# Seminars (GRAW02)

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Event When Speaker Title Presentation Material
GRAW02 27th January 2020
10:10 to 11:00
Meinolf Geck Computing Green functions in small characteristics
Green functions for finite groups of Lie type were introduced by Deligne and Lusztig in the 1970's, using cohomological methods. The computation of these functions is a crucial step in the more general programme of determining the whole character tables of those groups. Despite of a long tradition of work on Green functions, there are still open cases for groups in small characteristic. We report on some recent progress, which essentially relies on a combination of Lusztig's theory of character sheaves and computer algebra methods.
GRAW02 27th January 2020
11:30 to 12:20
David Craven Constructing subgroups of exceptional algebraic groups
When trying to understand the subgroup structure of the exceptional algebraic groups and groups of Lie type, one often needs to explicitly construct simple subgroups of large finite groups. In this talk, we will discuss some relatively simple, yet powerful, ideas and algorithms that can decide the existence and number of classes of various maximal subgroups of the larger exceptional groups (of types E and F). We will apply these methods to construct some sporadic and cross-characteristic Lie type maximal subgroups of E7 and E8.

This forms part of the speaker's programme to complete the classification of maximal subgroups.
GRAW02 27th January 2020
13:45 to 14:35
Heiko Dietrich Constructive recognition of matrix groups
After decades of work by many people, the successes of Matrix Group Recognition Projects and their implementations in the CompositionTree package allow, for the first time, to compute efficiently with large matrix groups (defined over finite fields). A crucial ingredient in those algorithms is the constructive recognition of classical groups. I will survey some of those results and then comment in more detail on my work (with Eamonn O'Brien and Charles Leedham-Green) for finding standard generators in classical groups.
GRAW02 27th January 2020
14:45 to 15:35
Joanna Fawcett Base sizes of permutation groups
A base of a permutation group G acting on a set X is a subset of X whose pointwise stabiliser in G is trivial. The base size of G is the minimal cardinality of a base for G. Bases have proved to be very useful, both theoretically (in bounding the order of a primitive permutation group in terms of its degree) and computationally (in many algorithms for permutation groups). Recently, much progress has been made on understanding the base sizes of primitive permutation groups. This talk will survey some of these results.
GRAW02 27th January 2020
16:05 to 16:35
Melissa Lee Base sizes of permutation groups: an encore
We describe some recent work towards classifying primitive groups of affine type with base size 2, specifically those with almost quasisimple point stabilisers.
GRAW02 27th January 2020
16:40 to 17:30
Thomas Breuer Connections between Group related Databases in GAP

Various databases of groups and related structures (representations, character tables, tables of marks, presentations, ...) are available in electronic form. Since it is often useful to combine information from different such databases, it is valuable to connect/integrate them tightly. The talk will report about some aspects of this integration in the computer algebra system GAP.

GRAW02 28th January 2020
09:10 to 10:00
Nicolas Thiery Musing on implementing semigroup representation theory and software integration
Extending representation theory from finite groups to finite semigroups brings interesting challenges, combinatorics, and applications. Almost a decade ago, I proposed an algorithm to compute the Cartan Matrix of a semigroup algebra -- a combinatorial invariant that contains information on how projective modules are built from simple modules. It boils down to computing with finite semigroups, characters of groups, and combinatorics. Despite this relative simplicity, and much to my embarrassment, a full production-grade implementation is only finally in reach.

In this talk, I will report on ongoing joint work with my PhD student Balthazar Charles to implement this algorithm and adapt it to modular representations, and use this occasion to illustrate the evolution of our computational landscape toward an ecosystem of interoperable software, thanks to large scale collaborations.
GRAW02 28th January 2020
10:10 to 11:00
Frank Lübeck Computing Brauer character tables of groups of Lie type in defining characteristic
I will sketch how to compute Brauer character values of a group of Lie type in its defining characteristic. The method uses weight multiplicities of irreducible representation of the underlying algebraic group, parameterizations of semisimple conjugacy classes of the finite groups, and ad hoc arguments to relate the resulting table with the ordinary character table of the finite group.
GRAW02 28th January 2020
11:30 to 12:20
Madeleine Whybrow An algorithm to construct dihedral axial algebras
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 called dihedral and are fundamental in the study of these algebras in general. We present an algorithm to classify and construct dihedral axial algebras. This work represents a significant broadening in our understanding of axial algebras.
GRAW02 28th January 2020
13:45 to 14:35
Klaus Lux The 5-modular Character Table of the Lyons Group
We will talk about the determination of the 5-modular character table of the sporadic simple Lyons group Ly. This table was computed jointly with Alexander Ryba, Queens College, CUNY, New York. As a starting point of our computations we will take the 111-dimensional representation over the field with 5 elements, conjectured to exist by Meyer, Neutsch and constructed by Meyer, Neutsch, and Parker. An important ingredient in our computations will be the interplay between modular character theory and the theory of condensation of representations, in particular condensations of tensor products and symmetrizations.
GRAW02 28th January 2020
14:45 to 15:35
David Stewart Between the sheets: rigid nilpotent elements in modular Lie algebras
(Joint with Sasha Premet) Let G be a reductive algebraic group over an algebraically closed field. Lusztig and Spaltenstein provided a method for inducing a nilpotent orbit from a Levi subgroup to the group G. Any orbit not obtained from a proper Levi subgroup is called rigid. These were classified by Kempken (for G classical) and Elashvili (for G exceptional). The latter was double-checked computationally by De Graaf. It turns out that this classification remains valid in characteristic p. I will explain the proof of this, obtained by extending the Borho-Kraft description of the sheets of the Lie algebra to positive characteristic and supported by a few computer calculations.
GRAW02 28th January 2020
16:05 to 16:55
Richard Parker 10 years of meataxe development.

Myself, Steve Linton and Jon Thackray have been working for nearly 10 years on a fairly major overhaul of matrix multiplication and Gaussian elimination over finite fields of order (mainly) up to 1,000 or so, aiming to make good use of modern processors - specifically the ubiquitous x86-64 from Intel and AMD. With clock speeds approaching a plateau we now need to use multiple cores, utilize the various levels of cache to reduce memory bandwidth demands, use the vector registers and avoid unpredictable branches, but by doing all of these, speed improvements in excess of a factor of 100 are readily obtained over the methods of a couple of decades ago.
This talk will explain some of the changes in technique that are needed to achieve this - both algorithmic and technological - that seem quite radical at the moment, but which I expect to become more mainstream in future.

GRAW02 29th January 2020
09:10 to 10:00
Adam Thomas Computational methods for exceptional groups
This talk will contain no new algorithms or methods for computation in groups. Instead, I will describe my (ab)using of existing algorithms (in Magma) to help study the structure of exceptional algebraic groups, their Lie algebras and the exceptional finite groups of Lie type. Such computational methods have guided me and often formed part of the argument in lots of my work, including joint projects with Tim Burness, Alastair Litterick and David Stewart.

These applications have included representation theory, especially calculating cohomology groups, intersection of subgroups and construction of subgroups. The talk will include many open-ended (and impossible-to-answer) questions about the finite groups of Lie type machinery and related algorithms, which will hopefully be helpful to other users in the audience.

GRAW02 29th January 2020
10:10 to 11:00
Allan Steel Constructing Ordinary Representations of Finite Groups via Extension
I will describe practical methods for the construction of ordinary representations of a finite group G via the extension of existing representations defined on a proper subgroup of G. I will also describe how, using an implementation of this algorithm within Magma, I was able to construct for the first time minimal-degree faithful ordinary representations of most of the large sporadic simple groups, such as the Baby Monster.
GRAW02 29th January 2020
11:30 to 12:20
Willem de Graaf Real forms of complex embeddings of maximal reductive Lie algebras in semisimple Lie algebras
Since the work of Dynkin the reductive subalgebras of a semisimple complex Lie algebra are divided in two groups: those that are contained in a proper
regular subalgebra, and those that are not (these are called S-subalgebras). I will describe computational methods to obtain real forms of the complex
embeddings of reductive Lie algebras in semisimple subalgebras. There is one algorithm for the regular subalgebras and one for the S-subalgebras.
Recently we have used these to obtain the maximal reductive subalgebras of the simple real Lie algebras of ranks up to 8. This is joint work with Heiko Dietrich,
Paolo Faccin and Alessio Marrani.
GRAW02 30th January 2020
09:10 to 10:00
Mohamed Barakat Chevalley’s Theorem on constructible images made constructive

Chevalley proved that the image of an algebraic morphism between algebraic varieties is a constructible set. Examples are orbits of algebraic group actions. A constructible set in a topological space is a finite union of locally closed sets and a locally closed set is the difference of two closed subsets. Simple examples show that even if the source and target of the morphism are affine varieties the image may neither be affine nor quasi-affine. In this talk, I will present a Gröbner-basis-based algorithm that computes the constructible image of a morphism of affine spaces, along with some applications.

GRAW02 30th January 2020
10:10 to 11:00
Alexander Hulpke Towards a nonsolvable Quotient Algorithm

[This is joint work with Heiko Dietrich from Monash U.]
Quotient algorithms have been a principal tool for the computational
investigation of finitely presented groups as well as for constructing groups.
We describe a method for a nonsolvable quotient algorithm, that extends a
known finite quotient with a module.
Generalizing ideas of the $p$-quotient algorithm, and building on results of
Gaschuetz on the representation module, we construct, for a finite group
$H$, an irreducible module $V$ in characteristic $p$, and a given number of
generators $e$ a covering group of $H$, such that every $e$-generator
extension of $H$ with $V$ must be a quotient thereof. This construction uses
a mix of cohomology (building on rewriting systems) and wreath product methods.
Evaluating relators of a finitely presented group in such a cover of a known
quotient then yields a maximal quotient associated to the cover.
I will describe theory and implementation of such an approach and discuss
the scope of the method.

GRAW02 30th January 2020
11:30 to 12:20
Bettina Eick Conjugacy problems in GL(n,Z)
The talk describes a practical algorithm to solve the conjugacy and the centralizer problems in GL(n,Z) in full generality; that is, given two matrices A and B in GL(n,Q) these algorithms allow to check if A and B are conjugate in GL(n,Z) and, if so, then to determine a conjugating element, and they allow to compute generators for the centralizer of A in GL(n,Z).  The talk also discusses possible extensions of this algorithm to finitely generated abelian or nilpotent subgroups of GL(n,Z). The latter are open problems in computational group theory and they have interesting applications.
GRAW02 30th January 2020
13:45 to 14:35
Ulrich Thiel Minimal models of symplectic quotient singularities

Namikawa associated to any conic symplectic singularity a hyperplane arrangement which is deeply intertwined with its geometry. For example, Bellamy proved that for a symplectic quotient singularity the cohomology of the complement of this arrangement encodes the number of minimal models of the singularity. For the symplectic singularity associated to a complex reflection group we were able to prove that the Namikawa arrangement coincides with the degenericity locus of the number of torus fixed points of the corresponding Calogero-Moser deformation. This has a series of remarkable consequences, especially it proves a conjecture by Bonnafé and Rouquier. Using representation theory and sophisticated computer algebraic methods, we could compute this arrangement explicitly for several exceptional complex reflection groups. The arrangements seem to be of a new kind, and many more are out there. This is joint work with Gwyn Bellamy (Glasgow) and Travis Schedler (London), and with Cédric Bonnafé (Montpellier).

GRAW02 30th January 2020
14:45 to 15:35
Dane Flannery Classifying finite linear groups in prime degree
We describe the complete, computational solution of an enduring problem in linear group theory. Specifically, we describe the classification of the finite irreducible monomial groups of prime degree (over the complex numbers). An exhaustive and self-contained solution of this problem has been obtained for all such solvable groups, and for all such groups of reasonably small degree (say, at most 23). We note obstacles that prevent a full solution of the problem for non-solvable groups in arbitrary prime degree.

This is joint work with Zolt\'an B\'acskai and Eamonn O'Brien.
GRAW02 30th January 2020
16:05 to 16:35
Eilidh McKemmie Invariable generation of finite classical groups
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 $n \rightarrow \infty$, 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 $S_n$ to the finite classical groups using the correspondence between classes of maximal tori of classical groups and conjugacy classes of their Weyl groups.
GRAW02 30th January 2020
16:35 to 17:05
Mun See Chang Computing normalisers of highly intransitive permutation groups
In general, there is no known polynomial-time algorithm for computing the normaliser $N_{S_n}(H)$ of a given group $H \leq S_n$. In this talk, we will consider the case when $H$ is a subdirect product of permutation isomorphic non-abelian simple groups. In contrast to the case with abelian simple groups, where only practical improvements have been made, here we show that $N_{S_n}(H)$ can be computed in polynomial time.
GRAW02 31st January 2020
09:10 to 10:00
Tobias Rossmann Growth of class numbers of unipotent groups
This talk is devoted to the symbolic enumeration of conjugacy classes in infinite families of finite p-groups attached to a given unipotent algebraic group. I will report on recent joint work with Christopher Voll which solves this problem for unipotent groups associated with graphs. As a by-product, we obtain polynomiality results in the spirit of Higman's conjecture on class numbers of unitriangular matrix groups.
GRAW02 31st January 2020
10:10 to 11:00
Jay Taylor Classifying Isomorphism Classes of Algebraic Groups

This talk will concern connected reductive algebraic groups (CRAGs) defined over an algebraically closed field. To each CRAG one can associate a combinatorial invariant known as its root datum. A classic result of Chevalley states that the isomorphism classes of CRAGs are in bijective correspondence with the isomorphism classes of root data. This begs the question, when are two root data isomorphic? In this talk we will describe an algorithmic solution to this problem. Part of this is joint work with Jean Michel.

GRAW02 31st January 2020
11:30 to 12:20
Christopher Jefferson Backtrack Search in Permutation Groups
While there are many problems can be solved in polynomial time, some important fundamental problems can only be solved by backtrack searches, which are often exponential time. These include many important permutation group problems including group and coset intersection, stabilizer, normaliser, and canonical image problems.

This talk will give an overview of backtracking algorithms in permutation groups, explaining both the fundamental ideas, and the most improvements. In particular this will cover Leon's Partition Backtrack algorithm and the more recent Graph Backtracking algorithm.

[This talk includes joint work with Rebecca Waldecker, Wilf Wilson and others]
GRAW02 31st January 2020
13:45 to 14:35
Cheryl Praeger Classical groups, and generating small classical subgroups
I will report on on-going work with Alice Niemeyer and Stephen Glasby. In trying to develop for finite classical groups, some ideas Akos Seress had told us about special linear groups, we were faced with the question:

"Given two non-degenerate subspaces U and W, of dimensions e and f respectively, in a formed space of dimension at least e+f, how likely is it that U+W is a non-degenerate subspace of dimension e+f?"

Something akin to this question, in a similar context is addressed in Section 5 of "Constructive recognition of classical groups in even characteristic" (J. Algebra 391 (2013), 227-255, by Heiko Dietrich, C.R.Leedham-Green, Frank Lubeck, and E. A. O’Brien). We wanted explicit bounds for this probability, and then to apply it to generate small classical subgroups.
GRAW02 31st January 2020
14:45 to 15:35
Martin Liebeck Computing with conjugacy classes in classical groups
I will discuss some theory and algorithms for performing the following tasks with unipotent elements of finite classical groups:
(1) writing down conjugacy class representatives
(2) computing centralizers
(3) solving the conjugacy problem, and finding conjugating elements.