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.