Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Christine Bessenrodt
Leibniz Universität Hannover
Kronecker products of characters of the symmetric groups with few components
One of the central open problems in the character theory of the symmetric groups is the problem of
decomposing the Kronecker products of their characters into irreducible characters. As James and Kerber
remark in their book The representation theory of the symmetric group, a product of irreducible characters
of Sn is “in general reducible”. Jointly with Kleshchev, we have shown in 1999 that such a product is even
only homogeneous in the (trivial) case of multiplying with a character of degree 1; we have also classified
products with two homogeneous components for determining the homogeneous character products for
the alternating groups. Furthermore, we conjectured in 1999 a classification of products with at most four
homogeneous components. This conjecture has now been verified; on the way towards this result new
information on constituents in Kronecker products has been obtained.
Literatur
Bessenrodt, C. and A. Kleshchev (1999). On Kronecker products of complex representations of the symmetric
and alternating groups. Pacific J. Math. 190, 201 - 223.
Bessenrodt, C. and S. van Willigenburg (2011). On Kronecker products of characters of the symmetric
groups with few components, arXiv:1105.3170
Michael Cuntz
Universität Kaiserslautern
From Hopf algebras to toric varieties via the Weyl groupoid
In the classification of finite dimensional Nichols algebras of diagonal type, the so-called Weyl groupoids
played an important role. Meanwhile, the finite Weyl groupoids have been thoroughly investigated,
including a complete classification. It turns out that they correspond to simplicial arrangements satisfying
a certain integrality axiom. This integrality translates to the smoothness of the toric varieties associated
to the corresponding fans.
Susanne Danz
University of Oxford
On the double Burnside ring of a finite group
In recent years, the double Burnside ring B(G,G) of a finite group G has proved to have important
connections with modular representation theory of finite groups, with topology, and with the theory of
fusion systems on finite p-groups. For these and other applications, one needs to carry out explicit
computations in the ring B(G,G), in order to determine certain units or idempotents. However, the
multiplicative structure of B(G,G) is, in general, non-commutative and rather involved. In this talk I will
focus on distinguished subrings of the double Burnside ring B(G,G), and present a candidate for a
non-commutative analogue of the well-known mark homomorphism of the commutative Burnside ring.
This homomorphism translates the complicated multiplication rule of these subrings into a much more
transparent one. If time permits, I will also outline some applications to the theory of fusion systems. This
is joint work with Robert Boltje.
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