This semester we meet again with the IRTG-Seminar. The meeting will take place in Saarbrücken.

#### Meeting: Wednesday, 29th January 2020, SR 6, Math Building, Saarland University. We start at 14:00.

Time | Name | Title |
---|---|---|

14:00-14:45 | Yvonne Weber | P-part of the class group |

15:00-15:45 | Bernhard Boehmler | Trivial source character tables of small finite groups |

16:00-16:45 | Simon Schmidt | On the quantum symmetry of distance-transitive graphs |

#### Abstracts of the meeting

Yvonne Weber:

Title : P-part of the class group

Abstract:

The class group is a fundamental invariant in number theory. Its computation is of high interest. For some applications it suffices to compute the p-part of the class group, this is the subgroup of the class group consisting of all elements whose order is a p-power. Taking on a geometric perspective the divisor class group of plane algebraic curves can be considered an analogue to the classical class group in number theory. We aim to effectively calculate the p-part of the divisor class group of plane algebraic curves over function fields with positive characteristic p. From these results we investigate p-class field towers and abelian extensions coming back to number theory again in that way.

Bernhard Böhmler:

Title: Trivial source character tables of small finite groups

Abstract: Trivial source modules, also called p-permutation modules, arise naturally in the representation theory of finite groups. They are, by definition, the indecomposable direct summands of the permutation modules. Trivial source modules are building pieces for Puig equivalences, endo-permutation source equivalences, p-permutation equivalences, and splendid Rickard equivalences of blocks. They are also relevant to Alperin’s weight conjecture as, for example, a simple module is a weight module if and only if it is a trivial

source module. In order to do calculations with trivial source modules the ordinary characters of their lifts from positive characteristic p to characteristic zero are of particular interest. The “trivial source character tables” or “species tables” collect information about the character values of trivial source kG-modules with all possible vertices and those of their Brauer constructions. They also implicitly contain information about decomposition matrices.

Simon Schmidt:

Title: On the quantum symmetry of distance-transitive graphs

Abstract: An important task in the theory of quantum automorphism groups of finite graphs is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative.

Focusing on distance-transitive graphs, we will discuss tools for proving that the generators of the quantum automorphism group commute.

Then, using those tools, we will show that certain families of distance-transitive graphs have no quantum symmetry.