# Algebra seminars, 2019-20

## In 'Past algebra seminars'

### A counterexample to a conjecture of Steinberg

#### Mikko Korhonen

**Tuesday 8 October 2019, 17:00-18:00 PHYW-SR2 (106)**

Let *G* be a semisimple algebraic group over an algebraically closed field *K*. At the 1966 ICM in Moscow, Robert Steinberg conjectured that two elements of *G* are conjugate if and only if their images are conjugate under every rational irreducible representation of *G*. The conjecture was proven by Steinberg in the case where *K* has characteristic zero, and also in the case where the two elements are semisimple. In this talk, I will present a counterexample which was discovered by computational methods.

### The maximal dimensions of simple modules over restricted Lie algebras

#### Lewis Topley, University of Birmingham

**Tuesday 15 October 2019, 16:00-17:00 Watson Building, Lecture Room C**

Restricted Lie algebras were introduced by Jacobson in the 1940s and ever since the first investigations into their representation theory, it has been understood that the simple modules of a given such algebra have bounded dimensions. In 1971 Kac and Weisfeiler made a striking conjecture (KW1) giving a precise formula for the maximal dimension *M*() of a restricted Lie algebra . In this talk I will give a general overview of this theory, and then I will describe a joint work with Ben Martin and David Stewart in which we apply the Leftschetz principle, along with classical techniques from Lie theory, to prove the KW1 conjecture for all restricted Lie subalgebras of the general linear algebra _{n}, provided the characteristic of the field is large compared to *n*.

### Embedding of PSL_{2}(*q*) in exceptional groups of Lie type

#### Andrea Pachera, Birmingham

**Tuesday 22 October 2019, 16:00-17:00 Watson Building, Lecture Room C**

We describe a technique used to construct groups of type PSL_{2}(*q*) inside the finite exceptional group *G*(*p*), and count the number of conjugacy classes. In some cases this gives information about the embedding in *G*(ℂ), which can be used to say something about the embedding in *G*(*k*) for a field *k* of certain characteristic. An example where this fails is Alt(6) ≅ PSL_{2}(9); we briefly explain how it can be studied using the Dickson form on the *E*_{6} natural module. This work is used to make progress in the classification of the maximal subgroups of exceptional groups of Lie type.

### Classifying 2-blocks with an elementary abelian defect group

#### Cesare Ardito, University of Manchester

**Tuesday 29 October 2019, 16:00-17:00 Watson Building, Lecture Room C**

Donovan’s conjecture predicts that given a *p*-group *D* there are only ﬁnitely many Morita equivalence classes of blocks of group algebras with defect group *D*. While the conjecture is still open for a generic *p*-group *D*, it was proved in 2014 by Eaton, Kessar, Külshammer and Sambale when *D* is an elementary abelian 2-group, and in 2018 by Eaton and Livesey when *D* is any abelian 2-group. The proof, however, does not describe these equivalence classes explicitly. A classiﬁcation up to Morita equivalence over a complete discrete valuation ring has been achieved for *D* with rank 3 or less, and for *D* = (*C*_{2})^{4}. I have done (*C*_{2})^{5} and, I have partial results on (*C*_{2})^{6}. I will introduce the topic, give the relevant deﬁnitions and then describe the process of classifying these blocks, with a particular focus on the individual tools needed to achieve a complete classiﬁcation.

### Higher Modular Representations of Lie Algebras

#### Matthew Westaway, University of Warwick

**Tuesday 5 November 2019, 16:00-17:00 Watson Building, Lecture Room C**

In positive characteristic, a representation of an algebraic group *G* gives rise to a restricted representation of Lie(*G*). These restricted representations are more connected to the representation theory of the first Frobenius kernel of *G* - a certain infinitesimal group scheme - than that of *G* itself. In fact, one can form a family of associative algebras by deforming the distribution algebra of the first Frobenius kernel in some way, and the irreducible representation theory of these algebras totally captures the irreducible (not necessarily restricted) representations of Lie(*G*). It is this theory which will be explained during this talk, as well as a consideration of how the theory changes when we instead look at higher Frobenius kernels.

### The structure of axial algebras

#### Justin McInroy, University of Bristol

**Tuesday 12 November 2019, 16:00-17:00 Watson Building, Lecture Room C**

Axial algebras are a new class of non-associative algebra, introduced recently by Hall, Rehren and Shpectorov, which generalise some properties found in vertex operator algebras and the Griess algebra. Axial algebras are generated by axes which are idempotents which decompose the algebra as a direct sum of eigenspaces. The multiplication of eigenvectors is controlled by a so-called fusion law. When this is graded, it leads naturally to a subgroup of automorphisms of the algebra called the Miyamoto group. The prototypical example is the Griess algebra which has the Monster simple sporadic group as its Miyamoto group. We will discuss some recent developments about the structure of such algebras: their ideals, sum decomposition and an alternating bilinear form.

### Maximal subgroups of Grigorchuk-Gupta-Sidki (GGS-)groups

#### Anitha Thillaisundaram

**Tuesday 19 November 2019, 16:00-17:00 Watson Building, Lecture Room C**

The GGS-groups were some of the early positive answers to the famous Burnside problem. These groups act on infinite rooted trees and are easy to describe, plus possess interesting properties. A natural aspect of these groups to study is their maximal subgroups, and in particular, whether these groups have maximal subgroups of infinite index. It was proved by Pervova in 2005 that the torsion GGS-groups do not have maximal subgroups of infinite index. In this talk, I will consider the remaining non-torsion GGS-groups. This is joint work with Dominik Francoeur.

### The compressed word problem in groups

#### Derek Holt, University of Warwick

**Tuesday 26 November 2019, 16:00-17:00 Watson Building, Lecture Room C**

Let *G* be a finitely generated group. The word problem WP(*G*) of *G* is the problem of deciding whether a given word w over the elements of some finite generating set and (their inverses) of *G* represents the identity element of *G*. It was proved in the 1950s that there are groups with unsolvable word problem, but for groups with solvable word problem, it is interesting both in theory and in practice to study the time and space complexity of WP(*G*) as a function of the length of the input word w. Many of the groups that arise in geometric group theory, including nilpotent groups, hyperbolic groups, Coxeter groups, Artin groups, braid groups, and mapping class groups are known to have word problem solvable in low degree polynomial time (usually linear or quadratic). For the compressed word problem CWP(*G*), words are input in compressed form as straight line programs or, equivalently, as context-free grammars. For some words, such as powers of generators, the uncompressed word can be exponentially longer than its compressed version. So the complexity of CWP(*G*) could conceivably be exponentially greater than WP(*G*). As motivation for studying this question, we observe that it is can be shown that the ordinary word problem in finitely generated subgroups of Aut(*G*) is polynomial time reducible to CWP(*G*). It turns out that for some classes of (finitely generated) groups in which WP(*G*) is solvable in polynomial time, including nilpotent groups, Coxeter groups, and right-angled Artin groups, CWP(*G*) is also solvable in polynomial time, whereas for others, such as the wreath product of a nonabelian finite group by an infinite cyclic group, CWP(*G*) has been proved to be NP-hard. In this talk, we will discuss a (fairly) recent result of Lohrey and Schleimer that, for hyperbolic groups *G*, CWP(*G*) is solvable in polynomial time, and speculate on whether this result can be extended to include groups that are hyperbolic relative to a collection of abelian subgroups.

### Linear characters of Sylow subgroups of the symmetric group

#### Stacey Law, University of Oxford

**Tuesday 3 December 2019, 16:00-17:00 Watson Building, Lecture Room C**

Let *p* be an odd prime and *n* a natural number. We determine the irreducible constituents of the permutation module induced by the action of the symmetric group *S _{n}* on the cosets of a Sylow

*p*-subgroup

*P*. In the course of this work, we also prove a symmetric group analogue of a well-known result of Navarro for

_{n}*p*-solvable groups on a conjugacy action of

*N*(

_{G}*P*). Before describing some consequences of these results, we will give an overview of the background and recent related results in the area.

### Bases for primitive permutation groups

#### Melissa Lee, Imperial College, London

**Thursday 16 January 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

Let *G* ≤ Sym(Ω) be a primitive permutation group. A base for *G* is a subset *B* ⊆ Ω such that the pointwise stabiliser *G _{B}* = 1. In this talk, after outlining the history and uses of bases, I will describe some recent work towards two prominent problems in the area - namely the solution to Pyber's conjecture and the classification of primitive groups with base size two.

### Isomorphic subgroups of finite solvable groups

#### George Glauberman, University of Chicago

**Tuesday 21 January 2020, 16:00-17:00 Arts Lecture Room 3**

In 2014, Moshe Newman asked the following question: If two subgroups of a finite solvable group are isomorphic and one is a maximal proper subgroup of *G*, must the other also be a maximal proper subgroup of *G*? This question is still open. I plan to discuss recent results with Geoffrey Robinson that give some sufficient conditions for an affirmative answer.

### Constructing the automorphism group of a finite group

#### Eamonn O'Brien, University of Auckland

**Thursday 6 February 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

Constructing the automorphism group of a finite group remains challenging. The critical hard case is that of a finite *p*-group *P* where much effort has been invested over the past 20 years in developing recursive algorithms which work down a central series for *P*. If we can locate characteristic structure in *P*, then we can often readily solve the problem. The real challenge remains class 2 *p*-groups of exponent *p*. In this lecture we will outline algorithmic approaches and report on recent joint work which offer new hope of progress on this intractable problem.

### Groups of card shuffles

#### Luke Morgan, University of Primorska

**Thursday 13 February 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

There are two standard ways to shuffle a deck of cards, the in and out shuffles. For the in shuffle, divide the deck into two piles, hold one pile in each hand and then perfectly interlace the piles, with the top card from the left hand pile being on top of the resulting stack of cards. For the out shuffle, the top card from the right hand pile ends up on top of the resulting stack. Standard card tricks are based on knowing what permutations of the deck of cards may be achieved just by performing the in and out shuffles. Mathematicians answer this question by solving the problem of what permutation group is generated by these two shuffles. Diaconis, Graham and Kantor were the first to solve this problem in full generality - for decks of size 2*n*. The answer is usually 'as big as possible', but with some rather beautiful and surprising exceptions. In this talk, I'll explain how the number of permutations is limited, and give some hints about how to obtain different permutations of the deck. I’ll also present a more general question about a 'many handed dealer' who shuffles kn cards divided into k piles.

### Diagonal structures and primitive permutation groups

#### Cheryl Praeger, University of Western Australia

**Thursday 20 February 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

Many maximal subgroups of finite symmetric groups arise as stabilisers of some structure on the point set: for example the maximal intransitive permutation groups are subset stabilisers. The primitive groups of diagonal type for a long time have seemed exceptional in this respect. Csaba Schneider and I have introduced diagonal structures which, for the first time, give a combinatorial interpretation to these primitive groups of simple diagonal type. In further work together also with Peter Cameron and Rosemary Bailey, we’ve exhibited these groups as automorphism groups of 'diagonal graphs'.

### The general Sakuma Theorem

#### Sergey Shpectorov, University of Birmingham

**Thursday 27 February 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

The original Sakuma Theorem classifies vertex operator algebras (VOAs) generated by two Ising vectors. The properties it relies on were turned by Ivanov into the axioms of a new class of non-associative algebras called Majorana algebras and thus axial algebras were born. The first axial version of the Sakuma theorem (for Majorana algebras) was published by Ivanov, Pasechnik, Seress and Shpectorov in 2010 and it was followed in 2015 by a more general version due to Hall, Rehren and Shpectorov, where many Majorana-specific assumptions were removed. The same year, Rehren attempted an even more general version, completely parting with Majorana restrictions and allowing arbitrary parameters a and b in the fusion rules to substitute the Majorana-specific values *a*=1/4 and *b*=1/32. He did not manage to obtain a complete classification, but he did show that the dimension of a 2-generated algebra is bounded by eight except when *a*=2*b* or *a*=4*b*. In a joint project with Franchi and Mainardis, we reprove and generalise Rehren's theorem to cover also the exceptional cases. In the case *a*=2*b*, we obtain the same bound, 8, on the dimension of a 2-generated algebra, although for a different spanning set. Even more interesting is the other exceptional case, where *a*=4*b*. Here we also have the upper bound eight, except when *a*=2 and *b*=1/2. In this final case, we found an unexpected example of an infinite-dimensional 2-generated algebra.

### Invariable generation of finite classical groups

#### Eilidh McKemmie, University of Southern California

**Thursday 5 March 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

We say a group is invariably generated by a subset if it forms a generating set even if an adversary is allowed to replace any elements with their conjugates. Eberhard, Ford and Green built upon the work of many others and showed that, as *n* tends to infinity, the probability that *S _{n}* is invariably generated by a random set of elements is bounded away from zero if there are four random elements, but goes to zero if we pick three random elements. This result gives rise to a nice Monte Carlo algorithm for computing Galois groups of polynomials. We will extend this result for

*S*to the finite classical groups using the correspondence between classes of maximal tori of classical groups and conjugacy classes of their Weyl groups.

_{n}### Examples of Degenerations of Frobenius Algebras

#### Daniel Kaplan, University of Birmingham

**Thursday 12 March 2020, 16:00-17:00 Lecture Theatre B, Watson Building**

Deformation theory of algebras was initiated by Gerstenhaber following Hochschild and his ideas are now ubiquitous across mathematics. Unfortunately, the subject can be uninviting -- often cited for its most formidable results: Kontsevich formality, Belinson-Bernstein localization, and Drinfeld and Jumbo's pioneering work on quantum groups. In this talk, I'll give an elementary introduction to deformation theory of algebras through examples. Then I'll give a crash course on Frobenius algebras and observe that: (1) a Frobenius algebra always deforms to a Frobenius algebra but (2) a Frobenius algebra can degenerate to a non-Frobenius algebra. Finally, I'll show that every Frobenius algebra can be realized as a deformation of a "most degenerate" class of Frobenius algebras

### Saturated fusion systems on a Sylow *p*-subgroup of a rank 2 Lie type group

#### Martin Van Beek, University of Birmingham

**Thursday 26 March 2020, 16:00-17:00 Zoom**

Saturated fusion systems are a means to capture and abstract the conjugacy present in Sylow p-subgroups of finite groups. After a gentle introduction to the theory of fusion systems and their applications, we will provide a general methodology for treating fusion systems on Sylow 2-subgroups of rank 2 groups of Lie type. As an application, we classify all saturated fusion systems on a Sylow *p*-subgroup of *G*_{2}(*p ^{n}*) and

*U*

_{4}(

*p*) for all primes

^{n}*p*.

### Representations of algebraic groups and their Lie algebras

#### Simon Goodwin, University of Birmingham

**Thursday 2 April 2020, 16:00-17:00 Zoom**

Starting with representation theory of symmetric groups, I'll aim to give an overview of some key ideas used in representation theory. We'll mainly consider representations of general linear groups and Lie algebras, and plan to finish with recent results on representations of reduced enveloping algebras in my joint work with Thomas and Topley.

### Twists in representation theory

#### Yuri Bazlov, University of Manchester

**Thursday 9 April 2020, 16:00-17:00 Zoom**

Very roughly, quantisation in algebra is about taking a commutative structure and using ideas and constructions inspired by physics make it noncommutative; then finding out what does not work and what does (and what works even better than in the commutative case). I will talk about one type of quantisation known as cocycle twisting, when the product on an associative algebra *A* is replaced by a new, "deformed" product. I am interested in how the representation theory of this deformed version of *A* differs from that of *A*. Cocycles which can be used for twisting arise from equations which are very difficult to solve, but in my talk I will focus on a simple case when the cocycle arises from an action on *A* by a finite non-cyclic abelian group - something referred to as discrete torsion by Vafa and Witten. I will show how such discrete torsion can deform a Coxeter group, leading to "mystic reflection groups" which have well-behaved invariants when they act on polynomials in skew-commuting variables. If time permits, I will mention recent work of Berenstein, Jones-Healey, McGaw and myself on representations on twisted Cherednik algebras.

### Unitriangularity of Decomposition Matrices of Unipotent Blocks

#### Jay Taylor, University of Southern California

**Wednesday 22 April 2020, 16:00-17:00 Zoom**

One of the distinguished features of the representation theory of finite groups is the ability to take a representation in characteristic zero and reduce it to obtain a representation over a fixed field of positive characteristic (a modular representation). If one starts with a representation that is irreducible in characteristic zero then its modular reduction can fail to be irreducible. The decomposition matrix encodes the multiplicities of the modular irreducible representations in this reduction. In this talk I will present recent joint work with Olivier Brunat and Olivier Dudas establishing a fundamental property of the decomposition matrix for finite reductive groups, namely that it has a unitriangular shape. The solution to this problem involves the interplay between Lusztig’s geometric theory of character sheaves and a family of representations whose construction was originally proposed by Kawanaka.

### Fusion systems on maximal class *p*-groups

#### Chris Parker, University of Birmingham

**Thursday 7 May 2020, 16:00-17:00 Zoom**

### An algorithm to construct 2-generated axial algebras

#### Madeleine Whybrow, University of Primorska

**Thursday 14 May 2020, 16:00-17:00 Zoom**

Axial algebras are non-associative algebras generated by semisimple idempotents, called axes, that obey a fixed fusion law. Important examples of axial algebras include the Griess algebra and Jordan algebras. Axial algebras that are generated by two axes are crucial in the study of these algebras in general. We present an algorithm to classify and construct 2-generated axial algebras.

### The maximal subgroups of *E*_{7}(*q*)

#### David Craven, University of Birmingham

**Thursday 21 May 2020, 16:00-17:00 Zoom**

This talk will survey the work that goes into the (nearly complete) list of maximal subgroups of *E*_{7}(*q*). We will start with some general information about simple groups, their maximal subgroups, then some preliminary facts about *E*_{7}(*q*) and its algebraic counterpart. After describing what is previously known about its maximal subgroups, we will then move on to new work of Alex Ryba and myself, which at the present time classifies all maximal subgroups save PSL_{2}(*q*) for *q*=7,8,9,11,13.

### Generic Computations with Reductive Groups and Finite Groups of Lie Type

#### Frank Lübeck (Aachen)

**Thursday 28 May 2020, 16:00-17:00 Zoom**

I will introduce some basic ideas how to encode reductive algebraic groups and finite groups of Lie type in a data structure which makes them accessible to "generic" computations (e.g., of subgroups, conjugacy classes, representations, ...). "Generic" means, that we are interested in algorithms which work for all such groups and which are as much as possible independent of the characteristic or field defining the group.

### Universal quantizations of nilpotent Slodowy slices

#### Lewis Topley

**Thursday 4 June 2020, 16:00-17:00 Zoom**

Finite dimensional complex simple Lie algebras are classified by Dynkin diagrams. The simply laced diagrams also classify the finite subgroups of SL_{2} which, in turn, give rise to a nice family of singular algebraic varieties: the Kleinian singularities. A conjecture of Grothendieck, proven by Brieskorn, states that the Kleinian singularity of Dynkin type *X* can be constructed as the subregular transverse slice to the nilpotent cone in the corresponding Lie algebra. In fact the transverse slice gives rise to a versal deformation of the Kleinian singularity. At the same time the transverse slices are quantized by finite *W*-algebras. Recently, in a joint work with Ambrosio, Carnovale, Esposito, we have proven a quantum analogue of Brieskorn's theorem, proving a universal property for the finite *W*-algebra. In this talk I will give a gentle introduction to the theory and explain our main result and applications.

### Groups acting with low fixity

#### Rebecca Waldecker

**Thursday 11 June 2020, 16:00-17:00 Zoom**

Abstract not available

### Extremely primitive groups

#### Adam Thomas, University of Warwick

**Thursday 18 June 2020, 16:00-17:00 Zoom**

Let *G* be a finite primitive permutation group acting on a set *X* with nontrivial point stabiliser *G _{x}*. We say that

*G*is extremely primitive if

*G*acts primitively on every orbit in

_{x}*X*\ {

*x*}. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. After surveying previous results we will discuss recent joint work with Tim Burness completing this classification. Most of the work is in dealing with almost simple groups with socle an exceptional group of Lie type. We will describe the various techniques used in the proof and in particular, discuss the results we proved on bases for primitive actions of exceptional groups.

### On ﬁnite groups as products of normal subsets

#### Attila Maroti, Alfréd Rényi Institute of Mathematics

**Thursday 25 June 2020, 16:00-17:00 Zoom**

Let *G* be a non-abelian ﬁnite simple group. A famous result of Liebeck and Shalev is that there is an absolute positive constant *c* such that whenever *S* is a non-trivial normal subset in *G* then *S ^{k}* =

*G*for any integer

*k*at least

*c*·(log |

*G*|/ log |

*S*|). In the ﬁrst part of the talk this result will be generalised by showing that there exists an absolute positive constant

*c*such that whenever

*S*

_{1}, ...,

*S*are normal subsets in

_{k}*G*with Π

_{i=1}

^{k}|

*S*| ≥ |

_{i}*G*|

*c*then S

_{1}...S

*=*

_{k}*G*. This is joint work with Laci Pyber. An ingredient of the proof is that there exists a constant δ with 0 < δ < 1 such that if

*S*

_{1}, ...,

*S*

_{8}are normal subsets in the alternating group

*A*each of size at least |

_{n}*A*|δ, then

_{n}*S*

_{1}...

*S*

_{8}=

*A*. In the second part of the talk we will discuss how the 8 in the previous statement can be reduced to 4. This is joint work with Martino Garonzi.

_{n}### The highwater algebra

#### Sergey Shpectorov, University of Birmingham

**Thursday 16 July 2020, 16:00-17:00 Zoom**

Abstract: By a result of Rehren, for general values of α and β, a 2-generated primitive algebra of Monster type (α,β) is of dimension at most 8. In this talk, I will present an exceptional example of an infinite-dimensional 2-generated primitive algebra of Monster type (2,1/2). The algebra has an elementary definition and its properties can be developed from scratch.

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