Algebra and Number Theory Seminar Winter 2018
Friday, January 12, 2018
No seminar
Friday, January 19, 2018
Britta Späth, University of Wuppertal and MSR
About the Inductive Conditions for Global/Local Conjectures: Origins, Motivation, Past and Future
In the last 15 years many global-local conjectures have been reduced to simple groups, more precisely the conjectures hold if all simple groups satisfy a strong version of that conjectures, the corresponding so-called inductive condition. This way the McKay conjecture on characters of odd degree was prove. I will try to explain the origin and the nature of the inductive conditions. At the end I sketch when those conditions could be
In 2004, G. Navarro proposed a refinement of the McKay conjecture: the actions of a particular group of Galois automorphisms on the two character sets involved in the McKay conjecture are permutation isomorphic. The Navarro conjecture implies that the local condition that a Sylow p-subgroup of a finite group G is self-normalizing can be characterized in terms of the character theory of G; an implication that has been recently verified for all primes p. In the same spirit, we restrict our attention to the character theory of the principal p-block of G and analyze how it is connected with the structure of the normalizer of P.
Categories and functors in the representation theory of vertex operator algebras
The category of all vertex operator algebras and the category of modules of a vertex operator algebra are discussed. A homomorphism between two vertex operator algebras should preserve the Virasoro vectors, which is equivalent to commuting with the operators L(n) for all n ∈ Z. We expand the morphisms of this category so that morphisms are semi-conformal in the sense that they commute with those L(n) with n>-1. This expansion does not change the classification of problem and makes the category into a tensor category. The coset construction becomes more natural in this category and relations between the module categories of vertex operator algebras can be described in terms of Hom-functors. As an application, we also construct the corresponding Jacquet functors. The semi-conformal vertex operator subalgebras plays the role of the Levi subgroups of a reductive group
Fusion Product for Permutation Orbifolds
The permutation orbifolds study the action of permutation groups on the tensor products of vertex operator algebras. This talk will report our recent progress on the fusion products of twisted modules for general permutation orbifolds. The motivation is to understand the module categories for the fixed point vertex operator subalgebras under the permutation groups. This is a joint work with Chongying Dong and Feng Xu.
Malle and Navarro have recently proposed a refined version of Brauer's celebrated height zero conjecture on blocks of finite groups. In this talk I will demonstrate that the refined version actually follows from Brauer's original conjecture.
A hypergeometric series with rational parameters has rational Taylor coefficients. The series is said to be p-adically bounded if its coefficients are p-adically bounded. The study of boundedness of hypergeometric series goes back to work of Dwork and Christol, who identified necessary and sufficient conditions for a series to be p-adically bounded. In joint work with Terry Gannon and Geoff Mason, we identified a new necessary and sufficient condition for boundedness and used it to show that, with finitely many exceptions, the p-adic boundedness of a hypergeometric series only depends on p modulo the denominators of the parameters defining the series. Thus, the set of bounded primes for a fixed rational hypergeometric series has a Dirichlet density. In this talk we will describe these results, we will give a formula for the density of bounded primes, and we will discuss the generic behaviour of the density of bounded primes as the rational parameters defining the hypergeometric series vary.
Real representations of finite simple groups
We finish the classification of finite simple groups with the property that all of their irreducible complex representations may be realized over the reals. As a result we conclude that a finite simple group has this property if and only if every element in the group is a product of two involutions. The final cases of interest are the simple symplectic and orthogonal groups over a field of characteristic 2, and in the process of covering these cases we obtain some nice combinatorial identities for the sum of the degrees of unipotent characters for these groups (for odd or even characteristic). We will also discuss some conjectures and results regarding the reality properties of other finite simple groups.
