Irreducible components in Hochschild cohomology of flag varieties
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913652675706880 |
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| author | Jeralds, Sam |
| author_facet | Jeralds, Sam |
| contents | Let $G$ be a simple, simply-connected complex algebraic group with Lie algebra $\mathfrak{g}$, and $G/B$ the associated complete flag variety. The Hochschild cohomology $HH^\bullet(G/B)$ is a geometric invariant of the flag variety related to its generalized deformation theory and has the structure of a $\mathfrak{g}$-module. We study this invariant via representation-theoretic methods; in particular, we give a complete list of irreducible subrepresentations in $HH^\bullet(G/B)$ when $G=SL_n(\mathbb{C})$ or is of exceptional type (and conjecturally for all types) along with nontrivial lower bounds on their multiplicities. These results follow from a conjecture due to Kostant on the structure of the tensor product representation $V(ρ) \otimes V(ρ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducible components in Hochschild cohomology of flag varieties Jeralds, Sam Representation Theory Algebraic Geometry 14M15, 17B10, 14F43 Let $G$ be a simple, simply-connected complex algebraic group with Lie algebra $\mathfrak{g}$, and $G/B$ the associated complete flag variety. The Hochschild cohomology $HH^\bullet(G/B)$ is a geometric invariant of the flag variety related to its generalized deformation theory and has the structure of a $\mathfrak{g}$-module. We study this invariant via representation-theoretic methods; in particular, we give a complete list of irreducible subrepresentations in $HH^\bullet(G/B)$ when $G=SL_n(\mathbb{C})$ or is of exceptional type (and conjecturally for all types) along with nontrivial lower bounds on their multiplicities. These results follow from a conjecture due to Kostant on the structure of the tensor product representation $V(ρ) \otimes V(ρ)$. |
| title | Irreducible components in Hochschild cohomology of flag varieties |
| topic | Representation Theory Algebraic Geometry 14M15, 17B10, 14F43 |
| url | https://arxiv.org/abs/2404.10266 |