| Time | Speaker | Title | Resources | |
|---|---|---|---|---|
| 09:20 to 10:00 | Wolfgang Kimmerle (University of Stuttgart, Germany) |
On torsion subgroups of the units of ZG and related questions. For a finite group $G$ denote by $\mathbb{Z}G$ its integral group ring and by $V(\mathbb{Z}G)$ its normalized unit group, i.e the units with augmentation 1. Object of the talk is the following. Theorem. Let $G$ be a finite nilpotent-by-abelian group. Then exists a finite group $E$ containing $G$ such that the following holds. (i) Torsion units of $\mathbb{Z}G$ are conjugate in $\mathbb{Q}E$. (ii) Let $p$ be a prime. Then $p$ - subgroups of the normalized unit group $V(\mathbb{Z}G)$ are conjugate within $\mathbb{Q}E$ to a subgroup of $G$. (iii) Different group bases of $\mathbb{Z}G$ are conjugate within $\mathbb{Q}E$. The theorem holds in particular for supersoluble groups and contains a Sylow like theorem for $V(\mathbb{Z}G)$ . Some parts of the theorem hold for larger classes of finite groups.With respect to Sylow like theorems inthe unit group of integral group rings and related questions forcharacter tables this will be discussed in the second part of the talk. References 1. A.B ̈achle, W.Kimmerle and M.Serrano, On the First Zassenhaus Conjecture and Direct Products, Can. J. Math, https://doi.org/10.4153/S0008414X18000044 2018. 2. W. Kimmerle and I. K ̈oster, Sylow Numbers from Character Tables and Group Rings, Stuttgarter Mathematische Berichte 2015 - 012. 3. W. Kimmerle and L. Margolis, p-subgroups of units in ZG. In Groups, Rings, Group Rings and Hopf algebras, Contemp. Math. 688, AMS, Providence, RI, 2017, 169 – 179. 4. I. K ̈oster, Sylow Numbers in Character Tables and Integral Group rings, Dissertation Universit ̈at Stuttgart 2017. |
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| 10:10 to 10:50 | Eric Jespers (Vrije Universiteit Brussels, Belgium) |
Set theoretic solutions of the Yang-Baxter equation and its associated structures A set-theoretic solution $(X,r)$ of the Yang-Baxter equation consists of a (finite) set $X$ and a map $r:X^2\rightarrow X^2: (x,y) \mapsto (\lambda_x(y), \rho_{y}(x))$ such that the braided relation $r_{12}r_{23}r_{12}=r_{23}r_{12}r_{12}$ is satisfied. Here $r_12$ denotes the map $r_{12}:X^3 \rightarrow X^3: (x,y,z) \mapsto (r(x,y),z)$ and similarly $r_{23} (x,y,z)=(x,r(y,z))$. The following algebraic structures are associated to such a solution: the structure group $G(X,r)=\gr(X\mid xy=uv\text{ if }r(x,y)=(u,v))$, the structure monoid $M(X,r)=\free{X \mid xy=uv\text{ if }r(x,y)=(u,v)}$ and the structure algebra $A(K,X,r)=K[M(X,r)]$, that is the $K$-algebra generated by the set $X$ with the ``same’’ defining relations as the monoid $M(X,r)$. In case $(X,r)$ is finite involutive (i.e. $r^2=\mbox{id}$) and non-degenerate (i.e. each $\lambda_x$ and $\rho_y$ is bijective), Etingof, Schedler and Soloviev, and Gateva-Ivanova and Van den Bergh proved some fundamental rich algebraic properties of these structures. In this lecture we present recent results, joint with Kubat and Van Antwerpen, and Cedo and Okninski, for non-degenerate bijective bijective solutions that are not necessarily involutive. It is has been shown that all bijective solutions on a finite set are determined by another algebraic structure, called a skew left brace. This notion was introduced by Rump for the involutive case and by Guarnieri and Vendramin for the general case. If time permits we will present some brief overview of some recent results and also show that this structure also is important for the study of the unit group of the integral group ring of a finite group. |
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| 10:50 to 11:20 | Break | Tea/coffee | ||
| 11:20 to 12:00 | Angel del Rio (University of Murcia, Spain) |
The Zassenhaus Conjecture for cyclic-by-nilpotent groups Let G be a finite group. The Zassenhaus Conjecture states that every torsion unit of augmentation 1 in the integral group algebra is conjugate in the rational group algebra to an element of G. Although a counterexample has been found recently in the class of metabelian groups, the conjecture has been proved for same large classes of groups including nilpotent groups and cyclic-by-abelian groups, and it is open for the class of supersolvable groups. We will present some positive results for the class of cyclic-by-nilpotent groups obtained recently in cooperation with Mauricio Caicedo. |
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| 12:10 to 12:40 | Andreas Bachle (Vrije Universiteit Brussels, Belgium) |
Central Units of Integral Group Rings A finite group G is called cut if the center of its integral group ring only contains the ‘obvious’ central units, i.e., Z(U(ZG)) = ±Z(G). This is a rich class of groups, yet restricted enough to obtain interesting structural results. In this talk we will survey recent results due to a nice blend of group theory and representation theory and highlight several challenging open problems. Vrije Universiteit Brussel, DWIS, Pleinlaan 2, 1050 Brussel, Belgium Email address: Andreas.Bachle@vub.be |
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| 12:50 to 13:20 | Leo Margolis (Vrije Universiteit Brussels, Belgium) |
The influence of the Luthar-Passi method on the study of units in group rings In 1989 Luthar and Passi proved the First Zassenhaus Conjecture for the alternating group of degree 5 using a character theoretic argument. I.e. they showed that for $G=A_5$ any unit of finite order in the integral group ring $\mathbb{Z}G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to a trivial unit of the form $\pm g$, for some $g \in G$. Their argument was reproduced for other groups and generalized by Hertweck and became known as the Luthar-Passi method or Hertweck-Luthar-Passi method in its extended version. In this talk I will revise the influence of the method on the study of the Zassenhaus Conjecture and other questions concerning units in integral group rings, the interaction of the method with other arguments and how it stimulated new ideas in the area. |
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| 13:20 to 15:00 | Break | Lunch | ||
| 15:00 to 15:40 | Allen Herman (University of Regina, Canada) |
Semisimple $\mathbb{Q}$-algebras in algebraic combinatorics Recently several new types of algebras have emerged in algebraic combinatorics that have taken a central role in important advances in graph theory, design theory, coding theory, and conformal field theory. The representation theory of these algebras is an emerging area of research that relies heavily on the techniques and ideas developed by those working in group rings and computational algebra. I will introduce the most prevalent of these: coherent algebras, adjacency algebras of association schemes, fusion rule algebras, and Terwilliger algebras. I will give an overview of their basic algebraic and categorical properties, the current state of their classification, and survey recent contributions to their ordinary, rational, integral, and modular representation theories. |
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| 15:40 to 16:00 | Break | Tea/coffee | ||
| 16:00 to 16:40 | L R Vermani (Kurukshetra University, India) |
Polynomial groups, polynomial maps, dimension subgroups and related problems The purpose of this talk is to review the work of Prof. Passi on polynomial groups, polynomial maps, dimension subgroups and related problems. This talk will cover his early work till 1990. In the process I have not done full justice to other workers in this area since it is not a survey on these topics as such. |
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| 16:50 to 17:20 | Sugandha Maheshwari (IISERM, India) |
On the pioneering works of Professor I.B.S. Passi Professor Passi is a well known algebraist. His main research area is group rings, for which he has contributed significantly. The pioneering results on the dimension subgroups, augmentation powers in group rings, and related problems have received wide recognition. In this talk, dedicated to the works of Professor Passi, I would like to discuss some of his major contributions, during last three decades. |