Basis and representation structure of the NAF coalgebra

Determine whether the \(\mathbb{Z}[t_n]\)-subcoalgebra spanned by the classes \(ev_{\sigma}\partial_F\), for \(\sigma\in S_n\) and noncrossing alternating forests \(F\), admits a \(\mathbb{Z}[t_n]\)-basis, and characterize this span as a representation of \(S_n\).

Background

The paper identifies an $S_n$-invariant subcoalgebra of equivariant homology generated by the classes $ev_{\sigma}\partial_F$, with $F$ ranging over noncrossing alternating forests. The generating family has nonnegative integer structure constants, but the authors do not show that it is linearly independent or provide a canonical basis.

The stated problem asks both whether a basis over the equivariant coefficient ring exists in the intended generating span and how the span behaves as an $S_n$-representation. These structural questions are left unresolved after the coproduct positivity results.