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

TL;DR

This paper introduces a new basis and presentation for the loop Hecke algebra, explores its combinatorial structure, and connects it to a Schur-Weyl duality involving quantum superalgebra, advancing understanding of its algebraic and representation-theoretic properties.

Contribution

It provides a new presentation and basis for the loop Hecke algebra, and establishes a Schur-Weyl duality interpretation involving quantum superalgebra.

Findings

New basis and presentation for the loop Hecke algebra.

Combinatorial enumeration using Dyck paths.

Representation-theoretic interpretation via Schur-Weyl duality.

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-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 $\mathfrak{gl}_{1|1}$.