Hidden symmetric group actions

The project is in algebraic topology, bringing in ideas from equivariant homotopy theory, together with algebra and combinatorics related to representation theory of the symmetric group. It will study some related situations in topology, algebra and combinatorics where the obvious action of the n-th symmetric group is in fact the restriction of a "hidden" action of the (n+1)-st symmetric group. This situation arises for spaces of trees and certain configuration spaces. There is a known equivariant homotopy equivalence between the n-th space of trees and the nerve of the lattice of partitions of n. The hidden extra action on the partition lattice will be made explicit and studied. Consequences for representations of the symmetric group will be explored.