December 2014 Disc. Math. 339 (2016), 2574–2766. Finds the minimum number of questions needed to find a spy, a nominated person's identity, or at least one person's identity in the knights and spies game. We consider both spies who always lie, and spies who answer as they see fit. The paper ends with some open problems. This Haskell code verifies the computational claims in the paper.
On signed Young permutation modules and signed p-Kostka numbers
Coauthors: Eugenio Giannelli, Kay Jin Lim and William O'Donovan
May 2015, extended September 2016 J. Group Theory 20 (2017), 637–679. Gives a symmetric group construction of signed Young modules and proves new reduction formulae for the multiplicities of these modules as direct summands of signed Young permutation modules. We also classify all indecomposable signed Young permutation modules.
Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D
Coauthor: John R. Britnell
July 2015 J. Combin. Theory Ser. A. 148 (2017) 116–144. We define generalized Bell and Stirling numbers that count certain sequences of random-to-top shuffles and box moves on Young diagrams. The proofs use Solomon's Descent Algebra in Type A. We generalize the results to Types B and D and give generating functions and asymptotic formulae for these numbers. This Magma code may be used to verify the computational claims in Section 7.
Indecomposable summands of Foulkes modules
Eugenio Giannelli
August 2015 J. Pure Appl. Alg. 220 (2016) 2969–2984. We characterize the vertices of the Foulkes modules corresponding to the action of S2n on the 2-subsets of a 2n set over fields of odd prime characteristic and give a complete description of all summands in blocks of weight at most two.
Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions
Coauthor: Rowena Paget
August 2016 Proc. Lond. Math. Soc. 118 (2019) 1153–1187. Gives a combinatorial characterization of the maximal and minimal constituents of a general plethysm of two Schur functions. The proof is carried out in the symmetric group gives an explicit homomorphism corresponding to each maximal or minimal constituent. This Haskell source code may be used to check Example 8.3 and perform Algorithm 9.5.
A generalized SXP rule proved by bijections and involutions September 2016 Ann. Comb. 22 (2017) 885–905. This paper gives a largely self-contained proof of a generalization of the SXP rule expressing the plethysm of a Schur function and a power sum symmetric function as a linear combination of Schur functions.
Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside May 2017 Math. Proc. Camb. Phil. Soc. 168 (2020) 613–633. A group K is said to be a B-group if a permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. Burnside correctly stated that any cyclic group of composite degree is a B-group but his two proofs both have serious flaws. We explain these flaws, give a correct proof in Burnside's setting, and survey the situation post CFSG. Update: expanded Section 5 breaks down the argument and includes steps that I should have made explicit in the published paper.
The multistep homology of the simplex and representations of symmetric groups April 2018 Math. Proc. Camb. Phil. Soc. 160 (2020) 231–253. Working over a field of characteristic two we generalize the boundary maps from simplicial homology to jump by several dimensions at once and explore the rich structure of the generalized homology modules. We obtain a new construction of the basic spin modules for the symmetric group and categorify several alternating-sum binomial identities. Magma code for checking base cases in Theorem 1.1 and the conjectures in Section 7.
A proof of the Murnaghan–Nakayama rule by Specht modules and tableau combinatorics
Coauthor: Jasdeep Kochhar
May 2018 Annals of Combinatorics. 24 (2020) 149–170. We give a new combinatorial proof of the Murnaghan–Nakayama rule by explicitly computing the trace of the matrix representing the action of an n-cycle on the standard basis of a skew-Specht module.
Plethysms of symmetric functions and highest weight representations
Coauthors: Melanie de Boeck, Rowena Paget
September 2018 To appear in Trans. Amer. Math. Soc. We generalize four results on plethysms due to Bruns—Conca—Varbaro, Brion, Ikenmeyer and the authors. In particular we characterize all maximal and minimal partitions in the dominance order that appear in an arbitrary plethysm and determine the multiplicity as counting a certain set of 'plethystic' semistandard tableaux. Extra: Computational examples.
Plethysms of symmetric functions and representations of SL2(C)
Coauthor: Rowena Paget
July 2019 Algebr. Comb. 4 (2021) 27–68. Let E be the natural representation of SL2(C). We make a systematic study of when the representations ∇λ SymrE and ∇μ SymsE are isomorphic. We obtain generalizations of earlier results of King and Manivel, many new isomorphisms and several new obstacles to such isomorphisms. We make a connection with MacMahon's enumeration of plane partitions and prove a new q-binomial identity in this setting. Haskell code and Mathematica notebook for checking Sections 10 and 11.
Involutive random walks on total orders and the anti-diagonal eigenvalue property
Coauthor: John R. Britnell
January 2021 To appear in Linear Algebra and its Applications. We study a family of random walks defined on the the finite ordinals: given x ∈ {0,1,… n - 1}, an element yx is chosen according to a prescribed probability distribution; the walk then steps to n - 1 - y. We characterise when these walks are reversible, and find the invariant distributions, eigenvalues and eigenvectors for a distinguished family whose eigenvalues are related to the Jacobi polynomials. Some extras.
Modular plethystic isomorphisms for two-dimensional linear groups
Coauthor: Eoghan McDowell
April 2021 To appear in J. Algebra. We use the two dual constructions of the symmetric power to construct an explicit isomorphism SymmSymlE and SymlSymmE where E is the natural two dimensional representation of SL2(K) and K is an arbitrary field. This generalizes Hermite reciprocity to arbitrary fields. We prove a similar generalisation of the classical Wronskian isomorphism and several further such modular plethysms isomorphisms.

Other papers not intended for publication.

A short proof of the existence of Jordan Normal Form December 2007 Not for publication as a kind referee pointed out the main idea has already appeared in print (despite this, it doesn't seem to be as widely known as it should). An earlier version is available, which proves the analogous result for finitely-generated abelian groups.
Computing derangement probabilities of the symmetric group acting on k-sets
Coauthor: John R. Britnell
October 2015 Peter Cameron has conjectured that the limiting proportion of derangements in the symmetric group Sn acting on k sets is an increasing function of k. We give an algorithm for computing these limits and use it to prove Cameron's conjecture for k ≤ 30. Haskell source code.
A corollary of Stanley's Hook Content Formula October 2018 We prove an attractive bijection between the entries in two rectangular tableaux defined using the hook lengths of a partition. I am grateful to Christine Bessenrodt for pointing out that the main result has a shorter proof using Corollary 3.2 in her paper On hooks of skew Young diagrams and bars in shifted diagrams. I do not plan submit this paper as it stands, but it may be useful as an introduction to hooks and contents of partitions.
A lower bound for the partition function from Chebychev's inequality applied to a coin flipping model for the random partition June 2019 (revised April 2021) We use a coin flipping model for the random partition and basic arguments from probability theory to prove that log p(n) ≥ Cn for all sufficiently large n and an explicit constant C. This Mathematica notebook may be used to simulate the model and check the explicit lower bound. Not for publication as the result is not very strong, but I hope the method will still be of interest.
Last modified: 28/3/22. Email: