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Mark Wildon

Publications and source records attributed to Mark Wildon.

47 records · Page 3Linked to original sources
arXiv2012

The probability that a pair of elements of a finite group are conjugate

Let $G$ be a finite group, and let $κ(G)$ be the probability that elements $g$, $h\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $κ(G) \geq 1/4$, and shows that $G$ is abelian whenever $κ(G)|G| < 7/4$. It is also shown that $κ(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $κ(S_n) \leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that $κ(S_n) \sim A/n^2$ as $n\rightarrow \infty$ for some constant $A$. The same techniques provide analogous results for $ρ(S_n)$, the probability that two elements of the symmetric group have conjugates that commute.

math.GR↗
arXiv2010

On types and classes of commuting matrices over finite fields

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of nilpotent classes. Classes which are represented in the centralizer of every nilpotent matrix are classified--this result holds over any field. Nilpotent classes are parametrized by partitions; we find pairs of partitions whose corresponding nilpotent classes commute over some finite fields, but not over others. We conclude by classifying all pairs of classes, parametrized by two-part partitions, that commute. Our results on nilpotent classes complement work of Košir and Oblak.

math.GR↗
arXiv2009

Vertices of Specht modules and blocks of the symmetric group

This paper studies the vertices, in the sense defined by J. A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. We also use it to find the Green correspondents of a particular family of simple Specht modules; as a corollary, we get a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, as developed by M. Brou{é}, together with combinatorial arguments using Young tableaux.

math.RT↗
arXiv2009

Multiplicity-free representations of symmetric groups

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic. Remark: parallel work of Godsil and Meagher (arXiv:math/0612567) gives an independent classification of the multiplicity-free permutation characters of symmetric groups of all degrees.

math.RT↗
arXiv2009

Knights, spies, games and ballot sequences

This paper presents a solution to the Knights and Spies Problem: In a room there are n people, each labelled with a unique number between 1 and n. A person may either be a knight or a spy. Knights always tell the truth, while spies may either lie or tell the truth, as they see fit. Each person in the room knows the identity of everyone else. Apart from this, all that is known is that strictly more knights than spies are present. Asking only questions of the form: `Person i, what is the identity of person j?', what is the least number of questions that will guarantee to find the true identities of all n people? The analysis of a related two-player game is critical to the proof. Some probabilistic aspects are also explored. The paper ends by presenting three open questions concerned with generalisations of the problem.

math.CO↗
arXiv2008

Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.

math.GR↗
arXiv2007

On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.

math.GR↗
arXiv2006

Labelling the character tables of symmetric and alternating groups

Let $X$ be a character table of the symmetric group $S_n$. It is shown that unless $n = 4$ or $n=6$, there is a unique way to assign partitions of $n$ to the rows and columns of $X$ so that for all $λ$ and $ν$, $X_{λν}$ is equal to $χ^λ(ν)$, the value of the irreducible character of $S_n$ labelled by $λ$ on elements of cycle type $ν$. Analogous results are proved for alternating groups, and for the Brauer character tables of symmetric and alternating groups.

math.RT↗
arXiv2006

Character values and decomposition matrices of symmetric groups

By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions $λ$ and $μ$ are equal if and only if $λ$ and $μ$ are conjugate. An analogous result is proved for Hecke algebras. 2. A Specht module for the symmetric group $S_n$, defined over an algebraically closed field of odd characteristic, is decomposable on restriction to the alternating group $A_n$ if and only if it is simple, and the labelling partition is self-conjugate. This result is generalised to an arbitrary field of odd characteristic.

math.RT↗
arXiv2006

Counting Partitions on the Abacus

In 2003, Maroti showed that one could use the machinery of l-cores and l-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case l=2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss.

math.CO↗