Temperley-Lieb Immanants

Abstract.

We use the Temperley-Lieb algebra to define a family of totally nonnegative polynomials of the form $\Sigma _{{\upsigma \in S_{n} }} f(\upsigma )x_{{1,\upsigma (1)}} \cdots x_{{n,\upsigma (n)}}$. The cone generated by these polynomials contains all totally nonnegative polynomials of the form $\Delta _{{J,J' }} (x)\Delta _{{L,L' }} (x) - \Delta _{{I,I' }} (x)\Delta _{{K,K' }} (x)$, where, $\Delta _{{I,I' }} (x), \ldots ,\Delta _{{K,K' }} (x)$ are matrix minors. We also give new conditions on the sets I,...,K′ which characterize differences of products of minors which are totally nonnegative.