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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.04669 |
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| _version_ | 1866929530033143808 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$ of symmetric functions in noncommuting variables, this motivates our introduction of a lifting of $X_{G}$ to the dual of $\textsf{QSym}$, i.e., the algebra $\textsf{NSym}$ of noncommutative symmetric functions, as opposed to $\textsf{NCSym}$. For an unlabelled directed graph $D$, our extension of chromatic symmetric functions provides an element $\text{X}_{D}$ in $\textsf{NSym}$, in contrast to the analogue $Y_{G} \in \textsf{NCSym}$ of $X_{G}$ due to Gebhard and Sagan. Letting $G$ denote the undirected graph underlying $D$, our construction is such that the commutative image of $\text{X}_{D}$ is $ X_{G}$. This projection property is achieved by lifting Stanley's power sum expansion for chromatic symmetric functions, with the use of the $Ψ$-basis of $\textsf{NSym}$, 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 $\textsf{NSym}$ consisting of expressions of the form $\text{X}_{D}$, building on the work of Cho and van Willigenburg on chromatic generating sets for $\textsf{Sym}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04669 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lift of chromatic symmetric functions to $\textsf{NSym}$ Campbell, John M. Combinatorics 05E05 If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$ of symmetric functions in noncommuting variables, this motivates our introduction of a lifting of $X_{G}$ to the dual of $\textsf{QSym}$, i.e., the algebra $\textsf{NSym}$ of noncommutative symmetric functions, as opposed to $\textsf{NCSym}$. For an unlabelled directed graph $D$, our extension of chromatic symmetric functions provides an element $\text{X}_{D}$ in $\textsf{NSym}$, in contrast to the analogue $Y_{G} \in \textsf{NCSym}$ of $X_{G}$ due to Gebhard and Sagan. Letting $G$ denote the undirected graph underlying $D$, our construction is such that the commutative image of $\text{X}_{D}$ is $ X_{G}$. This projection property is achieved by lifting Stanley's power sum expansion for chromatic symmetric functions, with the use of the $Ψ$-basis of $\textsf{NSym}$, 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 $\textsf{NSym}$ consisting of expressions of the form $\text{X}_{D}$, building on the work of Cho and van Willigenburg on chromatic generating sets for $\textsf{Sym}$. |
| title | A lift of chromatic symmetric functions to $\textsf{NSym}$ |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2410.04669 |