# Algebra seminars, 2014-15

## In 'Past algebra seminars'

### Defining an affine partition algebra

### Spectral multipliers and group representations: analysis of the Kohn Laplacian on complex spheres

#### Alessio Martini, University of Birmingham

**Saturday 4 October 2014, 16:00-17:00 Physics West 117**

Abstract not available

### Schubert calculus and Gelfand-Zetlin polytopes

#### Evgeny Smirnov, Independent University of Moscow

**Thursday 16 October 2014, 16:00-17:00 Physics West 117**

We describe a new approach to the Schubert calculus on complete flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. We also prove a formula for all Demazure characters of a given representation of GL(*n*) via exponential sums over integral points in faces of the Gelfand-Zetlin polytope associated with the representation. This is a joint work with V. Kiritchenko and V. Timorin.

### Permutation puzzles and finite simple groups

#### Jason Semeraro, University of Bristol

**Thursday 23 October 2014, 16:00-17:00 Physics West 117**

The 15 puzzle is a sliding puzzle that consists of a frame of square tiles in random order with one tile missing (the hole), and where the aim is to obtain an ordered arrangement through an appropriate sequence of moves. The set of sequences of moves which leave the hole in a fixed position forms a finite group (the puzzle group) which is easily seen to be isomorphic to the alternating group Alt(15). Various generalisations of the 15-puzzle have already been studied. For example, Wilson considers an analogue for finite connected and non-separable graphs. More recently, Conway introduced a version of the puzzle which is played with counters on 12 of the 13 points in the finite projective plane P(3). The 13th point *h* (called the hole) may be interchanged with a counter on any other point *p*, provided the two counters on the unique line containing *h* and *p* are also interchanged. It turns out that the group of move sequences which fix the hole is isomorphic to the Mathieu group M_{12}. In this talk, we extend Conway's game to arbitrary simple 2-(*n*,4,λ)-designs with the property that any two lines intersect in at most two points. We obtain a plethora of examples of puzzle groups including the symplectic and orthogonal groups in characteristic 2. We completely classify puzzle groups when λ < 3, and show that there are finitely many isomorphism classes of puzzle groups for each λ > 0. We also apply Mihailescu's theorem (formally Catalan's conjecture) to give a new characterisation of M_{12}.

### Dual singularities in exceptional type nilpotent cones

#### Paul Levy, University of Lancaster

**Thursday 6 November 2014, 16:00-17:00 Physics West 117**

It is well-known that nilpotent orbits in sl_{n}(ℂ) correspond bijectively with the set of partitions of *n*, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type *A* there is an order-reversing involution on the poset of nilpotent orbits. More generally, if * g* is any simple Lie algebra over ℂ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in

*to the set of special nilpotent orbits in the Langlands dual Lie algebra*

**g**

**g**^{L}. It was observed by Kraft and Procesi that the duality in type

*A*is manifested in the geometry of the nullcone. In particular, if two orbits

*O*

_{1}<

*O*

_{2}are adjacent in the partial order then so are their duals

*O*

_{1}

^{t}>

*O*

_{2}

^{t}, and the isolated singularity attached to the pair (

*O*

_{1},

*O*

_{2}) is dual to the singularity attached to (

*O*

_{2}

^{t},

*O*

_{1}

^{t}): a Kleinian singularity of type A

_{k}is swapped with the minimal nilpotent orbit closure in sl

_{k+1}(and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits. In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs O

_{1}< O

_{2}of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.

### Fusion systems on groups with an abelian subgroup of index *p*

#### David Craven, University of Birmingham

**Thursday 13 November 2014, 16:00-17:00 Physics West 117**

The search for exotic fusion systems at odd primes has turned up maybe new systems with a variety of properties. In an attempt to bring order to at least some of the chaos, we classify all exotic fusion systems on *p*-groups with an abelian subgroup of index *p*, which encompasses the Ruiz–Viruel examples of fusion systems on the extraspecial group 7_{+}^{1+2} and the Clelland–Parker examples coming from GL_{2}(*p*). Along the way we meet questions about finite simple groups and their representation theory, and end up classifying indecomposable modules for almost simple groups with the property that the Sylow *p*-subgroup has order *p*, and the module *M* has only one non-trivial summand when restricted to the Sylow *p*-subgroup. The theory then essentially provides a dictionary between these and new fusion systems, leading to a wealth of new examples, including another infinite family and several sporadic examples.

### Simple groups which are almost of Lie type

#### Chris Parker, University of Birmingham

**Thursday 27 November 2014, 16:00-17:00 Physics West 117**

One of the projects to provide an alternative proof of the classification of the finite simple groups intends to understand those simple groups which have local, or even parabolic, characteristic *p*. I'll explain what these notions mean and then discuss recent results which can be used in the final recognition of such groups. Along the way, or rather towards the end, I'll explain how two of the configurations lead to a construction of an infinite family of exotic fusion systems.

### Block Decompositions of Categories of Modules Determined By Varieties

#### Jeremy Rickard, University of Bristol

**Thursday 11 December 2014, 16:00-17:00 Physics West 117**

If *G* is a finite group, and we are interested in representations of *G* over a field *k* of prime characteristic, then an extremely useful tool has been the maximal ideal spectrum of the cohomology algebra of *G* over *k*, which is an algebraic variety of which every *kG*-module determines a subvariety that gives useful information about the module. If we fix a subvariety *V* and consider the class of modules whose variety is contained in *V*, then they form a category with many nice properties, especially if we work 'stably'. After giving an example-based introduction to the theory, I'll talk about some more recent joint work with Jon Carlson about how these categories decompose into blocks, as well as connections with Linckelmann's 'block varieties'.

### Cohomology for finite and algebraic groups

#### David Stewart, University of Cambridge

**Thursday 15 January 2015, 14:00-15:00 Watson Building, Lecture Room A**

The representation theory for finite groups of Lie type in defining characteristic is rather complicated, but owing to the link with algebraic groups, there are quite a few tools. I want to give an overview of some of the theory and show how some of these can be used to bound the cohomology of finite groups with coefficients in simple modules, motivated by an old conjecture of Guralnick.

### Foulkes modules for the symmetric group

#### Melanie de Boeck, University of Kent

**Thursday 15 January 2015, 15:00-16:00 Watson Building, Lecture Room A**

The action of the symmetric group Sym(*mn*) on set partitions of sets of size *mn* into *n* sets of size *m* gives rise to a permutation module called the Foulkes module. Structurally, very little is known about Foulkes modules, even over ℂ. In this talk, we will introduce Foulkes modules and their twisted analogues before presenting some results which shed light on the irreducible constituents of the ordinary characters of twisted Foulkes modules.

### Local-global conjectures in the representation theory of finite groups

#### Gunter Malle, TU Kaiserslautern

**Thursday 15 January 2015, 16:30-17:30 Watson Building, Lecture Room A**

More than 60 years ago Richard Brauer developed the theory of representations of finite groups over arbitrary fields. It showed a strong connection between the representation theory of a finite group and that of its *p*-local subgroups, for *p* a prime. Many more such connections have been observed in the meantime, but most of these are still conjectural.

Recently, a new reduction approach has offered the hope to solve all of these fundamental conjectures by using the classification of finite simple groups. In our talk we will try and explain the nature of these problems and will report on recent progress which might eventually lead to a solution of these long standing fundamental questions.

### Graded RoCK blocks and wreath products

#### Anton Evseev, University of Birmingham

**Wednesday 28 January 2015, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Invariants of Specht modules

#### Haralampos Geranios, University of York

**Wednesday 4 February 2015, 16:00-17:00 Watson Building, Lecture Room A**

This talk is based on a joint work with S. Donkin. In a research paper of D. Hemmer the author conjectures that the module of fixed points for the symmetric group Sym(*m*) of a Specht module for Sym(*n*) (with *n* > *m*), over an algebraically closed filed of positive characteristic *p*, has a Specht series when viewed as a Sym(*n*−*m*)-module. In this talk we will provide a counterexample for each prime *p*.

### The graded representation theory of the symmetric group and dominated homomorphisms

#### Liron Speyer, Queen Mary, University of London

**Wednesday 25 February 2015, 16:00-17:00 Watson Building, Lecture Room A**

I will give a brief overview of the cyclotomic KLR algebra and its use in studying the (graded) representation theory of the symmetric group. I will then talk about dominated homomorphisms between Specht modules and give examples to illustrate why they are so useful. If there is time at the end, I will briefly discuss a generalisation of the results and their application to Ariki-Koike algebras. This is joint work with Matthew Fayers.

### Modularity and the Fermat Equation over Totally Real Fields

#### Samir Siksek, University of Warwick

**Wednesday 4 March 2015, 16:00-17:00 Watson Building, Lecture Room A**

Just over twenty years ago Wiles proved Fermat's Last Theorem, by proving modularity of semistable elliptic curves over the rationals. Thanks to the efforts of many mathematicians, it is now possible to extend the modularity statement to almost all elliptic curves over totally real fields. We discuss what this means in down-to-earth language, and also some implications for the Fermat equation. This talk is based on joint work with Nuno Freitas (Bonn) and Bao Le Hung (Harvard).

### A generalization of a theorem of Camina and Herzog

#### Ellen Henke, University of Aberdeen

**Wednesday 11 March 2015, 16:30-17:30 Watson Building, Lecture Room A**

An old result of Camina and Herzog states that a finite group G has abelian Sylow 2-subgroups, provided |*G* : C* _{G}*(

*x*)| is odd for any 2-element

*x*∈

*G*. In my talk, I will report about a generalization of this theorem to saturated fusion systems, which was conjectured by Kühlshammer, Navarro, Sambale and Tiep. This does not only lead to a significant simplification of the proof of the theorem of Camina and Herzog, but also implies a new result in block theory. In my talk I will explain the proof of the main result and give an introduction to the theory of fusion systems along the way. The talk will be accessible to PhD students.

### A Rational Hilbert-Mumford Theorem

#### Michael Bate, University of York

**Wednesday 25 March 2015, 16:00-17:00 Watson Building, Lecture Room A**

The classical Hilbert-Mumford Theorem gives a way of identifying closed orbits for algebraic groups acting on varieties (think groups of matrices acting on vector spaces and you won't go far wrong). Identifying the closed orbits is very important if you want to form a quotient, for example. The theory is highly developed and works very well over algebraically closed fields, but problems arise as soon as you move to other fields (even the transition from complex to real numbers throws up some difficulties). In this talk I'll describe a new approach to such problems which works over arbitrary fields. I'll illustrate some of the key features using simple examples which shouldn't need much more than a bit of linear algebra to understand.

### Braid group of ℤ^{n}

#### Daan Krammer, University of Warwick

**Thursday 14 May 2015, 16:00-17:00 Physics West 117**

I will talk on my 2007 paper 'The braid group of ℤ^{n}'. The motivation for defining this group is that it is a missing link between the usual braid group and mapping class groups of higher genus surfaces. Braid groups were shown to be Garside in 1969. I shall show that the braid group of ℤ^{n} is Garside. Work in progress, that I will not discuss in detail, is a Garside structure on the mapping class groups.

### Relative asphericity and proving groups infinite

#### Martin Edjvet, University of Nottingham

**Thursday 21 May 2015, 16:00-17:00 Physics West 117**

We will give an introduction to relatively aspherical group presentations and show how this can be used when trying to prove groups infinite. In particular we study the example of the Fibonacci groups.

### Injective Schur modules

#### Stephen Donkin, University of York

**Thursday 28 May 2015, 16:00-17:00 Physics West 117**

Abstract not available

### The Thompson chain of perfect groups

#### Robert Curtis, University of Birmingham

**Thursday 4 June 2015, 16:00-17:00 Physics West 117**

Abstract not available

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.