Matthias K"unzer, Universit"at Bielefeld

Ties for the ZS_n

Thesis, University of Bielefeld

We interprete the ties (i.e. the congruences)
which describe the image of an embedding

ZS_n -> (Z)_{n_1} x ... x (Z)_{n_k}

as stemming from modular morphisms (modulo
prime powers). The image of such an embedding,
provided given in such a form that the 
localized Pierce decompositions can be read 
off, is a convenient tool to handle the ZS_n.

Several families of morphisms between 
Specht modules are provided via 
explicit formulas, one of them lifting
the one-box-shift part of a result of 
Carter and Payne to Z/(length of the box-shift). 

James lattices are combinatorially 
described generalized Specht lattices, equipped
with a Specht filtration. A system of 
morphisms between James modules is described 
via explicit formulas. Due to the shape
of the quiver over which (a slight modification of) 
this system furnishes a module, it is called 
the truss. It is sufficient to describe an
embedding as above via ties.

Therefore, in order to obtain a satisfactory 
embedding, it remains to exhibit a reasonable
normal form for the truss. In this generality,
the problem remains open. Tackling it 
using subsets of morphisms with known 
composition properties yields first modest 
partial results.