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Organizers |
Generalised dihedral quotients of finitely-presented groups
by
Matthew Auger
University of Auckland
A generalised dihedral group is a split extension of an abelian group A by the cyclic group C2 where the non-trivial element of C2 acts on A by inverting elements. I will discuss a method of determining which generalised dihedral groups are quotients of a given finitely-presented group.
Date received: October 30, 2008
On Generalised Inflations of Algebras
by
Graham Clarke
RMIT University
An easy way to enlarge a semigroup is to adjoin elements which mimic the behaviour of existing elements. This process creates an inflation of the semigroup, and associativity is preserved. More recently, Bob Monzo and I introduced the idea of a generalised inflation of a semigroup. Subsequently several papers have investigated this construction in various different contexts. I will discuss the different directions in which the research in this area has gone.
Date received: October 30, 2008
Recent progress in the study of regular maps on surfaces
by
Marston Conder
University of Auckland
A regular map is a 2-cell embedding of a connected graph (or multigraph) on a surface, such that the group of all its incidence-preserving automorphisms has a single orbit on flags (incident vertex-edge pairs). Following the computer-assisted determination of all regular and orientably-regular maps of characteristic -1 to -200 two years ago, a lot of new things have been discovered. We now have theorems about the genus spectra of such maps that are chiral, and such maps that have simple underlying graph. Also more is now known about regular Cayley maps (that is, orientably-regular maps whose underlying graph is a Cayley graph); for example, the curious result that a map for a cyclic group is reflexible if and only if it is anti-balanced. A detailed classification of all regular Cayley maps for cyclic groups is now in sight. I will report on many of these developments, some of which were obtained in joint work with Young Soo Kwon, Jozef Sirán and Tom Tucker.
Date received: October 30, 2008
Multiplicative structure in the centre of the Iwahori-Hecke algebra of type A.
by
Andrew Francis
University of Western Sydney
Coauthors: John Graham (Babcock & Brown), Weiqiang Wang (University of Virginia), Lenny Jones (Shippensburg University)
The connection between the centre of the Hecke algebra and the symmetric polynomials in Jucys-Murphy elements allows one to define an integral basis for the centre in terms of symmetric polynomials. This has a side effect of allowing one to show that the "diagonal" structure constants with respect to the minimal basis for the centre are independent of n. This in turn gives a filtration on the centre.
In general it would be nice if there were an integral basis for the centre that is multiplicative, but it turns out that this is not possible.
Date received: November 1, 2008
Fixed point free elements of prime order in primitive permutation groups
by
Michael Giudici
University of Western Australia
It is an easy consequence of the Orbit-Counting Lemma that every transitive permutation group of degree at least 2 has a fixed point free element. Using the Classification of Finite Simple Groups, Fein, Kantor and Schacher showed that there is actually one of prime power order. It has been proved that for primitive groups, except for a family involving M11 acting on 12 points, prime power order can be replaced with prime order. These two results do not give any information about which prime. This talk will discuss some recent work with Tim Burness which aims to provide such information for fixed point free elements of prime order in primitive groups.
Date received: October 29, 2008
Enhancing the nilpotent cone
by
Anthony Henderson
University of Sydney
Coauthors: Pramod N. Achar (Louisiana State University), Benjamin F. Jones (University of Georgia)
Many features of an algebraic group are controlled by the geometry of its nilpotent cone, which in the case of GLn(C) is merely the variety N of n×n nilpotent matrices. The study of the orbits of the group in its nilpotent cone leads to combinatorial data relating to the representations of the Weyl group, via the famous Springer correspondence. In the case of GLn(C), the basic manifestation of this correspondence is the fact that conjugacy classes of nilpotent matrices and irreducible representations of the symmetric group are both parametrized by partitions of n.
Pramod Achar and I have shown that studying the orbits of GLn(C) in the enhanced nilpotent cone Cn×N leads to exotic combinatorial data of type B/C (previously defined by Shoji under the name of "limit symbols"). This is closely related to Syu Kato's exotic Springer correspondence for the symplectic group.
I will review this story and report on more recent joint work with Achar and Ben Jones, in which we consider the question of whether the orbit closures in the enhanced nilpotent cone are normal varieties, as is known to be the case for the ordinary nilpotent cone N.
Date received: September 25, 2008
Equivalence classes of highly nonlinear functions between groups
by
Kathy Horadam
Mathematics, RMIT University
For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine (EA) equivalence and Carlet-Charpin-Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ-equivalent functions are EA-inequivalent because the equivalences are defined by different properties of functions.
I have recently solved the corresponding problem for functions between arbitrary finite groups, by relating graphs of functions to transversals and appealing to the theory of group extensions. There is a formula for all the functions in the generalised CCZ equivalence class of a given function, in terms directly comparable to the formula for all functions in its generalised EA equivalence class. As the EA classes are orbits under a group action, it is likely the former are too.
I will outline these results for the elementary abelian case, which is most useful for applications.
Date received: October 25, 2008
Monodromy groups of polytopes and self-invariance
by
Isabel Hubard
University of Auckland
Coauthors: Alen Orbanic and Asia Weiss
For every polytope P there is the universal regular polytope of the same rank as P corresponding to the Coxeter group C = [∞, . . . , ∞]. For a given automorphism d of C , using monodromy groups, we construct a combinatorial structure Pd. When Pd is a polytope isomorphic to P we say that P is self-invariant with respect to d, or d-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a polarity for self-dual polytopes.
Date received: October 30, 2008
Representation growth and alternating quotients
by
Ben Martin
University of Canterbury
Let G be a group. Given a positive integer n, we define rn(G) to be the number of isomorphism classes of complex n-dimensional representations of G (note that rn(G) can be infinite). We say that G has polynomial representation growth if there exist a, b ≥ 0 such that rn(G) ≤ anb for every n. In this talk I will discuss a question of Brent Everitt: does there exist a finitely generated group G such that
(1) G has polynomial representation growth; and
(2) G has the alternating group Am as a quotient for infinitely many m?
Date received: November 2, 2008
Blocks of generalized q-Schur algebras of type A
by
Andrew Mathas
University of Sydney
Coauthors: Marcos Soriano (Hannover)
Donkin introduced an analogue of the q-Schur algebras indexed by an arbitrary saturated set of weights. We classify the blocks of these algebras in type A. Quite surprisingly, these blocks are just the restrictions of the blocks of the corresponding (quantized) enveloping algebra. The proof is a slick combinatorial application of the Jantzen sum formula which gives new information even in the cases where this result was previously known.
Date received: November 6, 2008
Dualizing complexes via flat modules
by
Amnon Neeman
Australian National University
I will describe recent work, culminating in the PhD thesis of Daniel Murfet, giving a completely new approach to Grothendieck's dualizing complexes.
Date received: September 17, 2008
The Ore Conjecture
by
Eamonn O'Brien
University of Auckland
The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Recently, Liebeck, Shalev, Tiep and I used a combination of character-theoretic methods and computation to establish it. Here we summarise some of the key ideas.
Date received: October 29, 2008
Some results on the cohomology and representation theory of quantum groups
by
Brian Parshall
University of Virginia
Coauthors: L. Scott; C. Bendel, D. Nakano, C. Pillen
Let G be a semisimple, simply connected algebraic group over a field k of positive characteristic p. A family of rational representations of G can be constructed from the irreducible modules for the associated quantum enveloping algebra at a p-th root of unity. (They were first defined by Lusztig.) This talk will discuss some of the homological properties of these modules, applications, and open questions. We will also discuss recent results on the cohomology of quantum enveloping algebras at mth roots of unity for small values of m.
Date received: October 29, 2008
Beads on runners
by
Arun Ram
University of Melbourne
Coauthors: Alexander Kleshchev
Khovanov-Lauda algebras are a family of algebras whose representation theory provides a categorification of quantum groups. In this work we classify and construct homogeneous representations of these algebras. The construction generalises the construction of irreducible representations of the symmetric groups and the notions of partitions, skew shapes, and abaci.
Date received: September 22, 2008
Embeddings of finite unitary reflection groups
by
Don Taylor
University of Sydney
A unitary reflection is a linear transformation, of finite order, of a complex vector space, which fixes a hyperplane pointwise. The finite groups generated by unitary reflections were classified long ago: Mitchell (1914), Shephard and Todd (1954).
Given a reflection group one can ask for all subgroups generated by reflections and, more generally, for all subgroups that act as a reflection group on a subspace (as in recent work of Lehrer and Springer). In this talk I describe embeddings that are not covered by these methods. One consequence is that all `exceptional groups' of rank at least three are embedded in K6 or E8.
Date received: October 20, 2008
Projective modules over quantum symmetric algebras
by
Ruibin Zhang
University of Sydney
We study finitely generated projective modules for quantum analogues of symmetric algebras which arise from the theory of quantum groups. It is shown that finitely generated projective modules for quantum symmetric algebras over the function field C(q) are stably free. It is also shown that every finitely generated projective module for a quantum symmetric algebra over the power series ring C[[t]] is free. The latter case rather resembles the Quillen-Suslin theorem on Serre's problem for polynomial algebras.
Date received: November 9, 2008