2024/10/07 by Campbell, John M.
#05E05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.04669
If we consider previously introduced extensions of Stanley's chromatic symmetric function XG(x1, x2, …) for a graph G to elements in the algebra \textsfQSym of quasisymmetric functions and in the algebra \textsfNCSym of symmetric functions in noncommuting variables, this motivates our introduction of a lifting of XG to the dual of \textsfQSym, i.e., the algebra \textsfNSym of noncommutative symmetric functions, as opposed to \textsfNCSym. For an unlabelled directed graph D, our extension of chromatic symmetric functions provides an element XD in \textsfNSym, in contrast to the analogue YG ∈ \textsfNCSym of XG due to Gebhard and Sagan. Letting G denote the undirected graph underlying D, our construction is such that the commutative image of XD is XG. This projection property is achieved by lifting Stanley's power sum expansion for chromatic symmetric functions, with the use of the Ψ-basis of \textsfNSym, so that the orderings of the entries of the indexing compositions are determined by the directed edges of D. We then construct generating sets for \textsfNSym consisting of expressions of the form XD, building on the work of Cho and van Willigenburg on chromatic generating sets for \textsfSym.