Transplanting Trees: Chromatic Symmetric Function Results through the Group Algebra of S_n

2021/12/18 by Foley, Angèle M., Kazdan, Joshua, Kröll, Larissa +3
#05E05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.09937

Abstract

One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's. Though verified on graphs of order up to twenty-nine, this result has been proved only for certain subclasses of trees. Using the definition of the CSF that emerges via the Frobenius character map applied to ℂ[S_n], we offer new algebraic proofs of several results about the CSF's of trees. Additionally, we prove that a "parent function" of the CSF defined in the group ring of S_n can uniquely determine trees, providing further support for Stanley's conjecture.

Related

• A class of trees determined by their chromatic symmetric functions
• Chromatic symmetric functions and polynomial invariants of trees
• The chromatic symmetric function in the star-basis
• The Kneser chromatic function distinguishes trees
• On distinguishing special trees by their chromatic symmetric functions