Über Struktur und Kohomologie von Moduln symmetrischer Gruppen

• On structure and cohomology of symmetric group modules

Weber, Christian; Hiß, Gerhard (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2011)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2011

Abstract

Ordinary representation theory of the symmetric groups is quite well understood, but there are still many open questions concerning modular representation theory of the symmetric groups. About cohomology of S_n-modules, there is little known as well. This thesis has its starting point where the latter two fields meet. Possibilities of making statements about first and second cohomology of certain S_n-modules are explored. In the center of attention there are Specht modules, on the one hand over the integers, on the other hand over fields of prime characteristic p and also, as a link, over the p-adic integers. Over the field F_2, permutation modules, Young modules, irreducible modules and dual Specht modules are considered in addition. An important tool for this research is the Young graph, a directed graph whose vertices are given by all partitions of all nonnegative integers. Certain induced subgraphs provide - by their structure and interrelations - information about cohomology of the Specht modules corresponding to the contained partitions. In the case of odd primes p, a method of David Hemmer is taken up, that allows to decide on a combinatorial basis whether first cohomology of a Specht module over a field of characteristic p is trivial or not. The handling of this method gets improved, and in combination with the subgraphs of the Young graph, information about cohomology of Specht modules is obtained, that each of both approaches alone cannot give. In the case p = 2, Hemmer's method does not work. Here we go a different way: By subtle linking of appropriate exact cohomology sequences, several cohomology groups of certain F_2S_n-modules can be traced back to cohomologies that are already known. For this purpose we need to know the submodule lattices of certain F_2S_n-modules, especially Young modules. Some of them are determined in this thesis. A further central topic of this thesis is the search for partitions lambda with the following property: For a given pair of a prime p and a natural number iota with p*iota <= n, the second cohomology of the corresponding integral Specht module contains an element, that is not mapped to 0 by restriction to a cyclic subgroup of S_n, generated by a product of iota disjoint p-cycles. This problem is approached theoretically as well as computationally. The interest for such partitions is motivated by a conjecture of Andrzej Szczepanski, that - in the special case of symmetric groups - says: For every n there exists a Q-multiplicity free faithful ZS_n-lattice V, such that the second cohomology group H^2(S_n,V) contains an element, that is not mapped to 0 by restriction to any nontrivial subgroup of S_n. Via the methods used here, there are new ways to a potential proof of this statements highlighted; in particular, the conjecture is proved for n <= 13.