Title: All Kronecker coefficients are reduced Kronecker coefficients

Abstract: We settle the question of where exactly the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of a question by Stanley from 2000 and a question by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. This is joint work with Greta Panova, arXiv:2305.03003.

Title: Sylow branching coefficients for symmetric groups

Abstract: One of the key questions in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of G to a Sylow subgroup P of G, and have been recently shown to characterise structural properties such as the normality of P in G. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.

Title: Nice and Nasty Numerical Invariants

Abstract: For a permutation group G we can define the maximal irredundant base size and the relational complexity, denoted I(G) and RC(G) respectively. Roughly speaking the maximal irredundant base size is the size of the “worst” base for G, and relational complexity is a measure of when a local property extends to a global one.

We begin by defining these numerical invariants and then cover some examples which illustrate the “nice” behaviour of I(G) and the “nasty” behaviour of RC(G). We’ll then skim through the proof of the relational complexity of a family of groups.

Title: Self-extensions for irreducible representations of symmetric groups

Abstract: It has been conjectured that irreducible representations of symmetric groups have no non-trivial self-extensions in characteristic different from 2, that is that the only modules V with 2 composition factors isomorphic to D for some irreducible module D and no other composition factor are those of the form D + D. This conjecture has been proved for some classes of modules by Kleshchev-Sheth and Kleshchev-Nakano. I will present joint results with Harry Geranios and Sasha Kleshchev and current work with Harry Geranios considering reduction results and generalisations of the above mentioned papers.

Title: Exotic Fusion Systems Related to Sporadic Simple Groups

Abstract: Fusion systems offer a way examine and express properties of the p-conjugacy of elements in finite groups. However, not every fusion system may be constructed from a finite group in an appropriate way. This gives rise to exotic fusion systems. An important research direction involves the study of the behaviour of exotic fusion systems (in particular at odd primes).

In this talk, we describe several exotic fusion systems related to the sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow 3-subgroups of the Conway group Co₁ and the Thompson group F₃, and a Sylow 5-subgroup of the Monster M, as well as a particular maximal subgroup of the latter two p-groups. This work is supported by computations in MAGMA.

Title: Irredundant bases for the symmetric and alternating groups

Abstract: An irredundant base of a group G acting faithfully on a finite set Γ is a sequence of points in Γ that produces a strictly descending chain of pointwise stabiliser subgroups in G, terminating at the trivial subgroup. I will give an overview of known results about the irredundant base size, before focusing on the case where G is the symmetric or alternating group of degree n with a non-standard primitive action. It was proved in 2011 that an irredundant base of size 2 exists for such an action in all but finitely many cases. I will speak about the recent work by me and my supervisor, where we have shown that the maximum size of an irredundant base for the action is O(√n) and in most cases O((log n)^2). These upper bounds are also best possible in their respective cases, and I will present some interesting examples constructed to prove their optimality.

Title: Composition multiplicities of Verma modules for truncated current Lie algebras

Abstract: The problem of computing the composition multiplicities of Verma modules for a semisimple Lie algebra was famously solved by the proof of the Kazhdan-Lusztig conjecture, which gives the multiplicities in terms of certain polynomials known as the Kazhdan-Lusztig polynomials. In this talk, I will discuss this problem for a related class of Lie algebras, known as truncated current Lie algebras. I will also discuss the BGG category O of modules for a semisimple Lie algebra and an analogue of this category for truncated current Lie algebras.

Title: Growth in simple algebraic groups via unipotent elements

Abstract: Let G be a simple algebraic group over an algebraically closed
field and let A be a generating subset of G. We are interested in the
smallest integer m such that G=A^m. We discuss some aspects of this
problem with emphasis on unipotent elements. In particular we discuss
the case where A is a unipotent conjugacy class and recent contributions
to bounds on covering numbers.