turkmath.org

For any finite group G and a fixed prime p, the topology of the poset of nontrivial elementary abelian p-subgroups contains significant information about the representations of G in characteristic p. For the symmetric group, this poset and its homology remain rather mysterious. A reasonable conjecture is that larger symmetric groups should have topologically more connected posets. I will rephrase this conjecture as a representation stability phenomenon ala Church-Ellenberg-Farb, and provide evidence for it by exhibiting high acyclicity of certain subposets.