# Algebra seminars, 2017-18

## In 'Past algebra seminars'

### Semiprimitive groups

#### Michael Giudici, University of Western Australia

**Wednesday 27 September 2017, 16:00-17:00 Physics West 117**

A transitive permutation group is called semiprimitive if every normal subgroup is either transitive or semiregular. This class of groups includes all primitive, quasiprimitive and innately transitive groups. They have received recent attention due to a conjecture of Potočnik, Spiga and Verret that generalises the Weiss conjecture for locally primitive graphs. In this talk I will discuss recent work with Luke Morgan that develops a general theory for the structure of semiprimitive groups and give some applications of the new theory.

### Generating graphs

#### Ben Fairbairn, Birkbeck, University of London

**Wednesday 4 October 2017, 16:00-17:00 Watson Building, Lecture Room C**

Let *G* be a 2-generated group. Recall that the generating graph Γ(*G*) is defined as follows: the vertices are the non-trivial elements of *G* with two vertices adjoined by an edge if the corresponding elements generate the group. This has been much studied in the case of *G* a finite simple group. In this talk we discuss a closely related graph introduced by the speaker.

### Homomorphic encryption and black box algebra

#### Alexandre Borovik, University of Manchester

**Wednesday 11 October 2017, 16:00-17:00 Watson Building, Lecture Room A**

I will discuss some recent results in probabilistic recognition of black box groups in the wider context of algebraic cryptography. (Joint work Sukru Yalcinkaya)

### Complete reducibility in exceptional algebraic groups: the good, the bad and the ugly

#### Adam Thomas, University of Bristol

**Wednesday 1 November 2017, 16:00-17:00 Watson Building, Lecture Room C**

The notion of complete reducibility was introduced by J. P. Serre in 1998. It generalises the notion of a completely reducible module in classical representation theory. After giving an introduction to complete reducibility for algebraic groups, we discuss its impact on questions about the subgroup structure of exceptional algebraic groups. This includes recent and ongoing work with A. Litterick on classifying the subgroups of exceptional algebraic groups that are not completely reducible. The techniques used are a mix of standard representation theory, non-abelian cohomology and computational group theory.

### Two problems involving the divisor functions

#### Julio Bueno de Andrade, University of Exeter

**Thursday 22 February 2018, 17:00-18:00 Watson Building, Lecture Room A**

In this talk, I will discuss two problems involving the divisor functions. The first problem is about the auto-correlation of the values of the divisors functions and I will describe how we can completely solve the problem in the function field setting using a combination of analytic and algebraic techniques. The second problem is about the maxima pairwise of the divisors functions and I will present some recent results that improve some old results of Erdos and Hall.

### On decomposition numbers and bad primes

#### Alessandro Paolini, TU Kaiserslautern

**Thursday 1 March 2018, 17:00-18:00 Watson Building, Lecture Room A**

Let *G* be a finite group of Lie type defined over the field with *q* elements, with *q* = *p*^{f}, and let *U* be a Sylow *p*-subgroup of *G*. The problem of determining the (shape of the) *l*-modular decomposition matrices of *G* when *l* ≠ *p* is more difficult in the case where *p* is a bad prime for *G*. For instance, if *p* is good then one is allowed to use the theory of GGGRs, which is often crucial to obtain the unitriangularity of such matrices. The degrees of the irreducible characters of *U* are all powers of *q* when *p* is a good prime for *G* and rk(*G*) ≤ 6. On the other hand, if *p* is a bad prime for *G* then one always finds irreducible characters of *U* of degree of the form *q*^{n}/*p*. The goal of this talk is to explain why such characters seem to play a major role towards the determination of the *l*-decomposition numbers of *G* when *p* is a bad prime, and how they have already been used to obtain new results on the *l*-modular decomposition matrices of SO^{+}(8,*p*^{2f}).

### Weight conjectures for fusion systems

#### Jason Semeraro, University of Leicester

**Wednesday 9 May 2018, 16:00-17:00 Lecture room B, Watson building**

We will explain a result which shows that Alperin's weight conjecture implies an equality between two natural numbers, each of which is a function of a pair (ℱ,α) where ℱ is a saturated fusion system and α is a compatible family of Kulshämmer-Puig classes in the system. The statement does not make reference to blocks, but if (ℱ,α) comes from a block, the natural numbers in question are supposed to equal the number of ordinary irreducible characters in the block. The result leads naturally to a variety of other counting conjectures for fusion systems.

### The mod 2 homology of the simplex and representations of symmetric groups

#### Mark Wildon, Royal Holloway, University of London

**Thursday 17 May 2018, 16:00-17:00 Watson Building, Lecture Room A**

Abstract: The *k*-faces of an (*n*-1)-dimensional simplex correspond to *k*-subsets of {1,...,*n*}. These subsets are permuted transitively by the symmetric group S_n. The boundary maps from simplicial homology, defined with mod 2 coefficients, give homomorphisms between the corresponding permutation modules. In recent work I consider the generalized boundary maps, defined by jumping down by two or more dimensions at once. These give 'higher' homology groups, affording a family of intriguing representations of *S _{n}*. In my talk, I will characterize when the homology is zero. The special case of two-step boundary maps gives a new construction of the basic spin representations of the symmetric groups. We will see that the corresponding chain complex categorifies the binomial coefficient identity binom{4

*m*}{0} - binom{4

*m*}{2} + binom{4

*m*}{4} - \cdots + binom{4

*m*}{4

*m*} = (-2)

^{m}. I will end with some much deeper identities that, conjecturally, are categorified by an extension of these results to odd characteristic.

### Uniform Domination for Simple Groups

#### Scott Harper, University of Bristol

**Wednesday 30 May 2018, 16:00-17:00 Watson Building, Lecture Room A**

It is well known that every finite simple group can be generated by just two elements. In fact, by a theorem of Guralnick and Kantor, there is a conjugacy class *C* such that for each non-identity element *x* there exists an element *y* in *C* such that *x* and *y* generate the entire group. Motivated by this, we introduce a new invariant for finite groups: the uniform domination number. This is the minimal size of a subset *S* of conjugate elements such that for each non-identity element *x* there exists an element *s* in *S* such that *x* and *s* generate the group. This invariant arises naturally in the study of generating graphs. In this talk, I will present recent joint work with Tim Burness, which establishes best possible results on the uniform domination number for finite simple groups, using a mix of probabilistic and computational methods together with recent results on the base sizes of primitive permutation groups.

### Fact Checking the Atlases of Finite Groups

#### John Cannon, University of Sydney

**Monday 6 August 2018, 14:00-15:00 TBA**

The talk will review recent developments in Magma in the areas of finite group structure and representation theory. This would be done in the context of their application to building confidence in published tables of properties of finite simple groups.

We address the challenges facing society and the economy, from shedding light on the refugee crisis, to character education in schools, through to developing leaders in the NHS.