## Groups in Galway 2020 - Online edition

#### Abstracts

### Matteo Cavaleri: Gain graphs, group algebra valued matrices and Fourier transform

I will give an introduction to gain graphs. These are finite oriented graphs whose oriented edges are labelled by group elements. In this context, particular group algebra valued matrices play the same role of adjacency matrices associated with the simple graphs. Balance of the gain graphs can be detected by means of these special matrices or the spectra of their Fourier transforms.

This is joint work with D. D'Angeli and A. Donno.

### Joanna B. Fawcett: Tree-homogeneous graphs

Let $X$ be a class of graphs. A graph $\Gamma$ is *$X$-homogeneous* if for any graph isomorphism $\varphi:\Delta_1\to \Delta_2$
between finite induced subgraphs $\Delta_1$ and $\Delta_2$ of $\Gamma$ such that $\Delta_1$ is isomorphic to a graph in $X$, there exists an automorphism of $\Gamma$ that extends $\varphi$. For example, if $X$ consists of the graph with one vertex, then $X$-homogeneity is vertex-transitivity. A graph is \emph{tree-homogeneous} if it is $X$-homogeneous where $X$ is the class of trees. We will discuss some recent progress on classifying the finite tree-homogeneous graphs.

### Meinolf Geck: What is bad about bad primes? Some remarks about unipotent classes

Let $G$ be a simple algebraic group over an algebraic closure of the field with $p$ elements where $p$ is a prime. It is is known that the classification of unipotent classes is more complicated for "bad" ($=$certain small) primes than for larger primes. In an attempt to extend Kawanaka's construction of generalised Gelfand-Graev representations to the "bad" prime case, we proposed a new characterisation of Lusztig's concept of special unipotent classes of $G$ (which is now a theorem).

### Radhika Gupta: Uniform exponential growth for CAT(0) cube complexes

Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has uniform exponential growth or it is virtually abelian. The behavior, in this sense, of a group that acts by isometries on a higher dimensional CAT(0) cube complex is not known. In this talk, I will present some generalizations of their theorem. On the one hand we allow the action to be proper instead of free and on the other hand we assume our space has isolated flats. I will define exponential growth and also present the general strategy to obtain a result like that of Kar–Sageev. This is joint work with Kasia Jankiewicz and Thomas Ng.

### Joshua Maglione: Isomorphism via derivations

By bringing in tools from multilinear algebra, we introduce a general method to aid in the computation of group isomorphism. Of particular interest are nilpotent groups where the only classically known proper nontrivial characteristic subgroup is the derived subgroup. Through structural analysis of the biadditive commutator map, we leverage the representation theory of Lie algebras to prove efficiency for families of nilpotent groups. We report on joint work with Peter A. Brooksbank, and James B. Wilson.

### John Murray: Clifford theory of $2$-Brauer characters

Clifford's theorem explores the relationship between the representations of a group $G$ and those of a normal subgroup $N$, over an arbitrary field $F$. In particular it applies in the modular case, when the characteristic of $F$ is not $0$. Then the irreducible representations of $G$ are classified by its irreducible $p$-Brauer characters.

In this talk we discuss three new Clifford theoretic results which are peculiar to characteristic 2. We begin by showing that if $\theta$ is an irreducible $2$-Brauer character of $N$, then $G$ has a self-dual irreducible $2$-Brauer character over $\theta$ if and only if $\theta$ is $G$-conjugate to its dual $\overline\theta$.

Now suppose that $\theta$ is self-dual. Then it is a remarkable fact that $\theta$ has a unique self-dual extension to its stabilizer in $G$. So $G$ has a unique self-dual irreducible $2$-Brauer character $\mu$ such that $\theta$ occurs with odd multiplicity in $\mu{\downarrow_N}$, and this multiplicity is 1.

The uniqueness of $\mu$ still holds if $N$ is merely subnormal in $G$. We give an example of $N,\theta,\mu$, with $N$ subnormal in $G$ such that $\theta$ occurs with arbitrarily large odd multiplicity in $\mu{\downarrow_N}$.

Our final result relates to the geometric type of a self-dual irreducible $2$-Brauer character $\mu$ of $G$. Fong's Lemma asserts that $\mu$ is of symplectic type. However it is a delicate question to determine whether $\mu$ is of orthogonal type. Suppose that $\mu$ is not of orthogonal type. Then we show that $\mu{\downarrow_N}$ is a sum of distinct real-valued non-orthogonal irreducible $2$-Brauer character of $N$.

### Emily Norton: Some decomposition matrices of finite classical groups

One problem in modular representation theory is to compute the decomposition matrices of finite groups of Lie type. In this talk, we explain a solution to this problem for the $\Phi_{2n}$ block of $B_{2n}(\mathbb{F}_q)$. The computation provides the first known infinite family of decomposition matrices in type $B$ for blocks with non-cyclic defect group. A similar result should also hold in type $D$. This is work in progress with Olivier Dudas.

### Anitha Thillaisundaram: Groups acting on rooted trees of growing degrees

Over the past few decades, groups acting on rooted trees have attracted much attention, starting with the easily describable constructions, such as the Grigorchuk group and the Gupta-Sidki groups, in response to the Burnside problem. The most well-studied examples involve groups acting on regular rooted trees, such as the p-adic tree, for p a prime. In this talk, we consider groups acting on irregular trees, namely trees of growing degree, and we show how properties of such groups differ from those of groups acting on p-adic trees. This is joint work with Khalid Bou-Rabee and Rachel Skipper.