arXiv subjects

Mark Wildon

Publications and source records attributed to Mark Wildon.

At least 37 records · Page 2Linked to original sources

arXiv2018

The multistep homology of the simplex and representations of symmetric groups

The symmetric group on a set acts transitively on its subsets of a given size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

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arXiv2017

Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later in 1921 he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this note we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside's character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

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arXiv2016

A generalized SXP rule proved by bijections and involutions

This paper proves a combinatorial rule expressing the product $s_τ(s_{λ/μ} \circ p_r)$ of a Schur function and the plethysm of a skew Schur function with a power sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm $s_λ\circ p_r$. Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono, A simple proof of the Littlewood--Richardson rule and applications, Discrete Mathematics 193 (1998) 257--266. The connections with two later combinatorial rules for special cases of this plethysm are discussed. Two open problems are raised. The paper is intended to be readable by non-experts.

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arXiv2016

On signed p-Kostka numbers and the indecomposable signed Young permutation modules

We prove the existence and main properties of signed Young modules for the symmetric group, using only basic facts about symmetric group representations and the Brou{é} correspondence. We then prove new reduction theorems for the signed $p$-Kostka numbers, defined to be the multiplicities of signed Young modules as direct summands of signed Young permutation modules. We end by classifying the indecomposable signed Young permutation modules and determining their endomorphism algebras.

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arXiv2015

Computing derangement probabilities of the symmetric group acting on k-sets

Let $i(\infty,k)$ be the limiting proportion, as $n \rightarrow \infty$, of permutations in the symmetric group of degree $n$ that fix a $k$-set. We give an algorithm for computing $i(\infty,k)$ and state the values of $i(\infty,k)$ for $k \le 30$. These values are consistent with a conjecture of Peter Cameron that $i(\infty,k)$ is a decreasing function of $k$.

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arXiv2015

Indecomposable summands of Foulkes modules

In this paper we study the modular structure of the permutation module $H^{(2^n)}$ of the symmetric group $S_{2n}$ acting on set partitions of a set of size $2n$ into $n$ sets each of size $2$, defined over a field of odd characteristic $p$. In particular we characterize the vertices of the indecomposable summands of $H^{(2^n)}$ and fully describe all of its indecomposable summands that lie in blocks of $p$-weight at most two. When $2n < 3p$ we show that there is a unique summand of $H^{(2^n)}$ in the principal block of $S_{2n}$ and that this summand exhibits many of the extensions between simple modules in its block.

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arXiv2015

Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D

Let $B_t(n)$ be the number of set partitions of a set of size~$t$ into at most $n$ parts and let $B'_t(n)$ be the number of set partitions of $\{1,\ldots, t\}$ into at most $n$ parts such that no part contains both $1$ and~$t$ or both $i$ and $i+1$ for any $i \in \{1,\ldots,t-1\}$. We give two new combinatorial interpretations of the numbers $B_t(n)$ and $B'_t(n)$ using sequences of random-to-top shuffles, %that leave a deck of cards invariant, and sequences of box moves on the Young diagrams of partitions. Using these ideas we obtain a very short proof of a generalization of a result of Phatarfod on the eigenvalues of the random-to-top shuffle. We also prove analogous results for random-to-top shuffles that may flip certain cards. The proofs use the Solomon descent algebras of Types A, B and~D. We give generating functions and asymptotic results for all the combinatorial quantities studied in this paper.

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arXiv2015

A combinatorial proof of a plethystic Murnaghan--Nakayama rule

This article gives a combinatorial proof of a plethystic generalization of the Murnaghan--Nakayama rule. The main result expresses the product of a Schur function with the plethysm $p_r \circ h_n$ as an integral linear combination of Schur functions. The proof uses a sign-reversing involution on sequences of bead moves on James' abacus, inspired by the arguments in N. Loehr, Abacus proofs of Schur function identities, SIAM J. Discrete Math. 24 (2010), 1356-1370.

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arXiv2014

Searching for knights and spies: a majority/minority game

There are n people, each of whom is either a knight or a spy. It is known that at least k knights are present, where n/2 < k < n. Knights always tell the truth. We consider both spies who always lie and spies who answer as they see fit. This paper determines the number of questions required to find a spy or prove that everyone in the room is a knight. We also determine the minimum number of questions needed to find at least one person's identity, or a nominated person's identity, or to find a spy (under the assumption that a spy is present). For spies who always lie, we prove that these searching problems, and the problem of finding a knight, can be solved by a simultaneous optimal strategy. We also give some computational results on the problem of finding all identities when spies always lie, and end by stating some open problems.

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arXiv2014

Sylow subgroups of symmetric and alternating groups and the vertex of $S^{(kp-p,1^p)}$ in characteristic $p$

We show that the Sylow $p$-subgroups of a symmetric group, respectively an alternating group, are characterized as the $p$-subgroups containing all elementary abelian $p$-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition $(kp-p,1^p)$ under the assumption that $k \equiv 1$ mod $p$ and $k \not\equiv 1$ mod $p^2$.

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arXiv2014

Foulkes modules and decomposition numbers of the symmetric group

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in characteristic zero. The main theorem of this paper gives a combinatorial description of certain columns of the decomposition matrices of symmetric groups in odd prime characteristic. The result applies to blocks of arbitrarily high weight. It is obtained by studying the p-local structure of certain twists of the permutation module given by the action of the symmetric group of even degree 2m on the collection of set partitions of a set of size 2m into m sets each of size two. In particular, the vertices of the indecomposable summands of all such modules are characterized; these summands form a new family of indecomposable p-permutation modules for the symmetric group. As a further corollary it is shown that for every natural number w there is a diagonal Cartan number in a block of the symmetric group of weight w equal to w+1.

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arXiv2014

Set families and Foulkes modules

We construct a new family of homomorphisms from Specht modules into Foulkes modules for the symmetric group. These homomorphisms are used to give a combinatorial description of the minimal partitions (in the dominance order) which label irreducible characters appearing as summands of the characters of Foulkes modules. The homomorphisms are defined using certain families of subsets of the natural numbers. These families are of independent interest; we prove a number of combinatorial results concerning them.

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arXiv2014

Minimal and maximal constituents of twisted Foulkes characters

We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms s_ν\circ s_(m). As a corollary we prove two conjectures of Agaoka on the lexicographically least constituents of the plethysms s_ν\circ s_(m) and s_ν\circ s_(1^m).

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arXiv2014

The majority game with an arbitrary majority

The $k$-majority game is played with $n$ numbered balls, each coloured with one of two colours. It is given that there are at least $k$ balls of the majority colour, where $k$ is a fixed integer greater than $n/2$. On each turn the player selects two balls to compare, and it is revealed whether they are of the same colour; the player's aim is to determine a ball of the majority colour. It has been correctly stated by Aigner that the minimum number of comparisons necessary to guarantee success is $2(n-k) - B(n-k)$, where $B(m)$ is the weight of the binary expansion of $m$. However his proof contains an error. We give an alternative proof of this result, which generalizes an argument of Saks and Werman.

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arXiv2013

Character deflations and a generalization of the Murnaghan--Nakayama rule

Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S_{mn} to the characters of S_n. This map is obtained by first restricting a character of S_{mn} to the wreath product S_m \wr S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m \times ... \times S_m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S_{mn} under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan--Nakayama rule and special cases of the Littlewood--Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.

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arXiv2013

On types of matrices and centralizers of matrices and permutations

It is known that that the centralizer of a matrix over a finite field depends, up to conjugacy, only on the type of the matrix, in the sense defined by J. A. Green. In this paper an analogue of the type invariant is defined that in general captures more information; using this invariant the result on centralizers is extended to arbitrary fields. The converse is also proved: thus two matrices have conjugate centralizers if and only if they have the same generalized type. The paper ends with the analogous results for symmetric and alternating groups.

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arXiv2012

Orbit coherence in permutation groups

This paper introduces the notion of orbit coherence in a permutation group. Let $G$ be a group of permutations of a set $Ω$. Let $π(G)$ be the set of partitions of $Ω$ which arise as the orbit partition of an element of $G$. The set of partitions of $Ω$ is naturally ordered by refinement, and admits join and meet operations. We say that $G$ is join-coherent if $π(G)$ is join-closed, and meet-coherent if $π(G)$ is meet-closed. Our central theorem states that the centralizer in $\Sym(Ω)$ of any permutation $g$ is meet-coherent, and subject to a certain finiteness condition on the orbits of $g$, also join-coherent. In particular, if $Ω$ is a finite set then the orbit partitions of elements of the centralizer in $\Sym(Ω)$ of $g$ form a lattice. A related result states that the intransitive direct product and the imprimitive wreath product of two finite permutation groups are join-coherent if and only if each of the groups is join-coherent. We also classify the groups $G$ such that $π(G)$ is a chain and prove two further theorems classifying the primitive join-coherent groups of finite degree, and the join-coherent groups of degree $n$ normalizing a subgroup generated by an $n$-cycle.

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arXiv2012

Finding a princess in a palace: A pursuit-evasion problem

This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess.

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