Computational and algorithmic methods
Monday 27th January 2020 to Friday 31st January 2020
09:30 to 10:00  Registration  
10:00 to 10:10  Welcome from Christie Marr (INI Deputy Director)  
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.

INI 1  
11:00 to 11:30  Morning Coffee  
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 crosscharacteristic Lie type
maximal subgroups
of E7 and E8.
This forms part of the speaker's programme to complete the classification of maximal subgroups. 
INI 1  
12:20 to 13:45  Lunch at Westminster College  
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 LeedhamGreen) for
finding standard
generators in classical groups.

INI 1  
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.

INI 1  
15:35 to 16:05  Afternoon Tea  
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.

INI 1  
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. 
INI 1  
17:30 to 18:30  Welcome Wine Reception at INI 
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 productiongrade 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. 
INI 1  
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.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:20 
Madeleine Whybrow An algorithm to construct dihedral axial algebras
Axial algebras are nonassociative 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.

INI 1  
12:20 to 13:45  Lunch at Westminster College  
13:45 to 14:35 
Klaus Lux The 5modular Character Table of the Lyons Group
We will talk about the determination of the 5modular 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 111dimensional 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.

INI 1  
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 doublechecked 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 BorhoKraft description of the sheets of the Lie algebra to positive characteristic and supported by a few computer calculations.

INI 1  
15:35 to 16:05  Afternoon Tea  
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 x8664 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. 
INI 1 
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 openended (and impossibletoanswer) questions about the finite groups of Lie type machinery and related algorithms, which will hopefully be helpful to other users in the audience. 
INI 1  
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 minimaldegree faithful ordinary representations of most of the large sporadic simple groups, such as the Baby Monster.

INI 1  
11:00 to 11:30  Morning Coffee  
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 Ssubalgebras). 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 Ssubalgebras. 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. 
INI 1  
12:30 to 13:30  Lunch at Westminster College  
13:45 to 17:00  Free afternoon 
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 quasiaffine. In this talk, I will present a Gröbnerbasisbased algorithm that computes the constructible image of a morphism of affine spaces, along with some applications. 
INI 1  
10:10 to 11:00 
Alexander Hulpke Towards a nonsolvable Quotient Algorithm [This is joint work with Heiko Dietrich from Monash U.] 
INI 1  
11:00 to 11:30  Morning Coffee  
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.

INI 1  
12:20 to 13:45  Lunch at Westminster College  
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 CalogeroMoser 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). 
INI 1  
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 selfcontained 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 nonsolvable groups in arbitrary prime degree. This is joint work with Zolt\'an B\'acskai and Eamonn O'Brien. 
INI 1  
15:35 to 16:05  Afternoon Tea  
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.

INI 1  
16:35 to 17:05 
Mun See Chang Computing normalisers of highly intransitive permutation groups
In general, there is no known polynomialtime 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 nonabelian 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.

INI 1  
19:30 to 22:00 
Formal Dinner at Corpus Christi (Hall) 
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 pgroups 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 byproduct, we obtain polynomiality results in the spirit of Higman's conjecture on class numbers of unitriangular matrix groups.

INI 1  
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. 
INI 1  
11:00 to 11:30  Morning Coffee  
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] 
INI 1  
12:30 to 13:30  Lunch at Westminster College  
13:45 to 14:35 
Cheryl Praeger Classical groups, and generating small classical subgroups
I will report on ongoing 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 nondegenerate 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 nondegenerate 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), 227255, by Heiko Dietrich, C.R.LeedhamGreen, Frank Lubeck, and E. A. O’Brien). We wanted explicit bounds for this probability, and then to apply it to generate small classical subgroups. 
INI 1  
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. 
INI 1 