Algebra & Number Theory Seminar

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Fall 2021 ANT Zoominar
Fridays @ 10AM California Time 
Meeting ID: 941 6843 5061, Passcode: 821454 [Direct link]
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Professor Robert Boltje

Schedule (click dates for title and abstract) 

10/1
10/8
10/15
10/22 Nadia Romero University of Guanajuato
10/29
11/5 Caroline Lassueur University of Kaiserslautern
11/12 Mandi Schaeffer Fry MSU Denver
11/19
11/26 BLACK FRIDAY
12/3
Friday, October 22, at 10 am PST

Nadia Romero, University of Guanajuato

title 

abstract

Friday, November 5, at 10 am PST

Caroline Lassueur, University of Kaiserslautern

Classifying trivial source modules in blocks with cyclic defect groups

In the modular representation theory of finite groups the class of trivial source modules -- by definition the direct summands of the permutation modules -- is omnipresent and it is essential to understand them better in order to shed new light on main open questions concerned with the structure of blocks up to different kind of categorical equivalences. The aim of this talk is to present a concrete classification of the indecomposable trivial source modules belonging to a block B of a finite group with non-trivial cyclic defect groups. To achieve this goal, we rely on the classification of the indecomposable B-modules (by Janusz and dating back to the 1960's) and of the indecomposable liftable B-modules (by Hiß and Naehrig, 2012), both of which are determined up to Morita equivalence, hence by the Brauer tree of the block. Trivial source modules are not invariant under Morita equivalences in general. However, they are invariant under the stronger source-algebra equivalences. In this respect, besides the Brauer tree, two further parameters come into play. In particular, we will see how to recover the trivial source modules from the source algebra of the block in this case. This is joint work with Gerhard Hiß.

Friday, November 12, at 10 am PST

Mandi Schaeffer Fry, MSU Denver

The McKay--Navarro Conjecture: the Conjecture That Keeps on Giving!

The McKay conjecture is one of the main open conjectures in the realm of the local-global philosophy in character theory. It posits a bijection between the set of irreducible characters of a group with p’-degree and the corresponding set in the normalizer of a Sylow p-subgroup. In this talk, I’ll give an overview of a refinement of the McKay conjecture due to Gabriel Navarro, which brings the action of Galois automorphisms into the picture. A lot of recent work has been done on this conjecture, but possibly even more interesting is the amount of information it yields about the character table of a finite group.  I’ll discuss some recent results on the McKay—Navarro conjecture, as well as some of the implications the conjecture has had for other interesting character-theoretic problems.