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 *d*^{2}, 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 *d*^{2} 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 *F*_{τ} where τ is the root of the polynomial equation 1 = *a*_{1}τ + *a*_{2}τ^{2}. 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 *F*_{τ}. Showing that every element of *F*_{τ} can be represented as a tree pair is non-trivial and requires that β = 1/τ is a Pisot number.