A unipotent realization of the chromatic quasisymmetric function
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909324222136320 |
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| author | Gagnon, Lucas |
| author_facet | Gagnon, Lucas |
| contents | This paper realizes of two families of combinatorial symmetric functions via the complex character theory of the finite general linear group $\mathrm{GL}_{n}(\mathbb{F}_{q})$: chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated $\mathrm{GL}_{n}(\mathbb{F}_{q})$ characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$. The proof of these results also gives a general Hopf algebraic approach to computing the induction map. Additional results include a connection between the relevant $\mathrm{GL}_{n}(\mathbb{F}_{q})$ characters and Hessenberg varieties and a re-interpretation of known theorems and conjectures about the relevant symmetric functions in terms of $\mathrm{GL}_{n}(\mathbb{F}_{q})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06981 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A unipotent realization of the chromatic quasisymmetric function Gagnon, Lucas Combinatorics Representation Theory This paper realizes of two families of combinatorial symmetric functions via the complex character theory of the finite general linear group $\mathrm{GL}_{n}(\mathbb{F}_{q})$: chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated $\mathrm{GL}_{n}(\mathbb{F}_{q})$ characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$. The proof of these results also gives a general Hopf algebraic approach to computing the induction map. Additional results include a connection between the relevant $\mathrm{GL}_{n}(\mathbb{F}_{q})$ characters and Hessenberg varieties and a re-interpretation of known theorems and conjectures about the relevant symmetric functions in terms of $\mathrm{GL}_{n}(\mathbb{F}_{q})$. |
| title | A unipotent realization of the chromatic quasisymmetric function |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2211.06981 |