RHUL Mathematics Seminar

Autumn 2020

October 14 Stacey Law (Cambridge)
Sylow branching coefficients for symmetric groups (abstract)
October 14 James McKee (Royal Holloway)
The set of Cassels heights (abstract)
November 4 Gene Kopp (Bristol)
Complex equiangular lines and the Stark conjectures (abstract)
November 18 Kamilla Rekvenyi (Imperial College)
The orbital diameter of primitive permutation groups (abstract)
December 2 Nick Winstone (Royal Holloway)
Irrational variants of Thompson's Group F (abstract)

Stacey Law: Sylow branching coefficients for symmetric groups
The relationship between the representation theory of a finite group and that of its Sylow subgroups is a key area of interest. For example, recent results of MalleNavarro and Navarro–Tiep–Vallejo have shown that important structural properties of a finite group G are controlled by the permutation character. We introduce so-called Sylow branching coefficients to describe multiplicities associated with these induced characters, and discuss some properties and applications in the case where G is a symmetric group.

James McKee: The set of Cassels heights
The Cassels height function is used in the study of cyclotomic integers. Cassels introduce it in 1969 when settling a conjecture of Robinson. The set of all Cassels heights has a curious structure, which we explore. This is joint work with Byeong-Kweon Oh (Seoul) and Chris Smyth (Edinburgh).

Gene Kopp: Complex equiangular lines and the Stark conjectures
In this talk, I'll draw a connection between a complex geometry problem of interest in quantum information theory and a number-theoretic conjecture about special values of L-functions and class field theory. The geometry problem is: How may lines can you draw through the origin in d-dimensional complex space so that all the Hermitian angles are the same? The upper bound and expected answer is d2, but a proof is not known. The number-theoretic conjecture is the real quadratic case of the Stark conjectures, which predicts that certain derivative L-values are logarithms of algebraic "Stark units" in a ray class field. I give a conjectural construction of d2 equiangular lines in dimension d in terms of Stark units and discuss some partial results in support of the conjecture.

Kamilla Rekvenyi: The orbital diameter of primitive permutation groups
Let G be a group acting transitively on a finite setΩ. Then G acts on Ω × Ω componentwise. Define the orbitals to be the orbits of G on Ω × Ω. The diagonal orbital is the orbital of the form Δ = {(α, α) | α ∈ Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital.Define an orbital graphto be the non-directed graph with vertex setand edge set (α, β) ∈ Γ with α, β ∈ Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected.The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In this talk I will outline some important background information and the progress made towards finding specific bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of diagonal type and their connection to the covering number of finite simple groups.

Nick Winstone: Irrational variants of Thompson's Group F
Thompson's group F is the set of all piecewise-linear maps of the unit interval with breakpoints in the dyadic rationals, and whose slopes have gradient which is a power of 2. Variants of F have been around for decades, and the majority of work completed on them has been accomplished through the use of tree pairs. We define irrational variants Fτ where τ is the root of the polynomial equation 1 = a1τ + a2τ2. We are able to subdivide the unit interval into powers of τ using this equation. Representing these subdivisions as trees with lopsided carets we can take pairs of these trees to create elements of Fτ. Showing that every element of Fτ can be represented as a tree pair is non-trivial and requires that β = 1/τ is a Pisot number.

Last modified: 19/01/20. Email: mark.wildon@rhul.ac.uk