The Representation of Symmetric Groups in Two-Dimensional Persistent Homology

The two dimensional persistent homology, that can be reduced to the homology of a family of parameterized real functions, can be found to have some issues due to a phenomenon of monodromy, which happens when a loop in the parameters' space doesn't translate into loops of the elements of the persistent diagram. Instead they switch place between them in a functorial way. In this paper we present a formalisation of this functor and we will make an use of it, representing an arbitrary symmetric group by means of a suitable filtering function.