Planar Rook Algebras and Tensor Representations of 𝔤𝔩(1 | 1)

Abstract

We establish a connection between planar rook algebras and tensor representations of the natural two-dimensional representation of the general linear Lie superalgebra 𝔤𝔩(1 | 1). In particular, we show that the centralizer algebra is the planar rook algebra for all k ≥ 1, and we exhibit an explicit decomposition of into irreducible 𝔤𝔩(1 | 1)-modules. We obtain similar results for the quantum enveloping algebra and its natural two-dimensional module .