TY - JOUR

T1 - On chromatic functors and stable partitions of graphs

AU - Liu, Ye

N1 - Publisher Copyright:
© 2016 Canadian Mathematical Society.

PY - 2017/3

Y1 - 2017/3

N2 - 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 with 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 with a finite graph restricted to the category 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 that is representation stable in the sense of Church-Farb.

AB - 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 with 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 with a finite graph restricted to the category 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 that is representation stable in the sense of Church-Farb.

UR - http://www.scopus.com/inward/record.url?scp=85009344271&partnerID=8YFLogxK

U2 - 10.4153/CMB-2016-047-3

DO - 10.4153/CMB-2016-047-3

M3 - Article

AN - SCOPUS:85009344271

SN - 0008-4395

VL - 60

SP - 154

EP - 164

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

IS - 1

ER -