A basis and Schur-Weyl duality for the loop Hecke algebra

Data

• Title: A basis and Schur–Weyl duality for the loop Hecke algebra
• Authors: Geoffrey Janssens, Abel Lacabanne, Léo Schelstraete and Pedro Vaz
• arXiv link: https://arxiv.org/abs/2507.12839
• Published in: Still a preprint
• DOI:

Abstract

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–Rowell. 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 $\mathfrak{gl}_{1|1}$.

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