Tensor Products with Verma Module and Restriction to Parabolic Subalgebra
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914043126611968 |
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| author | Merceron, Antoine |
| author_facet | Merceron, Antoine |
| contents | Using the tensor identity, we obtain decomposition results for the tensor product of a generalized Verma module with a module $M$ in the category $\mathcal{O}^{\mathfrak{p}}$, based on the decomposition of the restriction of $M$ to the parabolic subalgebra $\mathfrak{p}$. We show that this restriction admits an essentially unique decomposition into indecomposable $\mathfrak{p}$-modules and identify two particular types of direct summands: finite-dimensional quotients and tilting submodules. Finally, we give the complete $\mathfrak{b}$-decomposition of all indecomposable $M \in \mathcal{O}$ in $\mathfrak{sl}_2$, and of all Verma modules in the block of a dominant integral weight in $\mathfrak{sl}_3$, from which we derive explicit computations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor Products with Verma Module and Restriction to Parabolic Subalgebra Merceron, Antoine Representation Theory Using the tensor identity, we obtain decomposition results for the tensor product of a generalized Verma module with a module $M$ in the category $\mathcal{O}^{\mathfrak{p}}$, based on the decomposition of the restriction of $M$ to the parabolic subalgebra $\mathfrak{p}$. We show that this restriction admits an essentially unique decomposition into indecomposable $\mathfrak{p}$-modules and identify two particular types of direct summands: finite-dimensional quotients and tilting submodules. Finally, we give the complete $\mathfrak{b}$-decomposition of all indecomposable $M \in \mathcal{O}$ in $\mathfrak{sl}_2$, and of all Verma modules in the block of a dominant integral weight in $\mathfrak{sl}_3$, from which we derive explicit computations. |
| title | Tensor Products with Verma Module and Restriction to Parabolic Subalgebra |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2509.14220 |