On chromatic functors and stable partitions of graphs

TL;DR

This paper explores the properties of chromatic functors of graphs, revealing their automorphism groups and establishing a connection to representation stability within symmetric groups.

Contribution

It determines the automorphism group of the chromatic functor and links chromatic functors to representation stability in symmetric groups.

Findings

The automorphism group of the chromatic functor can be larger than the graph's automorphism group.

Chromatic functors induce representation stable sequences of symmetric group representations.

Stable partitions are key to understanding the properties of chromatic functors.

Abstract

The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed in \cite{Yoshinaga2015} that two finite graphs have isomorphic chromatic functors if and only if they have the same chromatic polynomial. The key ingredient in the proof is the use of stable partitions of graphs. The latter is shown to be closely related to chromatic functors. In this note, we further investigate some interesting properties of chromatic functors associated to simple graphs using stable partitions. Our first result is the determination of the group of natural automorphisms of the chromatic functor, which is in general a larger group than the automorphism group of the graph. The second result is that the composition of the chromatic functor associated to a finite graph restricted to the category $\mathrm{FI}$ of finite sets and injections with the free functor into the category of complex vector spaces yields a consistent sequence of representations of symmetric groups which is representation stable in the sense of Church-Farb \cite{Church2013}.