Majorana theory is an axiomatic framework to study objects related to the Monster group and its 196884 dimensional representation, the Griess algebra. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right, or as Majorana representations of certain groups. Inspired by a paper of A. Seress, and joint with M. Pfeiffer, I have developed an algorithm in GAP to construct the Majorana representations of a given group. I will present the methods of this work as well as some consequences and results. I will also discuss future theoretical and computational approaches to the study of Majorana and axial algebras.
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Talks 2018
<< 201711 entries found
- July
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12. July17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraMadeleine Whybrow, TU KL: Constructing Majorana representations
- June
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21. June17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraAlex Malcolm, University of Bristol: Strong reality in finite simple groups: on products of classes and characters
In a finite simple group, the strongly real elements are those that are conjugate to their inverse by an involution. A related notion exists in the ordinary representation theory of the group, where we call an irreducible character orthogonal if it is not only real-valued, but has an underlying real representation. It was recently shown that any element of a finite simple group decomposes as a product of two strongly real elements, motivating the question as to whether an analogous result holds for irreducible characters? I.e. does every irreducible representation appear in the tensor product of two real reps? We will discuss the methods used in answering both of these questions, as they range from the analysis of finite simple group sub-structure, to a classical problem of computing Kronecker coefficients in the symmetric group.
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14. June17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraIulian Simion, Babes-Bolyai University, Cluj: Minimal realizations of finite groups of isometries
Let G be a finite group. The minimal dimension in which G can be realized by isometries is known to be the dimension of a minimal faithful real representation. The minimal number of points which realize the group as their full group of isometries appears to be less well-understood. We discuss some known bounds (this is work in progress).
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07. June17:30 - 18:30
Location: 48-436AG Algebra, Geometrie und Computer AlgebraKivanc Ersoy, Freie Universität Berlin: Groups with certain conditions on fixed points of automorphisms
Thompson proved that a finite group with a fixed point free automorphism of prime order is nilpotent. Later Higman, Kreknin and Kostrikin proved that in this case the nilpotency class is indeed bounded by the order of the automorphism. On the other hand, Brauer and Fowler proved that order of a finite simple group is indeed bounded in terms of the order of centralizers of involutions. Since then, there has been a lot of questions of the following type: Let G be a group with a particular type of centralizer (of fixed point set). How does the structure of the centralizer affect the structure of the group? In the first part of the talk we will give a survey of results obtained by imposing certain conditions on the centralizers. In the second part we will prove new results about finite and locally finite groups and their automorphisms.
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07. June16:30 - 17:30
Location: 48-436AG Algebra, Geometrie und Computer AlgebraIpek Tuvay, Mimar Sinan Fine Arts University, Istanbul: Brauer indecomposability of Scott modules for the quadratic group Qd(p
Brauer indecomposability of Scott modules is important for constructing stable equivalences of Morita type between the principal blocks of two different finite groups. For an algebraically closed field k of prime characteristic p, we compute the Scott kG-module with vertex P when F is a constrained fusion system on P and G is Park’s group for F. In the case F is a fusion system of the quadratic group on a Sylow p-subgroup P of Qd(p) and G is Park’s group for F, we prove that the Scott kG-module with vertex P is Brauer indecomposable. In this talk these results will be presented. This is a joint work with Shigeo Koshitani.
- May
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24. May17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraMelissa Lee, Imperial College London: Bases of almost quasisimple groups and Pyber's conjecture
A base of a permutation group G acting on Ωis a subset of Ωwhose pointwise stabiliser in G is trivial. Bases have their origins in computational group theory, where they were used to efficiently store permutation groups of large degree into a small amount of computer memory. The minimal base size of G is denoted by b(G). When b(G)=1, we say that G has a regular orbit on Ω.
A well-known conjecture made by Pyber in 1993 states that there is an absolute constant c such that if G acts primitively on Ω, then b(G) < c log |G| / log n, where Ω= n. Pyber's conjecture was established in 2016 by Duyan, Halasi and Maroti, following on from contributions from a variety of authors.
In this talk, I will cover some of the history and uses of bases, and discuss Pyber's conjecture, as well as present some results on the determination of the constant c for bases of almost quasisimple linear groups. I will also outline some recent work on determining which irreducible modules of linear groups contain a regular orbit. -
17. May17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraZhicheng Feng, TU Kaiserslautern: On the inductive blockwise Alperin weight condition for special linear and unitary groups
The inductive blockwise Alperin weight condition is a system of conditions whose verification for all non-abelian finite simple groups would imply the blockwise Alperin weight conjecture. This talk will explain how one can obtain the weights of a special linear or unitary group X from the general linear or unitary group and verify the inductive blockwise Alperin weight condition for any unipotent block of X when the prime does not divide the order of Z(X).
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07. May17:00 - 18:00
Location: 48-519AG Algebra, Geometrie und Computer AlgebraThomas Gobet, University of Sydney: Dual and classical generators of Artin groups of spherical type
The n-strand braid group has two Garside structures, a classical and a dual one, leading to different solutions to the word problem in the group. The corresponding Garside monoids are the classical braid monoid and the Birman-Ko-Lee braid monoid. These monoids exist for any Artin group of spherical type (i.e., attached to a finite Coxeter group).
The aim of the talk is to explain how to express the generators of an Artin group of spherical type coming from the dual braid monoid (and more generally the simples, that is, the divisors of the Garside element of the dual monoid) in terms of the classical generators: we give a closed formula to express a simple of the dual braid monoid, using combinatorial objects called c-sortable elements (introduced by Reading). It gives a word of shortest possible length in the classical generators representing a simple element of the dual braid monoid. It has as an immediate consequence that the simples of the dual braid monoids are Mikado braids. -
03. May17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraKeivan Mallahi-Karai, Jacobs University, Bremen: Kirillov's orbit method and the polynomiality of the essential dimension of p-g
The faithful dimension of a finite group G is defined to be the smallest dimension of a faithful complex representation of G. Aside from its intrinsic interest, the problem of determining the faithful dimension of finite groups is intimately related to the notion of essential dimension, introduced by Buhler and Reichstein.
The problem of determining the faithful dimension of families of p-groups arising from F_p-points of a nilpotent algebraic group defined over the field of rational numbers has been studied in some special cases, e.g. the Heisenberg and the full upper-triangular unipotent group.
In this paper, we will use Kirillov’s orbit method to address this problem for a large family of groups. It will be shown that this function is always a piecewise polynomial function along certain “number-theoretically defined” sets, while, in some specific cases, it is given by a uniform polynomial in p.
This talk is based on a joint work with Mohammad Bardestani and Hadi Salmasian. - January
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25. January17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraPetra Schwer, KIT Karlsruhe: Reflection length in affine Coxeter groups
Affine Coxeter groups have a natural presentation as reflection groups on some affine space. Hence the set R of all its reflections, that is all conjugates of its standard generators, is a natural (infinite) set of generators. Computing the reflection length of an element in an affine Coxeter group means that one wants to determine the length of a minimal presentation of this element with respect to R. In joint work with Joel Brewster Lewis, Jon McCammond and T. Kyle Petersen we were able to provide a simple formula that computes the reflection length of any element in any affine Coxeter group. In this talk I would like to explain this formula, give its simple uniform proof and allude to the geometric intuition behind it.
