FI-sets with relations. (arXiv:1804.04238v1 [math.CO])

Let FI denote the category whose objects are the sets $[n] = \{1,\ldots, n\}$, and whose morphisms are injections. We study functors from the category FI into the category of sets. We write $\mathfrak{S}_n$ for the symmetric group on $[n]$. Our first main result is that, if the functor $[n] \mapsto X_n$ is "finitely generated" there there is a finite sequence of integers $m_i$ and a finite sequence of subgroups $H_i$ of $\mathfrak{S}_{m_i}$ such that, for $n$ sufficiently large, $X_n \cong \bigsqcup_i \mathfrak{S}_n/(H_i \times \mathfrak{S}_{n-m_i})$ as a set with $\mathfrak{S}_n$ action. Our second main result is that, if $[n] \mapsto X_n$ and $[n] \mapsto Y_n$ are two such finitely generated functors and $R_n \subset X_n \times Y_n$ is an FI-invariant family of relations, then the $(0,1)$ matrices encoding the relation $R_n$, when written in an appropriate basis, vary polynomially with $n$. In particular, if $R_n$ is an FI-invariant family of relations from $X_n$ to itself, then the 查看全文>>