The partial Temperley–Lieb algebra and its representations

In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra $\mathrm{End}\_{\mathbf{U}\_q(\mathfrak{gl}\_2)}(V^{\otimes k})$, where ${V = V(0) \oplus V(1)}$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $\mathbf{U}\_q(\mathfrak{gl}\_2)$. It is a proper subalgebra of the Motzkin algebra (the $\mathbf{U}\_q(\fraksl\_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.