Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS166, part 2: Computational aspects of finite groups and their representations
Time:
Thursday, 11/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F011
30 seats, 59m^2

Presentations
3:00pm - 5:00pm

Computational aspects of finite groups and their representations

Chair(s): Armin Jamshidpey (University of Waterloo, Canada), Eric Schost (University of Waterloo, Canada), Mark Giesbrecht (University of Waterloo, Canada)

The theory of finite groups and their representations is not only an interesting topic for mathematicians but also provides powerful tools in solving problems in science. New computational tools are making this even more feasible. To name a few, one may find applications in physics, coding theory and cryptography. On the other hand representation theory is useful in different areas of mathematics such as algebraic geometry and algebraic topology. Due to this wide range of applications, new algorithmic methods are being developed to study finite groups and their representations from a computational perspective.

Recent developments in computer algebra systems and more specifically computational linear algebra, provide tools for developments in computational aspects of finite groups and their representations. The aim of this minisymposium is to gather experts in the area to discuss the recent achievements and potential new directions.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Calculations with Symplectic Hypergeometric Groups

Alexander Hulpke
Colorado State University

We study monodromy groups of hypergeometric equations that can be considered as matrix groups generated by the companion matrices of pimitive pairs of integral polynomials. It is known that these groups preseve a nondegenerate symplectic form and are Zariski-dense in the respective symplectic group, and much recent work has concentrated on the question whether particular groups are arithmetic, that is have finite index in the symplectic group.
Using matrix group algorithms on congruence images, we are able to calculate indices of the arithmetic closures of these groups, The information obtained also enables us to prove arithmeticity in some cases through a coset enumeration.
This is joint work with Dane Flannery (Galway) and Alla Detinko (St Andrews).

 

Algorithmic factorization of noncommutative polynomials

Viktor Levandovskyy
RWTH Acchen University

We are interested in factorizing polynomials over non-commutative rings. Let us start with a field K and a finitely presented associative K-algebra A, which is a domain.

There are at least two distinct notions of a factorization of polynomials over A. One of them originates from the ring theory (N. Jacobson, P.M. Cohn) and uses a weak notion of association relation (called left or right similarity), what is at the same time hard to approach algorithmically. On the contrary, in applications we'd like to use the classical association relation, i.e. when two elements differ by a factor, which is nonzero central unit.

I will present long-seeked conditions for a given algebra A to be a finite factorization domain, i.e. a domain, where every nonunit has at most finite number of factorizations. Over such domains a factorization procedure thus becomes into an algorithm. Examples, bounds and counterexamples will be given. Over the well-known class of ubiquitous G-algebras (a.k.a. PBW a.k.a. algebras of solvable type), we provide a factorization algorithm, its' smarter graded-driven version for graded algebras and a factorizing Groebner algorithm. All of these are implemented in Singular:Plural (www.singular.uni-kl.de). We view the factorizing Groebner algorithm as the only general possibility to obtain a weaker analogon to the primary decomposition from the commutative algebra.
Recent complexity results and applications of the mentioned algorithms will be presented.

 

Finite groups of Lie type and computer algebra

Meinolf Geck
Universität Stuttgart

The classification of finite simple groups highlights the importance of studying the class of groups in the title. These are defined in terms of algebraic groups over algebraically closed fields of positive characteristic. We discuss a few recent examples where computer algebra methods have played a significant role in developing and establishing new results.

 

Classification of regular parametrized one-relation operads

Murray Bremner
University of Saskatchewan

I will discuss an application of representation theory of symmetric groups to algebraic operads and nonassociative algebra. J.-L. Loday introduced parametrized one-relation operads (POROs): symmetric operads generated by one binary operation subject to one relation showing how to reassociate a left-normed product into to a linear combination of right-normed products:

(ab)c =Σσ in S_3 xσ aσ ( bσ cσ ) ( xσ in Q ).

For some values of xσ, the operad is regular: for all n its homogeneous component of degree n is isomorphic to the regular representation of Sn. (Equivalently, the corresponding free algebra on a vector space V is isomorphic as a graded vector space to the tensor algebra of V.) The familiar examples of regular POROs are those governing associative, Poisson, Leibniz, and Zinbiel algebras. We use computer algebra based on a constructive version of representation theory of symmetric groups to classify all regular POROs. We show that in addition to the above four operads, the only other example is the nilpotent operad. This is joint work with Vladimir Dotsenko of Trinity College Dublin.