Towards plethystic $\mathfrak{sl}_2$ crystals
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908557332447232 |
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| author | Gutiérrez, Álvaro |
| author_facet | Gutiérrez, Álvaro |
| contents | To find crystals of $\mathfrak{sl}_2$ representations of the form $Λ^n\text{Sym}^r\mathbb{C}^2$ it suffices to solve the combinatorial problem of decomposing Young's lattice into symmetric, saturated chains. We review the literature on this latter problem, and present a strategy to solve it. For $n \le 4$, the strategy recovers recently discovered solutions. We obtain (i) counting formulas for plethystic coefficients, (ii) new recursive formulas for plethysms of Schur functions, and (iii) formulas for the number of constituents of $Λ^n\text{Sym}^r\mathbb{C}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards plethystic $\mathfrak{sl}_2$ crystals Gutiérrez, Álvaro Combinatorics Representation Theory 05E10, 17B10, 05A30 To find crystals of $\mathfrak{sl}_2$ representations of the form $Λ^n\text{Sym}^r\mathbb{C}^2$ it suffices to solve the combinatorial problem of decomposing Young's lattice into symmetric, saturated chains. We review the literature on this latter problem, and present a strategy to solve it. For $n \le 4$, the strategy recovers recently discovered solutions. We obtain (i) counting formulas for plethystic coefficients, (ii) new recursive formulas for plethysms of Schur functions, and (iii) formulas for the number of constituents of $Λ^n\text{Sym}^r\mathbb{C}^2$. |
| title | Towards plethystic $\mathfrak{sl}_2$ crystals |
| topic | Combinatorics Representation Theory 05E10, 17B10, 05A30 |
| url | https://arxiv.org/abs/2412.15006 |