Higher Specht bases and $q$-series for the cohomology rings of certain Hessenberg varieties
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914969920995328 |
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| author | Salois, Kyle |
| author_facet | Salois, Kyle |
| contents | It is conjectured (following the Stanley-Stembridge conjecture) that the cohomology rings of regular semisimple Hessenberg varieties yield permutation representations, but the decompositions of the modules are only known in some cases. For the Hessenberg function $h=(h(1),n,\ldots,n)$, the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. We define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function $h' = ((n-1)^{n-m},n^m)$, and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of $P$-tableaux, motivated by the work of Gasharov, illustrating the connections between the $\mathfrak{S}_n$ action on these cohomology rings and the Schur expansion of chromatic symmetric functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher Specht bases and $q$-series for the cohomology rings of certain Hessenberg varieties Salois, Kyle Combinatorics 05E10 (Primary), 05E05, 05E14, 05C15 (Secondary) It is conjectured (following the Stanley-Stembridge conjecture) that the cohomology rings of regular semisimple Hessenberg varieties yield permutation representations, but the decompositions of the modules are only known in some cases. For the Hessenberg function $h=(h(1),n,\ldots,n)$, the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. We define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function $h' = ((n-1)^{n-m},n^m)$, and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of $P$-tableaux, motivated by the work of Gasharov, illustrating the connections between the $\mathfrak{S}_n$ action on these cohomology rings and the Schur expansion of chromatic symmetric functions. |
| title | Higher Specht bases and $q$-series for the cohomology rings of certain Hessenberg varieties |
| topic | Combinatorics 05E10 (Primary), 05E05, 05E14, 05C15 (Secondary) |
| url | https://arxiv.org/abs/2410.08366 |