# Algebra seminars, 2009-10

## In 'Algebra'

Back to 'Mathematics Research'### Finite W-algebras and primitive ideals

#### Jonathan Brown, University of Birmingham

**Thursday 1 October 2009, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### On the geometry of extremal elements in Lie algebras

#### Hans Cuypers, Technische Universiteit Eindhoven

**Tuesday 6 October 2009, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Median spaces, properties (T) and Haagerup, applications to mapping class groups

#### Cornelia Drutu, University of Oxford

**Thursday 15 October 2009, 16:00-17:00 Watson Building, Lecture Room A**

The median structure is a metric structure generalizing the one of tree. Both Kazhdan's property (T) and Haagerup's property can be formulated in terms of actions of groups on median spaces. Moreover, it turns out that every mapping class group of a surface has a natural equivariant asymptotic structure of median space. This allows to study homomorphisms into mapping class groups of surfaces, up to conjugation. The talk is on joint work with I. Chatterji and F. Haglund, and with J. Behrstock and M. Sapir.

### Groups and Surfaces

#### Gareth Jones, University of Southampton

**Thursday 22 October 2009, 16:00-17:00 Watson Building, Lecture Room A**

A map on a compact surface can be described as a transitive finite permutation representation of a triangle group. There is a natural complex structure on the underlying surface of the map, making it a Riemann surface, or equivalently a complex algebraic curve. Belyi's Theorem states that the algebraic curves arising from maps are those defined over algebraic number fields, giving a faithful action of the absolute Galois group (the Galois group of the field of algebraic numbers) on maps. This motivates efforts to classify maps, especially in the regular (most symmetric) case, and to understand the action of the absolute Galois group on maps. I shall illustrate this in the case of the Fermat curves and their generalisations, where one can apply work of Huppert, Ito and Wielandt on groups factorising as products of cyclic groups, and of Hall on solvable groups. If there is time I will mention recent work on Beauville surfaces: these are complex algebraic surfaces with certain rigidity properties, obtained from finite groups acting on pairs of regular maps; it is conjectured that every non-abelian finite simple group except *A*_{5} can be used here.

### Mickelsson algebras and irreducible representations of Yangians

#### Maxim Nazarov, University of York

**Thursday 5 November 2009, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Equivariant Riemann-Roch theorems for curves

#### Bernhard Koeck, University of Southampton

**Thursday 12 November 2009, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Rank Polynomials

#### Sinead Lyle, University of East Anglia

**Thursday 19 November 2009, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Catalan numbers for complex reflection groups

#### Iain Gordon, University of Edinburgh

**Thursday 3 December 2009, 16:00-17:00 Watson Building, Lecture Room A**

(Joint with Stephen Griffeth) I will explain a construction of *q*-Catalan numbers for all finite complex reflection groups. The formula is elementary, involving only some basic invariant theory of complex reflection groups, but the construction is interesting because it features the monodromy of the monodromy of the *KZ* connection, and hence unusual symmetries of the irreducible representations of complex reflection groups.

### The decomposition matrix of the Brauer algebra over the complex field

#### Paul Martin, University of Leeds

**Thursday 10 December 2009, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### A class of groups universal for free R-tree actions

#### Thomas Müller, Queen Mary, University of London

**Thursday 14 January 2010, 16:00-17:00 Watson Building, Lecture Room A**

I report on a new construction in group theory giving rise to a kind of continuous analogue of free groups. More explicitly, given any (discrete) group *G*, we construct a group *RF*(*G*) equipped with a natural (real-valued) Lyndon length function, and thus with a canonical action on an associated ℝ-tree X_{G}, which turns out to be transitive. Analysis of these groups *RF*(*G*) is difficult. However, conjugacy of hyperbolic elements is understood, as are the centralizers and normalizers of hyperbolic elements; moreover, we show that *RF*-groups and their associated ℝ-trees are universal (with respect to inclusion) for free ℝ-tree actions. Furthermore, we prove that |*RF*(*G*)| = |*G*|^{2ℵ0}, and that non-trivial normal subgroups of *RF*(*G*) contain a free subgroup of rank |*RF*(*G*)|, as well as a number of further structural properties of *RF*(*G*) and its quotient by the span of the elliptic elements.

### What is a *q*-reflection group?

#### Yuri Bazlov, University of Bath

**Thursday 21 January 2010, 16:00-17:00 Watson Building, Lecture Room A**

If *V* is a vector space over a field *k*, a reflection of *V* is a linear transformation of *V* which fixes a hyperplane in *V* pointwise. A finite group generated by reflections of *V* is termed a reflection group. If *k*=ℚ, ℝ or ℂ, reflection groups are the same as Weyl groups, Coxeter groups or complex reflection groups, respectively, and are very important in Lie theory. We will look at two key properties of a reflection group *G* over a field of characteristic 0: # *G*-invariants in the polynomial ring *k*[*V*] are themselves a polynomial ring (the Chevalley-Shephard-Todd theorem); # there is a special family of commuting differential-difference operators on *k*[*V*], called Dunkl operators (they give rise to a rational Cherednik algebra of *G*). It is interesting to note that the Chevalley-Shephard-Todd theorem (1), which dates back to 1950s, was nontrivially extended to fields of positive characteristic in a more recent work of Serre and Kemper-Malle. In my talk, however, I will remain in characteristic 0 but will be interested in a generalisation of the C-S-T theorem where the polynomial ring is replaced with a noncommutative algebra. I will aim to describe results of myself and Berenstein (inspired by the theory of integrable systems) which lead to a *q*-commuting version of Dunkl operators and Cherednik algebras (2). They are attached to a family of finite groups which are no longer reflection groups; remarkably, the *q*-commutative Chevalley-Shephard-Todd theorem for our groups was proved almost immediately in an independent work by Zhang and collaborators. I will describe this family of finite groups and, if time permits, will mention possible further generalisations.

### Rational Cherednik algebras, Hilbert schemes of points and quantum Hamiltonian reduction

#### Toby Stafford, University of Manchester

**Thursday 28 January 2010, 16:00-17:00 Watson Building, Lecture Room A**

Type *A* Cherednik algebras *H _{c}*, which are particular deformations of the twisted group ring of the

*n*-th Weyl algebra by the symmetric group

*S*, form an intriguing class of algebras with many interactions with other areas of mathematics. A few years ago Gordon and I proved a sort of Beilinson-Bernstein equivalence of categories, thereby showing that

_{n}*H*(or more formally its spherical subalgebra

_{c}*U*) is a non-commutative deformation of the Hilbert scheme Hilb(

_{c}*n*) of

*n*points in the plane. This has significant applications to the representation theory of

*U*and

_{c}*H*. More recently the three authors have shown how to relate this to the notion of quantum Hamiltonian reduction due to Gan and Ginzburg and this again has significant applications to the structure of

_{c}*U*-modules and their associated varieties.

_{c}### Irreducible characters of Sylow p-subgroups of D_4(p^k)

#### Tung Le, University of Aberdeen

**Thursday 4 February 2010, 16:00-17:00 Watson Building, Lecture Room A**

Let *U*(*q*) be a Sylow *p*-subgroup of a Chevalley group *D*_{4}(*q*), where *q* is a power of a prime *p*. We describe a construction of all complex irreducible characters of *U*(*q*) and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in *q*−1 with nonnegative coefficients.

### Cuspidal representations for rational Cherednik algebras

#### Gwyn Bellamy, University of Edinburgh

**Thursday 18 February 2010, 16:00-17:00 Watson Building, Lecture Room A**

In this talk I'll introduce the rational Cherednik algebra and describe its rich, combinatorial representation theory 'at *t*=0'. In particular, I'll show that much of the representation theory is governed by certain cuspidal representations.

### Majorana representations of groups

#### Alexander Ivanov, Imperial College, London

**Thursday 25 February 2010, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Elementary constructions of Suzuki and Ree groups

#### Rob Wilson, Queen Mary, University of London

**Thursday 4 March 2010, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Torsion Units in Integral Group Rings of Sporadic Simple Groups

#### Alexander Konovalov, University of St Andrews

**Thursday 11 March 2010, 16:00-17:00 Watson Building, Lecture Room A**

(joint work with Victor Bovdi, Eric Jespers, Steve Linton, Salvatore Siciliano et al.) Let U(ℤ*G*) be the unit group of the integral group ring ℤ*G* of a finite group *G*. A unit of ℤ*G* of the form *a*_{1}*g*_{1}+*a*_{2}*g*_{2}+···+*a*_{n}*g*_{n} with *a*_{i} in ℤ and *g _{i}* in

*G*is called normalised, if

*a*

_{1}+

*a*

_{2}+···+

*a*= 1. Normalised units form a subgroup of U(ℤ

_{n}*G*) called the normalized unit group of ℤ

*G*and denoted by V(ℤ

*G*). The long-standing conjecture of H.Zassenhaus (ZC) says that every torsion unit from V(ℤ

*G*) is conjugate within the rational group algebra ℚ

*G*to an element of

*G*. One of its weakened variations can be formulated in terms of the Gruenberg-Kegel graph (also called the prime graph) of an arbitrary group

*X*, which has vertices labeled by primes

*p*for which there exists an element of order

*p*in

*X*and edges between distinct primes

*p*and

*q*if and only if

*X*has an element of order

*pq*. Clearly, if (ZC) holds for a finite group

*G*, then

*G*and V(ℤ

*G*) have the same prime graph. The criterion for ZC can be formulated in terms of vanishing of partial augmentations of torsion units (for an element of a group ring of the form

*a*

_{1}

*g*

_{1}+

*a*

_{2}

*g*

_{2}+···+

*a*

_{n}

*g*

_{n}with

*a*in ℤ and

_{i}*g*in

_{i}*G*, its partial augmentation with respect to the conjugacy class

*C*of elements of the group

*G*is the sum of coefficients

*a*over those

_{i}*g*which belong to the class

_{i}*C*). Therefore, it is useful to know for each possible order of a torsion unit in V(ℤ

*G*), which combinations of partial augmentations may arise. This motivated us to start the project to collect information about possible partial augmentations of torsion units of integral group rings of sporadic simple groups. As a consequence, at the time of this talk we proved that

*G*and V(ℤ

*G*) have the same prime graph for the following thirteen sporadic simple groups: * Mathieu groups M

_{11}, M

_{12}, M

_{22}, M

_{23}, M

_{24}; * Janko groups J

_{1}, J

_{2}, J

_{3}; * Held, Higman-Sims, McLaughlin, Rudvalis and Suzuki groups. In my talk I will summarise known information about orders and partial augmentations for these groups, explain enhancements of the Luthar-Passi method that were developed during the project, and highlight some challenges arising from the remaining sporadic simple groups.

### Symmetric generation of the Rudvalis group

#### Rob Curtis, University of Birmingham

**Thursday 25 March 2010, 16:00-17:00 Watson Building, Lecture Room A**

A free product of *n* copies of the cyclic group of order *m*, which we denote by *m*^{*n}, possesses many *monomial* automorphisms which permute the *n* generators and raise them to powers prime to *m*. Indeed the group of all such automorphisms is soon seen to have order φ(*m*)^{n}*n*!, where φ(*m*) denotes the number of natural numbers less than *m* and co-prime to it. If *N* is a subgroup of this group which acts transitively on the *n* cyclic subgroups then a semi-direct product of form *P* = *m*^{*n}:*N* is called a *progenitor*. Symmetric generation of groups is concerned with finding interesting finite images of such progenitors. It turns out that the case *m*=2 is particularly fruitful and many *symmetric presentations* of sporadic simple groups have been found needing just one short additional relation.Several of these will be mentioned briefly by way of motivation, but the the body of the talk will be devoted to the Rudvalis simple group Ru which proved surprisingly resistant to this approach. In the work we shall describe we take *N*=L_{4}(2) acting naturally on a 4-dimensional vector space over ℤ_{2}. It turns out that if we apply a simple lemma, which restricts the form of relators by which it is sensible to factor *P*, we are led directly to defining relators for Ru.

### Additive group invariants in positive characteristic

#### Emilie Dufresne, Universität Heidelberg

**Friday 9 April 2010, 16:00-17:00 Watson Building, Lecture Room A**

Additive group actions in positive characteristic Abstract: (joint work with Andreas Maurischat) Roberts, Freudenburg, and Daigle and Freudenburg have given the smallest counterexamples to Hilbert's fourteenth problem. Each arises as the ring of invariants of an additive group action on a polynomial ring over a field of characteristic zero, and thus, each corresponds to the kernel of a locally nilpotent derivation. In positive characteristic, additive group actions correspond to locally finite iterative higher derivations, a more restrictive notion. We set up characteristic-free analogs of the three examples mentioned above, and show that, contrary to characteristic zero, in every positive characteristic, the invariant rings are finitely generated.

### Metric dimensions of graphs arising from partial cubes

#### Sandi Klavzar, University of Ljubljana

**Tuesday 13 April 2010, 16:00-17:00 Watson Building, Lecture Room A**

Graphs isometrically embeddable into hypercubes are called partial cubes. Then naturally lead to the isometric dimension, the lattice dimension, and the Fibonacci dimension of a graph. These dimensions are defined as the smallest integer *d* such that a given graph admits an isometric embedding into the *d*-dimensional hypercube, the *d*-dimensional integer lattice, and the *d*-dimensional Fibonacci cube, respectively. In each of the three cases partial cubes are precisely the graphs having finite dimension. In the talk classical results about partial cubes as well as results on the recently introduced Fibonacci dimension (due to the speaker, Sergio Cabello and David Eppstein) will be presented.

### Periodic automorphisms of simple Lie algebras

#### Paul Levy, University of Lancaster

**Thursday 15 April 2010, 16:00-17:00 Watson Building, Lecture Room A**

Let θ be an automorphism of order m of the complex simple algebraic group *G*, and let *g* be the Lie algebra of G. Then there is a direct sum decomposition *g* = *g*(0)+*g*(1)+···+ *g*(*m*-1), where g(*j*) is the e^{2πij/m}-eigenspace for the action of the differential *d*θ on *g*. In fact, this is a ℤ/*m*ℤ-grading: [*g*(*i*),*g*(*j*)]< *g*(*i*+*j*) (where *i* and *j* should be considered as integers modulo *m*). Let *G*(0) be the connected component of the fixed point subgroup for the action of θ on *G*; then *G*(0) is reductive, Lie(*G*(0)) = *g*(0) and *G*(0) stabilizes each of the subspaces *g*(*i*). The first example to consider is the case *m*=2, that is, where θ is an involution. In this case *G*(0) is commonly denoted *K*, *g*(0) = *k* and *g*(1) = *p*. Then it is well known that the action of *K* on *p* shares many invariant-theoretic properties with the adjoint representation: closed orbits are orbits of semisimple elements, the invariants are polynomial, and so on. It is rather less well-known that (due to the seminal work of Vinberg) most of these properties also hold for the action of *G*(0) on *g*(1) for arbitrary *m*. In this talk I will give an overview of Vinberg's results, explain how they can be extended to positive characteristic and discuss a long-standing conjecture of Popov concerning the existence of an analogue of Kostant's slice to the regular orbits in the adjoint representation.

### Kac-Moody groups

#### Inna Capdeboscq, University of Warwick

**Monday 26 April 2010, 11:00-12:00 Watson Building, Lecture Room B**

Abstract not available

### The Diameter of the Monster Graph

#### Peter Rowley, University of Manchester

**Monday 26 April 2010, 12:00-13:00 Watson Building, Lecture Room B**

Abstract not available

*p*-Groups with Maximal Elementary Abelian *p*-Subgroups of Rank 2

#### Nadia Mazza, University of Lancaster

**Monday 26 April 2010, 15:00-16:00 Watson Building, Lecture Room B**

Abstract not available

### Characteristic Subgroups for Pushing Up

#### George Glauberman, University of Chicago

**Monday 26 April 2010, 16:00-17:00 Watson Building, Lecture Room B**

Abstract not available

### Mapping Class Groups and Complexes of Curves on Orientable/Nonorientable Surfaces

#### Elmas Irmak, Bowling Green State University

**Tuesday 18 May 2010, 16:00-17:00 Watson Building, Lecture Room A**

I will talk about the relation between the mapping class groups of surfaces and the automorphism groups and the superinjective simplicial maps of the complexes of curves on surfaces for both orientable and nonorientable surfaces. I will also talk about the proof that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface if *g*+*n* is not 4, where *g* is the genus and *n* is the number of boundary components of the surface.

### Group actions on curves and their Weierstrass points

#### Helmut Völklein, Universität Duisburg-Essen

**Thursday 10 June 2010, 16:00-17:00 Watson Building, Lecture Room A**

Abstract not available

### Minimax Sets in Some Classical Groups of Dimension 3

#### Philip Keen, University of Birmingham

**Thursday 17 June 2010, 16:00-17:00 Watson Building, Lecture Room A**

A set *S* in a group *G* is said to be *independent* if no element of *S* can be written as a word in the other elements of *S*. A set *S* in a group *G* is said to be a *minimax set* if it is an independent generating set of largest size in *G*. My current research seeks to find good upper bounds for sizes of minimax sets in the groups SL_{3}(*q*), SO_{3}(*q*) and SU_{3}(*q*). I will present some of the background for the research, some of the tools used, and some of the results already obtained.

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.