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

We establish a connection between planar rook algebras and tensor representations $\VV^{\otimes k}$ of the natural two-dimensional representation $\VV$ of the general linear Lie superalgebra $\gl$. In particular, we show that the centralizer algebra $\maths{End}_{\gl}(\VV^{\otimes k})$ is the planar rook algebra $\CC \mathsf{P}_{k-1}$ for all $k \geq 1$, and we exhibit an explicit decomposition of $\VV^{\otimes k}$ into irreducible $\gl$-modules. We obtain similar results for the quantum enveloping algebra $\UU_\qq(\gl)$ and its natural two-dimensional module $\VV_\qq$.

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