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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2511.00169 |
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| _version_ | 1866915879612055552 |
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| author | Doty, Stephen Giaquinto, Anthony Martin, Stuart |
| author_facet | Doty, Stephen Giaquinto, Anthony Martin, Stuart |
| contents | For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ (where $V_q$ is the $n$-dimensional vector representation) satisfies Schur--Weyl duality with respect to the commuting actions of the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_n)$ and the Iwahori--Hecke algebra $\mathbf{H}_q(\mathfrak{S}_r)$, with the latter action derived from the $R$-matrix. In the limit as $q \to 1$, one recovers classical Schur--Weyl duality.
Using a recursive construction of certain linear combinations $Ψ_j$ of Coxeter monomials in the negative part of $\mathbf{U}_q(\mathfrak{gl}_n)$, we give a combinatorial realization of the corresponding isotypic semisimple decomposition of $V_q^{\otimes r}$, indexed by paths in the Bratteli diagram. This extends earlier work (Journal of Algebra 2024) of the first two authors for the case $n =2$. Our construction works over any field containing a non-zero element $q$ which is not a root of unity.
The element $Ψ_j$ depends on a weight $λ$ and is the ``evaluation at $λ$'' of a certain $q$-lowering operator $\overlineΨ_j$ satisfying a similar recursion, up to renormalization. This simplifies the construction of lowering operators. Both $Ψ_j$ and $\overlineΨ_j$ are independent of a choice of root vectors. On the other hand, the $Ψ_j$ can be applied to construct root vectors (independent of the braid group action) as explicit linear combinations of Coxeter monomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lowering operators, orthogonal decomposition of tensor space, and quantized Schur--Weyl duality Doty, Stephen Giaquinto, Anthony Martin, Stuart Quantum Algebra Combinatorics Representation Theory 16T30, 16T20, 17B37 For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ (where $V_q$ is the $n$-dimensional vector representation) satisfies Schur--Weyl duality with respect to the commuting actions of the quantized enveloping algebra $\mathbf{U}_q(\mathfrak{gl}_n)$ and the Iwahori--Hecke algebra $\mathbf{H}_q(\mathfrak{S}_r)$, with the latter action derived from the $R$-matrix. In the limit as $q \to 1$, one recovers classical Schur--Weyl duality. Using a recursive construction of certain linear combinations $Ψ_j$ of Coxeter monomials in the negative part of $\mathbf{U}_q(\mathfrak{gl}_n)$, we give a combinatorial realization of the corresponding isotypic semisimple decomposition of $V_q^{\otimes r}$, indexed by paths in the Bratteli diagram. This extends earlier work (Journal of Algebra 2024) of the first two authors for the case $n =2$. Our construction works over any field containing a non-zero element $q$ which is not a root of unity. The element $Ψ_j$ depends on a weight $λ$ and is the ``evaluation at $λ$'' of a certain $q$-lowering operator $\overlineΨ_j$ satisfying a similar recursion, up to renormalization. This simplifies the construction of lowering operators. Both $Ψ_j$ and $\overlineΨ_j$ are independent of a choice of root vectors. On the other hand, the $Ψ_j$ can be applied to construct root vectors (independent of the braid group action) as explicit linear combinations of Coxeter monomials. |
| title | Lowering operators, orthogonal decomposition of tensor space, and quantized Schur--Weyl duality |
| topic | Quantum Algebra Combinatorics Representation Theory 16T30, 16T20, 17B37 |
| url | https://arxiv.org/abs/2511.00169 |