  A new spin-chain representation of the Temperley-Lieb algebra $TL_n(\beta=0)$
is introduced and related to the dimer model. Unlike the usual XXZ spin-chain
representations of dimension $2^n$, this dimer representation is of dimension
$2^{n-1}$. A detailed analysis of its structure is presented and found to yield
indecomposable zigzag modules.
