Kromatic quasisymmetric functions
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909539190702080 |
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| author | Marberg, Eric |
| author_facet | Marberg, Eric |
| contents | We provide a construction for the kromatic symmetric function $\overline{X}_G$ of a graph introduced by Crew, Pechenik, and Spirkl using combinatorial (linearly compact) Hopf algebras. As an application, we show that $\overline{X}_G$ has a positive expansion into multifundamental quasisymmetric functions. We also study two related quasisymmetric $q$-analogues of $\overline{X}_G$, which are $K$-theoretic generalizations of the quasisymmetric chromatic function of Shareshian and Wachs. We classify exactly when one of these analogues is symmetric. For the other, we derive a positive expansion into symmetric Grothendieck functions when $G$ is the incomparability graph of a natural unit interval order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16474 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kromatic quasisymmetric functions Marberg, Eric Combinatorics We provide a construction for the kromatic symmetric function $\overline{X}_G$ of a graph introduced by Crew, Pechenik, and Spirkl using combinatorial (linearly compact) Hopf algebras. As an application, we show that $\overline{X}_G$ has a positive expansion into multifundamental quasisymmetric functions. We also study two related quasisymmetric $q$-analogues of $\overline{X}_G$, which are $K$-theoretic generalizations of the quasisymmetric chromatic function of Shareshian and Wachs. We classify exactly when one of these analogues is symmetric. For the other, we derive a positive expansion into symmetric Grothendieck functions when $G$ is the incomparability graph of a natural unit interval order. |
| title | Kromatic quasisymmetric functions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.16474 |