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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2503.23597 |
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| _version_ | 1866910899210551296 |
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| author | Hikita, Tatsuyuki |
| author_facet | Hikita, Tatsuyuki |
| contents | By using level one polynomial representations of affine Hecke algebras of type $A$, we obtain a $(q,t)$-analogue of the chromatic symmetric functions of unit interval graphs which generalizes Syu Kato's formula for the chromatic symmetric functions of unit interval graphs. We show that at $q=1$, the $(q,t)$-chromatic symmetric functions essentially reduce to the chromatic quasisymmetric functions defined by Shareshian-Wachs, which in particular gives an algebraic proof of Kato's formula. We also give an explicit formula of the $(q,t)$-chromatic symmetric functions at $q=\infty$, which leads to a probability theoretic interpretation of $e$-expansion coefficients of chromatic quasisymmetric functions used in our proof of the Stanley-Stembridge conjecture.
Moreover, we observe that the $(q,t)$-chromatic symmetric functions are multiplicative with respect to certain deformed multiplication on the ring of symmetric functions. We give a simple description of such multiplication in terms of the affine Hecke algebras of type $A$. We also obtain a recipe to produce $(q,t)$-chromatic symmetric functions from chromatic quasisymmetric functions, which actually makes sense for any oriented graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(q,t)$-chromatic symmetric functions Hikita, Tatsuyuki Combinatorics Representation Theory By using level one polynomial representations of affine Hecke algebras of type $A$, we obtain a $(q,t)$-analogue of the chromatic symmetric functions of unit interval graphs which generalizes Syu Kato's formula for the chromatic symmetric functions of unit interval graphs. We show that at $q=1$, the $(q,t)$-chromatic symmetric functions essentially reduce to the chromatic quasisymmetric functions defined by Shareshian-Wachs, which in particular gives an algebraic proof of Kato's formula. We also give an explicit formula of the $(q,t)$-chromatic symmetric functions at $q=\infty$, which leads to a probability theoretic interpretation of $e$-expansion coefficients of chromatic quasisymmetric functions used in our proof of the Stanley-Stembridge conjecture. Moreover, we observe that the $(q,t)$-chromatic symmetric functions are multiplicative with respect to certain deformed multiplication on the ring of symmetric functions. We give a simple description of such multiplication in terms of the affine Hecke algebras of type $A$. We also obtain a recipe to produce $(q,t)$-chromatic symmetric functions from chromatic quasisymmetric functions, which actually makes sense for any oriented graphs. |
| title | $(q,t)$-chromatic symmetric functions |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2503.23597 |