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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.11601 |
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| _version_ | 1866916093779509248 |
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| author | Sun, Binni Zhao, Yufeng |
| author_facet | Sun, Binni Zhao, Yufeng |
| contents | Let $\mathfrak{g}$ be a complex simple Lie algebra and $Z(\mathfrak{g})$ be the center of the universal enveloping algebra $U(\mathfrak{g})$. Denote by $V_λ$ the finite-dimensional irreducible $\mathfrak{g}$-module with highest weight $λ$. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra $\text{End}_{U(\mathfrak{g})}(V_λ^{\otimes r})$ in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the $\mathfrak{g}$-invariant endomorphism algebras $R_λ(\mathfrak{g})= (\text{End} V_λ\otimes U(\mathfrak{g}))^\mathfrak{g}$ and $R_{λ,π}(\mathfrak{g})=(\text{End} V_λ\otimes π(U(\mathfrak{g})))^\mathfrak{g}.$ In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs $(\mathfrak{g}, V_λ)$, which are multiplicity free and for such pairs, $R_λ(\mathfrak{g})$ and $R_{λ,π}(\mathfrak{g})$ are generated by generalizations of the quadratic Casimir elements of $Z(\mathfrak{g})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_11601 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Descriptions of strongly multiplicity free representations for simple Lie algebras Sun, Binni Zhao, Yufeng Representation Theory Let $\mathfrak{g}$ be a complex simple Lie algebra and $Z(\mathfrak{g})$ be the center of the universal enveloping algebra $U(\mathfrak{g})$. Denote by $V_λ$ the finite-dimensional irreducible $\mathfrak{g}$-module with highest weight $λ$. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra $\text{End}_{U(\mathfrak{g})}(V_λ^{\otimes r})$ in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the $\mathfrak{g}$-invariant endomorphism algebras $R_λ(\mathfrak{g})= (\text{End} V_λ\otimes U(\mathfrak{g}))^\mathfrak{g}$ and $R_{λ,π}(\mathfrak{g})=(\text{End} V_λ\otimes π(U(\mathfrak{g})))^\mathfrak{g}.$ In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs $(\mathfrak{g}, V_λ)$, which are multiplicity free and for such pairs, $R_λ(\mathfrak{g})$ and $R_{λ,π}(\mathfrak{g})$ are generated by generalizations of the quadratic Casimir elements of $Z(\mathfrak{g})$. |
| title | Descriptions of strongly multiplicity free representations for simple Lie algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2304.11601 |