CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

On Chromatic Functors and Stable Partitions of Graphs

  Published:2016-08-19
 Printed: Mar 2017
  • Ye Liu,
    Department of Mathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, JAPAN
Format:   LaTeX   MathJax   PDF  

Abstract

The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed 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.
Keywords: chromatic functor, stable partition, representation stability chromatic functor, stable partition, representation stability
MSC Classifications: 05C15, 20C30 show english descriptions Coloring of graphs and hypergraphs
Representations of finite symmetric groups
05C15 - Coloring of graphs and hypergraphs
20C30 - Representations of finite symmetric groups
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/