A basis and Schur-Weyl duality for the loop Hecke algebra
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913945819807744 |
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| author | Janssens, Geoffrey Lacabanne, Abel Schelstraete, Léo Vaz, Pedro |
| author_facet | Janssens, Geoffrey Lacabanne, Abel Schelstraete, Léo Vaz, Pedro |
| contents | The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a basis. We use higher linear rewriting theory to show linear independence and the combinatorics of Dyck paths to compute the cardinality of the basis. This yields a conjecture of Damiani-Martin-Rowel. We also give a representation theoretic interpretation of the loop Hecke algebra in terms of (non-semisimple) Schur-Weyl duality involving the negative half of quantum $\mathfrak{gl}_{1|1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A basis and Schur-Weyl duality for the loop Hecke algebra Janssens, Geoffrey Lacabanne, Abel Schelstraete, Léo Vaz, Pedro Representation Theory Geometric Topology Quantum Algebra 20C08, 20F36, 17B37, 16T99, 16S15 The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a basis. We use higher linear rewriting theory to show linear independence and the combinatorics of Dyck paths to compute the cardinality of the basis. This yields a conjecture of Damiani-Martin-Rowel. We also give a representation theoretic interpretation of the loop Hecke algebra in terms of (non-semisimple) Schur-Weyl duality involving the negative half of quantum $\mathfrak{gl}_{1|1}$. |
| title | A basis and Schur-Weyl duality for the loop Hecke algebra |
| topic | Representation Theory Geometric Topology Quantum Algebra 20C08, 20F36, 17B37, 16T99, 16S15 |
| url | https://arxiv.org/abs/2507.12839 |