Overview of the talks

• Nicolas Faroß (Saarland University)

Title: Spatial partition quantum groups

Abstract: We begin with a short introduction to compact matrix quantum groups, which were first defined by Woronowicz and generalize classical groups of unitary matrices. Then we take a look at some concrete examples, like the free unitary quantum group and the quantum permutation group. In the second part of the talk, we will see that these examples belong to the class of so-called easy quantum groups, which can be described by the combinatorics of set partitions. These partitions can be manipulated using a diagrammatic calculus, which can be generalized to three dimensions to defines the class spatial partition quantum groups I am currently studying.

• Jonas Hetz (Universität Stuttgart)

Title: Characters and character sheaves of finite groups of Lie Type

Abstract: An important task in the representation theory of finite groups is the determination of their character tables. As the classification of finite simple groups shows, the main difficulties in this context concern the finite groups of Lie type, which arise as an infinite series of finite groups associated to a certain algebraic group over a field of positive characteristic. In order to generically tackle the problem of determining the character tables of finite groups of Lie type, Lusztig developed the theory of character sheaves in the 1980s. In this framework, due to the work of Lusztig and Shoji, the problem is in principle reduced to determining certain roots of unity. We report on some recent progress in this area.

• Linda Hoyer (RWTH Aachen University)

Title: Orthogonal determinants of principal series characters of $GL_n(q)$

Abstract: For a finite group G, it is well-known that an ordinary irreducible character can be afforded by a real representation, if and only if the representation has a G-invariant nondegenerate symmetric bilinear form. If the degree of the character is even, there is an up to a square unique element d of the character field, such that the Gram determinant of any such bilinear form is equal to d, up to a square. We will call this the orthogonal determinant of the character.

For p an odd prime, q a power of p, n a natural number, we will regard the group G=GL_n(q), the general linear group over the field with q elements. We will recall the representation theory of these groups, and then move on to discussing methods how to calculate the orthogonal determinants of its characters. As it turns out, the case of the character not belonging to the principal series is easily handled, so the main focus of the talk will be about the principal series case.

References: 

1. G. Nebe, R. Parker. "Orthogonal Stability" (J. Algebra, to appear)

2. I. G. Macdonald. "Symmetric Functions and Hall Polynomials". Oxford Mathematical Monographs. The Clarendon Press
   Oxford University Press, New York, second edition (1995)

3. W. Fulton, J. Harris. "Representation theory". Springer-Verlag (2004)

4. R. W. Carter. "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters". Pure Appl. Math. 44 (1985)

• Max Mayer (RPTU Kaiserslautern-Landau)

Title: Computations in polycyclic groups

Abstract: Polycyclic groups are computational wise an interesting class of groups as a lot of usually hard or even undecidable questions can be answered algorithmically in these groups. In this talk we have a look at the subgroup algorithm which is central to many other algorithms. We discuss what the issues of the currently used algortihm are and how to improve it by the use of an LLL-reduced hermite normal form computation. Afterwards we have a look at an application to this algorithm namely the computation of the intersection of (two) subgroups.

References:

1. "Polycyclic groups", Daniel Segal, 1983, Cambridge University Press

2. "Computation with Finitely Presented Groups", Charles C. Sims, 1994, Cambridge University Press

3. "Handbook of computational group theory", Derek F. Holt at al, 2005, Chapman and Hall/CRC Press, (mostly Chapter 8)

4. "Computing with infinite polycyclic groups", Bettina Eick, 2001

• Stevell Muller (Saarland University)

Title: On Symplectic Birational Transformations of OG10-type Hyperkähler Manifolds

Abstract: Hyperkaehler manifolds, also known as irreducible holomorphic symplectic manifolds, are one of the three building blocks of an important class of complex manifolds. Their symplectic birational self maps have been widely studied: one could cite a celebrated paper of Mukai regarding the case of K3 surfaces.

In this talk, I plan to give a non-specialised overview on what motivates the study of symplectic birational self-maps of hyperkaehler manifolds. I explain how one can use a computer, in particular Oscar, to answer some questions requiring expensive computations. Without going too much into technical details, I use as a support for my talk a work in progress with my collaborator Lisa Marquand, about a classification of subgroups of symplectic birational self-maps for OG10-type hyperkaehler manifolds.

• Liam Rogel (RPTU Kaiserslautern-Landau)

Title: Categorifications of two-sided Kazhdan Lusztig cells using Soergel diagrammatics

Abstract: We introduce the diagrammatic Hecke Category by Elias and Williamson and show how it categorifies the Hecke-Algebra. We further motivate Lusztig's $H$-cell reduction by showing fusion ring properties on the Grothendieck ring and imitate the construction on the category level. Small examples of dihedral groups are discussed throughout the talk and we end up with complete fusion data needed to implement the asymptotic Hecke Category into the OSCAR package TensorCategories.jl.