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On Chromatic Functors and Stable Partitions of Graphs

Published online by Cambridge University Press:  20 November 2018

Ye Liu
Ye Liu*
Affiliation:
Department ofMathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan e-mail: liu@math.sci.hokudai.ac.jp
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Abstract

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The chromatic functor of a simple graph is a functorization of the chromatic polynomial. M. Yoshinaga showed that two finitely 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 finitely graph restricted to the category FI of finitely 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.

Keywords

05C1520C30chromatic functorstable partitionrepresentation stability
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 154 - 164
Copyright
Copyright © Canadian Mathematical Society 2017

References

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