Three Examples of Quasisymmetric Compatible $\mathfrak{S}_n$-modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917668993368064 |
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| author | Hicks, Angela Miller-Brown, Samantha |
| author_facet | Hicks, Angela Miller-Brown, Samantha |
| contents | The Schur functions, a basis for the symmetric polynomials (Sym), encode the irreducible representations of the symmetric group, $\mathfrak{S}_n$, via the Frobenius characteristic map. In 1996, Krob and Thibon defined a quasisymmetric Frobenius map on the representations of $\mathcal{H}_n(0)$, mapping them to the quasisymmetric functions (QSym). Despite the obvious inclusion of Sym in QSym and the close relationship between $\mathfrak{S}_n$ and $\mathcal{H}_n(0)$, there is no known direct link between these two Frobenius characteristic maps and the related representations. We explore three specific situations in which a deformation of an $\mathfrak{S}_n$ action results in a valid $\mathcal{H}_n(0)$ action and gives a quasisymmetric Frobenius characteristic that is equal to the symmetric Frobenius characteristic. We introduce the concept of quasisymmetric compatibility, which formalizes a link between the two maps, and we show it applies to all $\mathfrak{S}_n$-modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three Examples of Quasisymmetric Compatible $\mathfrak{S}_n$-modules Hicks, Angela Miller-Brown, Samantha Combinatorics 05E05 (Primary) 05E10 (Secondary) The Schur functions, a basis for the symmetric polynomials (Sym), encode the irreducible representations of the symmetric group, $\mathfrak{S}_n$, via the Frobenius characteristic map. In 1996, Krob and Thibon defined a quasisymmetric Frobenius map on the representations of $\mathcal{H}_n(0)$, mapping them to the quasisymmetric functions (QSym). Despite the obvious inclusion of Sym in QSym and the close relationship between $\mathfrak{S}_n$ and $\mathcal{H}_n(0)$, there is no known direct link between these two Frobenius characteristic maps and the related representations. We explore three specific situations in which a deformation of an $\mathfrak{S}_n$ action results in a valid $\mathcal{H}_n(0)$ action and gives a quasisymmetric Frobenius characteristic that is equal to the symmetric Frobenius characteristic. We introduce the concept of quasisymmetric compatibility, which formalizes a link between the two maps, and we show it applies to all $\mathfrak{S}_n$-modules. |
| title | Three Examples of Quasisymmetric Compatible $\mathfrak{S}_n$-modules |
| topic | Combinatorics 05E05 (Primary) 05E10 (Secondary) |
| url | https://arxiv.org/abs/2403.16249 |